Representation theory and sheaves on the bruhat-tits building

Peter Schneider; Ulrich Stuhler

doi:10.1007/BF02699536

Representation theory and sheaves on the bruhat-tits building

Schneider, Peter; Stuhler, Ulrich 2007-08-31 00:00:00 REPRESENTATION THEORY AND SHEAVES ON THE BRUHAT-TITS BUILDING by PETER SCHNEIDER and ULvacri STUHLER The Bruhat-Tits building X of a connected reductive group G over a nonarchi- medean local field K is a rather intriguing G-space. It displays in a geometric way the inner structure of the locally compact group G like the classification of maximal compact subgroups or the theory of the Iwahori subgroup. One might consider X not quite as a full analogue of a real symmetric space but as a kind of skeleton of such an analogue. As such it immediately turned out to be an important technical device in the smooth representation theory of the group G. As a reminder let us mention that the irreducible smooth representations of G lie at the core of the local Langlands program which aims at understanding the absolute Galois group of the local field K. In this paper we develop a systematic and conceptional theory which allows to pass in a functorial way from smooth representations of G to equivariant objects on X. There actually will be two such constructions--a homological and a cohomological one. Since the building carries a natural C W-structure the notion of a coefficient system (or cosheaf) on X makes sense. In the homological theory we will construct functors from smooth representations to G-equivariant coefficient systems on X. It should be stressed that the definition of the coefficient system only involves the original G-repre- sentation as far as the action of certain compact open subgroups of G is concerned. One therefore might consider the whole construction as a kind of localization process. Our main result will be that the cellular chain complex naturally associated with a coefficient system provides (under mild assumptions) a functorial projective resolution of the G-representation we started with. In the cohomological theory we will associate, again functorially, G-equivariant sheaves on X with smooth G-representations. The main task which we will achieve then is the computation of the cohomology with compact support of the sheaves coming from an irreducible smooth G-representation. The result can best be formulated in terms of a certain duality functor on the category of finite length smooth G-represen- tations. As a major application we will prove Zelevinsky's conjecture in [Zel] that his duality map on the level of Grothendieck groups preserves irreducibility. For carrying 13 98 PETER SCHNEIDER AND ULRICH STUHLER out our computation we have to extend the sheaves under consideration in such a way to the Borel-Serre compactification X of X that the cohomology at the boundary becomes computable. Since the stabilizers of boundary points are parabolic subgroups it might not surprise that this can be achieved by using the Jacquet modules of the representation as the stalks at the boundary points. The cohomology at the boundary then is computed by adapting a strategy of Deligne and Lusztig ([DL]) for reductive groups over finite fields to our purposes. Apart from the theory of buildings we will very much rely on the beautiful results of Bernstein on the category of smooth representations in [Ber]. All the known homo- logical finiteness properties of this category follow already from his work. The point of our paper is rather that we construct nice projective resolutions in that category in a functorial as well as explicit manner. Exactly this explicitness enables us to apply our theory to the harmonic analysis of the group G. It turns out that Kottwitz' Euler- Poincard function in [Kot] is a special case of a general theory of Euler-Poincar6 functions for finite length smooth G-representations. Besides being pseudo-coefficients their main property is that their elliptic orbital integrals coincide with the Harish-Chandra character of the given representation. This leads to a Hopf-Lefschetz type trace formula for the Harish-Chandra character at an elliptic element. Combined with powerful results of Kazhdan in [Kal] it also leads to a proof of the general orthogonality formula for Harish- Chandra characters as conjectured by Kazhdan. Let us now describe the contents of the paper in some more detail. The first chapter contains most of the input which we need from the theory of buildings. The ceils of the natural C N-structure of X actually are polysimplices and are called facets. For any such facet F of X let PF denote its pointwise stabilizer in G. The technical heart of our theory is the construction of certain decreasing filtrations PF -~ U~ ) ~ 9 9 9 ~- U~, ~ 2 -- - of PF by compact open subgroups U~ '1. This is done in I. 2 where also the more basic properties of these filtrations are established. Since we work with an arbitrary connected reductive group G that construction involves more or less all of the finer aspects of the theory developed in the volumes [BT]. This unfortunately makes numerous references to [-BT] unavoidable so that any reader without an expert knowledge of the work [-BT] might find this section hard to read. We apologize for that. In order to make it a little easier we give in I. 1 a brief overview over the theory of buildings for reductive groups. In the section I. 3 we give those properties of the groups U~ '1 which later on are needed for the computation of the (co)homology. Notably we study how the groups U~ "~ behave if the facet F is moved along a geodesic in the building X. In case G is absolutely quasi- simple and simply connected similar filtrations appear in [PR] and [MP]. The second chapter contains the homological theory. In II. 1 we briefly recall the formalism of cellular chains. The section II. 2 contains the definition of the functor y, from smooth G-representations to equivariant coefficient systems on X. Here e/> 0 is a fixed " level ". The coefficient system y,(V) corresponding to a representation V is formed by associating with a facet F the subspace of U~'~-invariant vectors in V. In 99 REPRESENTATION THEORY AND SHEAVES addition properties of finite generation and projectivity of the chain complex of u are discussed. The main result is shown in II.3. It says that at least for any finitely generated smooth G-representation V we can choose the level e large enough so that the chain complex of y,(V) is an exact resolution of V in the category Alg(G) of all smooth G-representations. In the case that G is the general linear group we proved this already in [SS]. The strategy of the proof for arbitrary G is the same once the necessary properties of the groups U~ ~ are known. In the third chapter we develop that part of the duality theory which uses the chain complex of y,(V). Since the polysimplicial structure of the building X is locally finite we actually can associate with the coefficient system y,(V) also a complex of cochains with finite support. Let us fix a character X of the connected center of G, let Algx(G ) denote the category of all those smooth G-representations on which the connected center acts through X, and let ovg x denote the ?(-Hecke algebra of G. If V is an admissible representation in Algx(G ) then the functor HomQ(., 9fix) transforms the chain complex of y,(V) into the cochain complex of y,(V) where V is the smooth dual of V. Assume now that V even is of finite length and choose e large enough. Then we know that the chain complex of y,(V) is a projective resolution of V in Alg� It follows that the cochain complex of y,('~) computes the Ext-groups o"*(V) := Ext~lgx~Q~(V , Yt~ All this is shown in III. 1. Later on in IV. 1 wc will see that the same cochain complex computes the cohomology with compact support of a certain sheaf on X associated with the representation V. This fact will enable us in chapter IV to compute the groups d~'(V) in the case that V is a representation which is parabolically induced from an irreducible supercuspidal representation of a Levi subgroup. In III. 2 we briefly recall the theory of parabolic induction following [Gas]. Then in III.3 taking the computation of $r for induced V for granted we deduce the following result for an arbitrary irreducible smooth representation V: The groups g"(V) vanish except in a single degree d(V), 8(V):----- ~a~v~(v) again is an irreducible smooth representation, and moreover g'($*(V))= V. Standard techniques of homological algebra now allow to establish a general duality formalism which relates the Ext- and Tor-functors on the category Algx(G ). Loosely speaking one might say that o~r x is a " Gorenstein ring " Since our applications to harmonic analysis all come from the exactness of the chain complex of ye(V) we include them here as the section III.4 before we turn to the sheaf theory on X. For reasons of convenience we assume that the center of G is compact. Since in the paper [Kal] the field K is assumed to be of characteristic 0 we have to make the same assumption in most of our results of this section. We obtain: A general notion of Euler-Poinear6 functions, a formula for the formal degree, the existence of explicit pseudo-coefficients, the Harish-Ghandra character on the elliptic set as an explicit orbital integral, the general orthogonality relation for Harish-Chandra characters, and the 0-th Chern character on the Grothendieck group of finite length representations. 100 PETER SCHNEIDER AND ULRICH STUHLER In the fourth chapter we present the sheaf theory on X. In IV. 1 we functorially associate a sheaf V on X with any representation V in Alg(G). Of course this construction again depends on the choice of a level e >i 0 which is fixed once and for all and which, for simplicity, is dropped from the notation. The sheaf V is constant on each facet F having the U~e'-coinvariants of V as stalks. As promised earlier we show that the coho- mology with compact support of V is computable from the complex of cochains with tq~ finite support of the coefficient system %(V). We also rewrite our earlier formula for the Harish-Chandra character of a finite length representation V at an elliptic element h ~ G as a Hopf-Lefschetz trace formula: The character value at h is equal to the trace (in the sense of linear algebra) of h on the cohomology of the sheaf V restricted to the fixed point set X h. In IV. 2 we construct a " smooth" extension of V to a sheaf j., ~ V on the Borel-Serre compactification X of X. This extension is in some sense intermediate between the extension by zero j: V and the full direct image j. V. It requires a rather detailed and technical investigation of the geometry of X. Let X~o := X\X be the boundary. In IV.3 we compute the cohomology of j.,~ V restricted to X~ in the case where the representation V is parabolically induced from a supercuspidal representation. Then in IV. 4 we show that, for any finitely generated V and any e large enough, the sheaf j., ~ V in fact has no higher cohomology. The combination of these two results immediately leads to the computation of the cohomology with compact support of the original sheaf V provided V is parabolically induced as above. This is the fact which we had taken for granted in chapter III. So the duality theory, i.e. the investigation of the Ext-groups o~(V), now is complete. As an application we prove in IV. 5 Zele- vinsky's conjecture. At this point we want to mention that Bernstein has a completely different proof (unpublished) of this conjecture along with the fact that the Zelevinsky involution comes from the functor d'*(. ). The last chapter complements the discussion of coefficient systems. We show that a rather big subcategory of AIg(G) is a localization of the category of equivariant coefficient systems on X. As will be explained in a forthcoming paper of the first author the latter objects constitute something which one might call perverse sheaves on the building X. From this point of view our constructions bear a certain resemblance to the Beilinson-Bernstein localization theory from Lie algebra representations to perverse sheaves on the flag manifold. During this work we have profited from conversations with M. Harris, G. Henniart, M. Rapoport, M. Tadic, J. Tits, and M.-F. Vigneras for which we are grateful. We especially want to thank E. Landvogt whose expert knowledge of the building has helped us a lot. The support which we have received at various stages from the MSRI at Berkeley, the Newton Institute, the Tata Institute, and the Universit~ Paris 7 is gratefully acknowledged. REPRESENTATION THEORY AND SHEAVES 101 Added in proof. -- As the referee has pointed out, special cases of the Zelevinsky conjecture are treated in the papers [Kat] and [Pro]. Their methods are completely different from ours. Also in the meantime Aubert has given in [Au2] a proof of the Zelevinsky conjecture in the general case by studying on the Grothendieck group a certain involution wtfich is defined in terms of parabolic induction (compare our IV. 5.2). CONTENTS I. The groups U~ I ........................................................................ 102 I. 1. Review of the Bruhat-Tits building ................................................ 102 1.2. Definition of the groups U~ ) ...................................................... 105 1.3. Properties of the groups U~ I ...................................................... 117 II. The homologicai theory ................................................................. 119 II.1. Cellular chains ................................................................... 119 II.2. Representations as coefficient systems ............................................... 121 II.3. Homological resolutions ............................................................ 123 III. Duality theory ......................................................................... 126 III.l. Cellular cochalns ................................................................. 126 III.2. Parabolic induction ............................................................... 130 III.3. The involution ................................................................... 132 III.4. Euler-Poincar~ functions ........................................................... 135 IV. Representations as sheav~ on the Borel-Serre compactification ................................ 152 IV. 1. Representations as sheaves on the Bruhat-Tits building ............................... 152 IV.2. Extension to the boundary ......................................................... 155 Appendix: Geodesics in .X ........................................................ 166 IV.3. Cohomology on the boundary ..................................................... 167 IV.4. Cohomology with compact support ................................................. 173 IV.5. The Zelevinsk'y involution ......................................................... 183 V. The functor from equivariant coefficient systems to representations ............................. 185 LIST ov NOTATIONS ........................................................................... 188 REFE RENC-.ES ................................................................................. 190 NOTATIONS K a nonarchimedean locally compact field, o the ring of integers in K, a fixed prime element in o, ca : K � ~ Z the discrete valuation normalized by co(n) = 1, := o]~o the residue clas~ field of o, for any object X over o for which it makes sense to speak about its base change to K we denote this base change by X, G a connected reductive group over K, G := G(K). 102 PETER SCHNEIDER AND ULRICH STUHLER I. THE GROUPS U[g I 1.1. Review of the Bruhat-Tits building The (semisimple) Bruhat-Tits building X of the group G is the central object in this paper. Since the monumental treatise [BT] is not so easily accessible for the nonexpert we believe it to be necessary to briefly review the construction and basic properties of X. Most of the notations to be introduced for tiffs purpose will be needed later on anyway. We fix a maximal K-split torus S in G. (Strictly speaking S is the group of K-rational points of that torus. This kind of abuse of language will usually be made.) Let X*(S), resp. X.(S), denote the group of algebraic characters, resp. cocharacters, of S. Similarly let X.(C) denote the group of K-algebraic cocharactcrs of the connected center C of G. The real vector space A := (X,(S)/X.(C)) | R is called the basic apartment. Let Z, resp. N, be the centralizer, resp. normalizer, of S in G. The Weyl group W := N/Z acts by conjugation on S; this induces a faithful linear action of W on A. On the other hand let (,): X,(S) � X*(S) -~ Z be the obvious pairing; its R-linear extension also is denoted by (,). There is a unique homomorphism : Z --~ X.(S) | R such that z l s ) = - o(z(g)) for any g e Z and any K-algebraic character ) of Z. We let g e Z act on A by the translation gx:----x+image ofv(g) in A for xeA. The first important observation in this theory is that this translation action of Z on A can be extended to an action of N on A by affine automorphisms ([Tit] 1.2). We fix one such extension and simply denote it by x ~-, nx for n E N and x e A. One has: -- All possible other such extensions are given by x ~-, n(x + xo) --xo where x0 e A is a fixed but arbitrary point. -- IfweWis the image ofneN then (x ~ wx) = linear part of (x ~-, nx). In order to equip A with an additional structure we need the set 9 _ X*(S) of roots of G with respect to S. Any root e obviously induces a linear form e : A ~ R. Also REPRESENTATION THEORY AND SHEAVES 103 corresponding to any a ~ ~ we have the coroot ~ ~ A and the involution s, E W whose action on A is given by s,x=x--~(x).~ for xeA. For us the most important object associated to an ~ e 9 is its root subgroup U, ~_ G ([Bor] 21.9 where the notation Uc, ~ is used); in particular it is a unipotent subgroup normalized by Z. Let Om := {, e 9 : ~/2 r (I) } be the subset of reduced roots. Crucial is the following fact ([BoT] w 5): For 0c eel) ~a and each u e U,\{ 1 } the intersection U_, uU_ ~ n N = { m(u) } consists of a single element called m(u); moreover the image of m(u) in W is s~. A central assertion in the Bruhat-Tits theory now is the fact that the translation part of the affine automorphism of A corresponding to m(u) is given by -- g(u). ~ for some real number t(u), i.e. we have m(u) x = s~x --t(u).~ = x -- (~(x) +t(u)).~ for any xcA. We may view m(u) as the " reflection " at the affine hyperplane { x e A : ,(x) = -- t(u) } ([Tit] 1.4). Put F~ := {t(u) : u e U~\{ 1 } } _ R; this is a discrete subset in R unbounded in both directions and --F, = F_~ ([BT] 1.6.2.16). The affine functions ~(.)+g on A for ~ e @,,a and g e P, are called affine roots. Two points x and y in A are called equivalent if each affine root is either positive or zero or negative at both points; the corresponding equivalence classes are called facets. This imposes an additional geometric structure on the apartment A which is respected by the action of N. Parallel to this structure the root subgroup U~ for ~ e (I) "~ possesses the filtration U,.,:={ueU,k{1}:t(u)>~r}u{1} for reR. This is an exhaustive and separated discrete filtration of U, by subgroups ([BT] 1.6.2.12b)); put U,.~o := { 1 }. For any nonempty subset fl c-_ A we define fn:*~R u{oo} e~-- inf e(x) and U n : = subgroup of G generated by all U,. tn~ for e e ~,,a. This group has various important properties ([BT] 1.6.2.10, 6.4.9, and 7.1.3): 1. nUan -1 = U,n for any n eN; in particular Nn:={n eN:nx = x for any x etl} normalizes Un. 2. UnnN ___N n. 3. U n n U, ----- U~. ~'ac~l for any 0c e ~. 4. Let q) = q~+u q)- be any decomposition into positive and negative roots and put U e := subgroup of G generated by all U~ for c~ e ~+ n q)~a; then Un = (Ua n U-) (Ua n U +) (Un n N); 104 PETER SCHNEIDER AND ULRICH STUHLER moreover the product map induces bijections II U~, Ial~J -% Ua r~ U whatever ordering of the factors on the left hand side we choose. Define Pa :----- Na. Ua which contains Un as a normal subgroup by 1. By 2, we have Pan N = N n. (Warning: In [BT] our groups N a and Pa are denoted by iq a and Pa and our symbols have a different meaning.) In case ~ = { x } we write f,, U,, N,, and P, instead off~,~, ... We now are ready to define the Bruhat-Tits building X. Consider the relation on G x A defined by (g, x) --~ (h,y) if there is a n ~ N such that nx = y and g- 1 hn E U.; it is easily checked that tiffs is an equivalence relation. We put X:=G xA/,-~. It is straightforward to see that G acts on X via g.class of (h,y) := class of (gh,y) for g~G and (h,y) ~G x A and that the map A~X x ~ class of (1, x) is injective and N-equivariant. Viewing the latter map as an inclusion we can write gx for the class of (g, x). A first basic fact ([BT] I. 7.4.4) is that, for f~ _~ A nonempty, Pa={geG:gx-=x for any x~f~} holds true. The relation between the facet structure of A and the subgroup filtration in U~, for ~ e ~a is given by the fact that for u e U~\{ 1 } we have {xcA:ux=x}={x~A:~(x) +g(u)>/ 0} ([BT] 17.4.5). The subsets of X of the form gA with g ~ G are called apartments. A very important technical property of the G-action on X is the following: 5. For any g e G there exists a n EN such that gx = nx for any x ~A ng-lA ([BT] 1.7.4.8). REPRESENTATION THEORY AND SHEAVES 105 For example it implies that the partition into facets can be extended from A to all of X in the following way: A subset F' c X is called a facet if it is of the form F' = gF for some g 9 G and some facet F _c A. It also implies: 6. For any nonempty ~ ___ A the group Uo acts transitively on the set of all apartments which contain fL From the Bruhat decomposition G=-U, NU v for x,y 9 one concludes: 7. Any two points and even any two facets in X are contained in a common apartment ([BT] 1.7.4.18). For any nonempty subset ~2 _c X we define Pn:={geG:gz=z for any z 9 and P~:={g 9 and we abbreviate P, := P{,) = P~,} for any z 9 X. Finally we fix once and for all a W-invariant euclidean metric d on A. The action of N on A then automatically is isometric. As a simple consequence of 5-7, this metric extends in a unique G-invariant way to a metric d on all of X. The metric space (X, d) together with its isometric G-action and its partition into facets is called the Bruhat-Tits building of G. Further properties of this very rich structure will be recalled when they are needed. 1.2. DeAnition of the groups U~ ~ For any facet F in A let G ~ be the smooth affine o-group scheme with general fiber G constructed in [BT] II.5.1.30. By [BT] II.5.2.4 the group G~ is the subgroup of G generated by U F and .~o'~ where ~0 is the connected component of the " canonical " extension ~ of Z to a smooth affine o-group scheme ([BT] II. 5.2.1). Put H:={neN:nx=x for all xeA} and H 1 := { n 9 H : co(z(n)) = 0 for any K-algebraic character )( of G }. According to [BT] II.5.2.1 we have .~(0) = H 1. Therefore L~~ is of finite index in H 1. Since H ~ N F we see that u, G~ = PF. 14 106 PETER SCHNEIDER AND ULRICH STUHLER It follows from [BT] I1.4.6.17 that any inner automorphism g ~-~ ngn -1 of G with n e N extends to an isomorphism of o-group schemes G0 0 The closed fiber ~,o is a connected smooth algebraic group over I(; let R~ denote its unipotent radical. Put R F := { g E t3~ : (g mod ~) e RI,(K ) }; this is a compact open subgroup of G. Because of nR~ n-1 = R, F for n e N, it is nor- realized by Nt~:={neN:nF=F}. The property 1.5 implies that P; = N; P, = N; U.. Hence R F is a normal subgroup ofPtF. In the sequel we will construct a specific decreasing filtration 13~ ~ R, -----: U~'_~ U~'_~ ... by subgroups U~ *~ which are normal in Pt F and compact open in G. For doing this we need the concept of a concave function in [BT] 1.6.4.1-5. First we have to introduce the totally ordered commutative monoid R:=Ru{r+:r~R}u{oo}. Its total order is given by the usual total order on R and by r~< r+~<s~< co ifr<s; its monoid structure extends the addition on R and is given by r+ (s+) = (r+) + (s+) =- (r+s) + and r+co= (r+) +oo =co+co=co. We put 89 (89 + and 89 := oo. A function f: q) --~R is called concave if for any a,~,a+~q~, f(a) +f(~) >--f(a + ~) for any a e q~ and f(a) +f(-- a) >/ 0 hold. For a E ~i and r ~ R we define U,,,+ := O U~,,. s~R,.>r REPRESENTATION THEORY AND SHEAVES 107 Then, for any concave function f, the group U t :-----subgroup of G generated by all U,. tl-I for a e ~t and all U2~ n U~. tI~2~ for 0c, 2~ e has properties completely analogous to 1.1-1.4 ([BT] I. 6.4.9). Observe that Un = Urn. Starting from the concave function fF, for a facet F in A, we define a new function f;:r -+R by /f~(~)+ if ~IF is constant, f~(~) := ~ fF(~) otherwise; it is concave, too, by [BT] 1.6.4.23. In case ~, 2~ er we have f;(2~) -~ 2f~(~) so that UI; is the subgroup generated by all U~. ~,~1 for a e (I) '~i. Lemm I. 2.1. -- We have R~, n U~, I,~ = U~. g~j for any ~ e (I) ~. Proof. -- We have to introduce further notations. In case 2~ e ~) put := { 2t(u) : 1 } } (recall that U~, _~ U~) and F'~ := { g(u) ~ F, : u 9 U,\U2~ and g(u) = sup/(uU2, ) }; one has P~ ----- P', u 89 ([BT] 1.6.2.2) and F', 4 ~ ~ ([BT] II 4 2.21). In case 2~ r put F'~ := F~. The " optimization " gr : ~ ~ R of the function ft, is de ined by g,(~) := inf{ l e F; : t >t fF(~) }. Moreover we put (g,(~)+ ifgF(~) +gF(--~) =0, := { [ gF(~) otherwise. (The functions g~ and g~, in general are no longer concave but only quasi-concave in the sense of [BT].) In [BT] II.4.6.10 (compare in particular the third paragraph on p. 321) and 5.1.31 it is proved that U~.~ if 2~ r ~, R r c~ U~, tr = (U~ c~ holds true. Let us first consider the case 2a r (I). If g,(~) + g~(- ~) = 0 then clearly alsof~(a) +fl,(-- a) = 0, i.e. a IF is constant and gF(a) ----f~(a); we obtain U,.a~ =- U,.gr = U,.,,,). 108 PETER SCHNEIDER AND ULRICH STUHLER Assume now that gF(e) + gF(-- ~) ~e 0. If ~ [ F is not constant then by the definitions we have U~,.;~) = U~, ~,(~) = U~, ~.(~ = U~, ~(~1; the same holds if ~ [ F is constant since, in that case, fF(~ ) r I'~, which implies the last identity. We turn to the case 2~ ~ q). There are the following four possibilities: 1) g~(00 = g~(~)+ and g~(2~) = g~(200§ 2) g;(~) = gF(~) + and g;(2e) = gx,(200, 3) g~(~) = gF(e) and g~,(2e) = gr(2e), 4) g~(e) = gF(o0 and g;(2e) = gt(2e)+. In case 1) we have 1 , ] F is constant, ft(e) = g~(e) e r', n ~ r~, and gi,(2~) = 2ft(~) and hence U.. ~.(~. (U2. n U., toI(2~)) = U., t;.(~. (U2. n U., #(~) = U., ~.~. In case 2) we have 0c [ F is constant, fF(e) = gF(~) E r',, and ~ gr(2e) >fF(~) and hence In case 3) we have g)(~) = inf{ t e : t ~> fF(00 } 1 g~,(2~)=inflte~r;.:t>~f~(~) t. and Let us first assume that 89 ) 1> g~(~); then g~(e) ---- inf { t e r~ : t/> fF(e) }. This implies U=. 4(~," (U2~ n U=, to;~2=1) = U=, o;~=) = U=, I,(=~" Now assume that 89 < g~(a); then x g~(2~) = inf{t e P~ :e >~ fF(~) }. Z REPRESENTATION THEORY AND SHEAVES 109 We are going to use the following general fact which is a straightforward consequence of the definition of the set P'~: If r < s are values in F, such that rCF'~ and s=inf{g 9 r} then (,) U~,, _= U.,,. (Uz~ c~ U~,,). Applied to our situation this leads to U~, g;(~. (U2~ ca U~, i,;~) = U~, I~;~ = U~, ,,(~. Moreover in case 3) we always have U=,t.t=> = Ue, t;,~, since if el F is constant then both g~(e) and 89 are strictly bigger thanf.(e). Finally in case 4) we have 1 1 [ F is constant, f~(~) = ~ g~,(2~) e ~ P~, and gv(~) >f~(~) and hence U.,,lc.,. (U2. c~ U.,t,~,...~ ) = U.,~.I.~. (U2. n U.,,,I.,+) = U.,~ where the second identity again is a consequence of (*) sincefF(a ) r F',. [] There is a scheme-theoretic version of 1.4 ([BT] II.5.2.2-4): G ~ possesses smooth closed o-subgroup schemes o//,, F for e e ~ and o//~ for any fixed decomposition = ~+ t3 ~- into positive and negative roots such that ~'~,F(o) = U~,I.,. ~ and ~(o) = U~ n U (We have simplified the notation a little by writing q/=,r for ~., ~&l~l, &12.)~ in loc. cit.) Moreover the product map induces an isomorphism of o-schemes (whatever ordering of the factors on the left hand side we choose) as well as an open immersion of o-group schemes qG x ~rOx ~'~6 ~ We put Zr176 := { g e ~L~~ : (g mod r~) 9 R.(.~ ~ (K) } where R.(.~ ~ denotes the unipotent radical of .~o Proposition 1.2.2. -- The product map induces a bijection II U~.,~,~,) � Z '~ x ( II U~ ,:c~) _v~ RF. ~ ~. ~-t-h q) red r ~ ~" t'5 ~, red ' 110 PETER SCHNEIDER AND ULRICH STUHLER Proof. -- We recall from [BT] II. 4.6.4 and 5.1.31: If oq '~ denotes the connected component of the N6ron model over o of S then 5oo is a maximal K-split torus in ~o and .o~ ~ is its centralizer. Also the q/,,r are the root subgroups in G ~ By [BT] II. 1.1.11 the above open immersion therefore induces an isomorphism ( I/ ~,~ n ~) x ~(~) x ( II ~, n ~,) -~ ~,, ~ ~- ~ ~red ~ ~ ~* t~ ~red ' and hence also an isomorphism between formal completions (I/~,) x ~ x (II~,)-% G ~ where ?R denotes the formal completion of ? along ? n RF. It remains to observe that qz~ F(o),, = R~ n U,,,,,,, = U,,a,, ,, Lr~ = Z '~ and G~R(o) = R F. [] Corollary 1.2.3. -- We have R F = Uj~ Z ~~ With fx; also the functions f~ + e, for any integer e i> 0, are concave. Hence we have the descending sequence of subgroups Ut~ -~ UI~+I ~_ UI~+,, ~ ... We also need a corresponding filtration Z ~~ __ Z ~a~ ~ ... __ Z ~'~ ~ ... The subgroups we are looking for then will be defined to be U~ "~ := UI~+,.Z~'~; note that H normalizes U 1 for any concave function f ([BT] 1.6.2.10 (iii)). The properties which we want the subgroups U~ '1 to have impose certain conditions on the possible shape of the filtration Z ~'~. These conditions are axiomatized in [BT] I. 6.ff in the following way. First of all it is notationally convenient to define U2~:={1}in case ~e(l) but 2~r and U2~,k := U2~ c~ U~,t~ for any ~ e~ and any k ~ R. For any k eR put H~k I := set of all h e H such that (h, U~.,) := { (h, u) : u e U~., } ~ U~.,+k.U~.~,+~ for any ~ e(I) and any r e R. REPRESENTATION THEORY AND SHEAVES Ill The H~k ) form a decreasing family of subgroups in H which are normal in N; obviously H~k~ = H for k ~< 0. Another such family denoted by H[k ~ is given as follows: For k ~< 0 put H[,j := subgroup generated by all Hc~<U~,,wU_~_,) for 0~e~ and reR. In case 0<k<oo the commutator (u,u') for u eU=,,, u'eU_~.o with r+s-----k or u' e U_2~ ' , with 2r -t- s = k, r, s e It, and any a e r lies in a double coset U~ h~,., U_~ with a uniquely determined element h~,r e H ([BT] 1.6.3.9); we put H~l :-= subgroup generated by all those h,,~,. Finally we set Ht~l:= [7 H[~ 1. k<oo The H[~ 1 again form a decreasing family of subgroups of H which are normal in N. One has H[,+~ = [J H[0~ for any r e R. The key property of this latter family is the S>r following. Let us call a function f: 9 u { 0 } -+ R concave if f] 9 is concave and if f(~) +f(-- ~) >if(0)/> 0 for any ~ e~ holds. In this situation we have H c~ Utt o _ HEs(0)] ([BT] 1.6.4.17). A good filtration of H now by definition is a family of subgroups H, _ H for r ~ R such that -- H,-~H for r~ 0, -- H,_H. ifr>ls, -- H m_H,~H(,) for any r~R, and -- (H,, H,)_~ H,+, for any r,s~R. These properties together with [BT] I. 6.4.33 imply that the H, are normal in N. A necessary and sufficient condition for the existence of a good filtration is, according to [BT] 1.6.4.39, that H m ~ Hlk) holds true for any k e R. In [BT] I. 6.4.15 it is stated that this condition actually is fulfilled--see Proposition 6 below. For the moment we assume that a good filtration is given. We pose H,+:---- [J H, for reR and H~:= ~ H,. a>r r6R 112 PETER SCHNEIDER AND ULRICH STUI-ILER Also for any concave function f: 9 u { 0 } -+ R we define the subgroup U I := U11 | Ht<o~- Lemma 1.2.4. ---Let f,g:~ u{ 0}-+R be two concave functions such that g(p~ + q~) <<. pg(a) qf(~)foranya, ~ eOu{O}andp, q~Nsuchthatpoc + q~er u{ 0}; then U t normalizes Ug. Proof.--- [BT] 1.6.4.43. ~] Lemma 1.2.5. i) Hto4] ~ Z ~~ ~ H~o+); ii) Z c~ is normal in N. Proof. -- i) The subgroup Hco+j is generated by the h~, u, for u ~ U~.,+ and u'~U_~,_, with ~'~a and rER. Fix such u and u' and choose a vertex x in A (i.e. { x } is a facet in A) such that U.,, = U.,_~.,. Then u e Rt.~ by Lemma 1 and u' e U. __q P.. Since R~| I is normal in P. the commutator (u, u') lies in RI. ~ . It follows now from Proposition 2 that h~,.. e Z ~~ On the other hand we have to show that (Z ~~ _ U~,,! for any eeO~a and r e R. But choosing the vertex x as before we have (Z '~ U~, ,) ~ (Z ~~ U~, t,~) -~ Rr n U~. t,c~, = U~. t~,4 c_ U~. ,+. ii) By the very construction of the o-group scheme ~Lr ~ any automorphism g ~-, ngn-x of Z with n e N extends to an automorphism of the o-group scheme :Lr0. [] Proposition 1.9~.6. -- There exists a good filtration (tt,),~l~ of H such that: i) Ho+ =Z ~~ ii) H~ ={l}, and iii) H,+ is open in H for any r ~ R. Proof. - - As mentioned already this is a slightly sharpened version of [BT] I. 6.4.15. We are indebted to J. Tits for explaining to us the proof which is missing in [BT] and which we briefly sketch in the following. First of all we note that it suffices to find a good filtration H', which fulfills ii), iii), and the weaker condition i') Zl~ Ho+ , because then H for r~< 0, H,:= ZI~ for r>0 is a good filtration satisfying i)-iii). This follows from Lemma 5 i) and the fact that Z ~~ is open in Z. REPRESENTATION THEORY AND SHEAVES 113 Step 1: The split case. ~ IfG is split then we have S = Z -~ (K� and ~e ~ ~ G~,/o. We define H, := ker(.o~e~ -+ .o~~ n+x o)) if m < r ~< m 4- 1 with m ~ N u{ 0 }. The only thing which has to be checked is H~, 1~ H,~ H~,~ for r>0. The left inclusion, by [BT] II.3.2.1, can be checked in SL,(K) where it is straight- forward. For the right inclusion we use the following two identities. Let ~ e 9 be a root. -- ([BT] II.3.2.1) (h, u) = (0~(h) -- 1) u for h e Z and u e U~. -- ([BT] 1.6.1.3 b) and 6.2.3 b)) t(au) = ~o(a) + l(u) for a e K and u e U~ (here l(1) := oo). By definition any h 9 H, satisfies ~(a(h) -- 1) >/m + 1. For u 9 U~.0 we therefore obtain t((h,u)) =r l) +l(u)/> m4- 1 4-s>/r+s, i.e. (h,u)~U~.,48. This shows that h e H~,~. To deduce the general case we observe that Bruhat and Tits proceed by applying the descent theory in [BT] I. 9 in two steps: First from the split to the quasi-split case ([BT] II.4.2.3) and then from the latter to the general case ([BT] 11.5.1.20). Hence we may use [BT] 1.9.1.15 in order to see that our assertion descends as well. In each of the two steps we have to check that the assumption (DP) in loc. cit. is fulfilled and that the descent preserves the properties i'), ii), and iii). Step 9: From the split to the quasi-split case. ~ If G is quasi-split then Z is a maximal torus in G and ~e is that part of the N6ron model of Z over o which in the closed fibre consists of the connected components of finite order. The condition (DP) follows from the explicit computations in [BT] II.4.3.5. The filtration of H by construction is the intersection of H with a corresponding filtration over a splitting field of Z. From this it is obvious that the properties i'), ii), and iii) are preserved. Step 3: From the quasi-split to the general case. -- Note that the descent is along an unramified extension L]K. The condition (DP) (with t = 0) holds by [BT] II. 5.2.2. The descent of the properties i)-iii) is deduced from the following fact: Applying the argument in the proof of Proposition 2 to the group scheme ~r0 (compare [BT] II. 5.2.1) we obtain the decomposition Z'~ II U~o+ x 7-~x H U~,o+); here Z~ ~ denotes the analog of Z ~~ for G(L), ~ is the root system of G(L) with respect to some mammal L-split torus which contains S, and O0 is the subset of those reduced roots which restrict to 0 on S. [] 15 114 PETER SCHNEIDER AND ULRICH STUHLER We fix once and for all a good filtration of H as in Proposition 6. Define U (e) ZC'l:= He+ and U~; I:= ~+,. for e>~ 0. In other words we have U~ *1 = U~,+, where the concave function h r : 9 u { 0 } ~ R is defined by hr [~ :=jr; and hg(0) := 0-4-. The functions h r and fr extended by fF(0) := 0 fulfill the assumption of Lemma 4: This is straightforward if one of the ~, 9, P~ -4- q~ is equal to 0; otherwise it is shown in the proof of [BT] 1.6.4.23. Therefore U~ normalizes U~ '~ for any e >/ 0. Since N normalizes Z TM and N~ normalizes U~+, ([BT] 1.6.2.10 iii)) we obtain that U~ '~, for any e/> 0, is normal in P~. The same argument shows that nU~ n-1 = for n e N and e >i 0. Proposition 1.2.7. -- For any e >1 0 the product map induces a bijection ( II Ua+ . n U~) x z `~ x ( II Ur.+. n U,) --~ U~"; moreover we have U,~+, c~ U, = U,,t~(~+,.U~.~c,~+, for any at e~ ~a. Proof. -- Proposition 2 and [BT] I. 6.9 i). [] Corollary I. 9. .8. -- The following equality holds for any e >1 0: W' = (W' n u-) (W I n z) (W' n u+). Corollary 1.9..9. -- The U~r "' for e >1 0 (and F fixed) form a fundamental system of compact open nzighbourhoods of 1 in G. Proof. -- Since ~ x -~ x ~l ,+ ~ G~ is an open immersion the subset ( H u~.,,,~,) x .~(o) x ( H u,.,,,~,) ~t ~ ~- ~ ~red tt 6 4) + ~ ~red is compact open in G. Qlearly the U,+, n U= form a fundamental system of compact open neighbourhoods of I in U~ for any ~ e ~m. Similarly Proposition 6 implies that the Z ~*~ form a fundamental system of compact open neighbourhoods of 1 in 2L~'~ [] --',~zTc"~ REPRESENTATION THEORY AND SHEAVES 115 Using 1.5 we may define, for any facet F' in X and any e/> 0, a compact open subgroup U~):=gU~ '~g-~ if F'-=gF with g 9 and F a facet in A in G. By construction we have gU~, g-1 -- ~gF'lI~ for any g ~ G. If x is a vertex of X, i.e. { x } is a facet, then we replace similarly as before { x } by x in all our notations; e.g. we write U ~ instead of 11~ Lemma I. 9,. 10. -- There is a point Yo E A such that we have vet = (y0) + - z for any rta where n~, 9 N is a natural number which moreover is even in case 2o~ 9 ~. Proof'. Step 1. -- According to [BT] 1.6.2.23 the statement at least holds with some real number ~et > 0 instead of --.1 (The point --Y0 has to be a special point; in net loc. cit. it is assumed to be the origin and therefore does not appear. Also note that 9 '=~ by [BT] II.4.2.21 and 5.1.19.) Step ~. -- It suffices to show that Fet contains a subset of the form cet 4- n-- Z with some n" 9 N which is even in case 20t ~ 9 and some c~ ~ 11. Because then there has to be amap~:Z~Zsuchthat c~4-n~m=~(y0) 4-~et~(m) for any m 9 If m = 0 this means that r = ~(Y0) 4- s~ ~(0) which inserted back implies ~, v(0) 4- n', m set ,~(m) for any m 9 Z. We obtain so that our assertion holds with net := n'~. (v(1) -- ~(0)). Step 3. -- Let K ~ be the strict Henselization of K. The quasi-split group Gm,h possesses a maximal K'h-split torus T which is defined over K and contains S ([BT] II. 5.1.12) ; let ~h be the set of roots of G/K,h with respect to T. Restricting characters defines a surjecfive map ~Bhu{0}-~U{0}. By [BT] II.5.1.19 we have 116 PETER SCHNEIDER AND ULRICH STUHLER F~ c_ P~, _c F~, whenever ~ c ~'~ restricts to ~ e r The sets r~ are explicitly computed in [BT] II. 4.2.21 and 4.3.4: For any ~ c ~ one has r~ = c~ + -- z nff with appropriate constants c~ ~ R and n~ e N. Let now an a e ~,,d be given. If 2a r (I) then we choose an ~ e (1) ~ restricting to a and we obtain F~ ~_ P~ =c~ +--Z. n~ If 2e e q) then we choose a ~ e ~'~ restricting to 2e and we obtain 1 1 1 1 F~ = ' = c~ + Z. [] Proposition 1.2.11. -- On~ has: i) U~',' G U(;' for any two facets F, F' in X such that F' ~_ F; ii) U(~ '~ = II U(~ '~ for any facet F in X and any ordering of the factors on the right hand side. 9 vex'~x Proof. ~ We may assume that F __c A. First we consider the case that F' = { x } is a vertex. Then U"~ n U~ = U~,c_~+,~+.U~,~_~,~+,~+ for ~ e~ ~. If a(x) = inf ,t(y) then clearly (-- e(x)) + >>.f;(o:) and hence I1 ~*1 n U, c U[g' n U,. If > inf e(y) then el F is not constant so that vql* -- =(x) <fF(~) =f;(~). Furthermore, by the definition of facets, we then have --a(y) eP~ for anyyeF. Hence inf{ t ~ P~ : g > -- ~(x) } >i fF(~) and because of Lemma 10 also inf{ger~:g>--e(x) +e}/>fF(=) +e e. inf teP~:t>--c((x) +~ l>fF(~) +~ m case 2~e~. and This implies that again UI; I n U, _ U~ e~ n U,. Using Proposition 7 we obtain U TM ? a IT~'~ = the subgroup generated by all _, for x c vertex. REPRESENTATION THEORY AND SHEAVES 117 To get the reverse inclusion we fix an ~ e ~ and consider U~ "~ c~ U,~. Let x e F be a vertex such that a(x) = sup a(y). Then IrEF )~< -- l~ey inf,(y) =f~(~) if ~l F is not constant, (-- ~(x)) + = (-- infer(y))+ if ~[ F is constant; ~eF hence U~ '~ c~ U, _~ U~ ~ n U,. Again using Proposition 7 we see that Uk '~ = the subgroup generated by aU U~; ~ for x ~ F a vertex. This implies in particular the assertion i) for an arbitrary facet F' _ F. For ii) it remains to show that for any two vertices x,y ~ F the subgroup --~ II TM normalizes the subgroup --v II TM " But by i) we have UIs~ c U~ ') : PF = Pv z -- -- -- and _~N ~'~ is normal in P~. [] Finally we define, for any z e X, UC;~ := U~ '~ if z lies in the facet F of X. Note that U, = U F in this situation if F _~ A ([BT] 1.7.1.2). 1.3. Properties of the groups U~ > Here we will establish those properties of the groups U~ '~ which are responsible for our later results about the cohomology of the Bruhat-Tits building. We recommend the reader to skip this section at first reading and only come back to it when the results are needed. Fix an e/> 0. A first technical clue is the observation that the following representation-theoretic fact is at our disposal. The notion of a smooth G-representation will be recalled in II. 2. A vertex x in A is called special if ~(x) ~ -- r~ for any ~ e @~a. There always exists a special vertex ([BT] 1.6.2.15). Theorem (Bernstein). -- Let x be a special vertex in A. The category of smooth G-repre- sentations V which are generated (as a G-representation) by their U~'~-fixed vectors V~Z~ (') is stable with respect to the formation of G-equivariant subquotients. Proof. -- This is [Ber] 3.9 i). We only have to check that our group U~ ~ fulfills the assumptions made there. Since the vertex x is special the Iwasawa decomposition G = U, P = G~~ P for any parabolic subgroup P G G holds true ([BT] 1.7.3.2 ii)). Moreover the decomposition property (3.5.1) in [Ber] is a consequence of 2.7. 118 PETER SCHNEIDER AND ULRICH STUHLER Next we need some control over how U~, '~ changes if z varies along a geodesic line in X. We fix two different points x and x' in A. The geodesic [xx'] joining x and x' is [xx'] ={(1--r) x+rx':0<~ r<~ 1}. Proposition I. 3.1. -- Assume x to be a special vertex; for any point z ~ [xx'] we have Uce~ ~ 1)'r Tlce) Proof. -- We may assume z to be different from x and x'. Let F, resp. F', denote the facet in A which contains z, resp. x'. Define 9 " := { 9 : < }. There certainly exists a decomposition 9 = ~+ to O- into positive and negative roots such that ~" ~_ q~+. As a consequence of 2.7 it suffices to check that if ~ e O~a\~, Ucx, {- a(z) + el + "Ugot, (- 2a(z) + el + Ue, f,~r + e" U2~, 231~1 + e --- { if e c~F. U~. #,~ + ,. Uz~ ' 21;,~ +, Assume first that e c O~a\~F. Since x is special we have t := -- 0c(x) e F,. If ~(x) = ~(x') then -- e(y) = t for anyy e{ x } to F to F' and hencef;(~) = t+. If ~(x) > e(z) > ~(x') then -- ~(y) > t for anyy 9 F and hencef~;(~) >~ fF(~) >/t+. Now assume that e e W, i.e. that ~(x) < e(z) < ~(x'). If there exists an t' 9 F, such that -- ~(x') < l' ~< -- e(z), then Otherwise there are two successive values g' < t" in P~ such that e',< -- ~(x') < -- ~(z) < t"; hence g' <f~(~) ~< l" and t' <<.f~,(~) ~ t". Using 2.10 we see that in this case U,. r,~.~ + ." U2-. 2r,~ +, : U~,r,+ e. U2~,2r,+ e [] Because of 1.7 the assumption that x and x' are contained in the basic apartment A is unnecessary. Also the statement remains true even if x is not a special vertex. Since it is not needed we do not go into this. But see the proof of III.4.14. Consider the half-line s:--{ (1 --r) x+rx':r>~ 0} REPRESENTATION THEORY AND SHEAVES 119 in A and put Us := subgroup generated by all U~ for ~ e 9 such that c~(x') > o~(x). As will be explained in IV.2 this group is the unipotent radical of some parabolic subgroup of G. Proposition I. 3.9,. __ Assume x to be a special vertex; for any point z ~ ~ we have Proof. -- This is a straighforward (actually simpler) variant of the previous proof. The only additional fact to use is that the product map induces a bijecfion II U~ --% U, whatever ordering of the factors on the left hand side we choose ([Bor] 21.9). [] H. THE HOMOLOGICAL THEORY II. 1. Cellular chains Through its partition into facets the Bruhat-Tits building X acquires the structure of a d-dimensional locally finite polysimplicial complex ([BT] 1.2.1.12 and II. 5.1.32) where d :---= dim A is the semisimple K-rank of G. For 0 ~< q ~< d put Xg := set of all q-dimensional facets of X. In particular we may view X as a d-dimensional CW-complex the q-cells of which are the facets in Xq. The G-action on X is cellular. Let X~:= O FEXq denote the q-skeleton of X, also put X -1 := 0. With the composed maps ~: H~+,(X q+l, Xq; Z) a_~ H~(X~, Z) -+ Hq(X ~, Xq-'; Z) as boundary maps the augmented complex Hd(X d,xd-';z) 0a_g ... a0> Ho(X0, Z) = @ Z r.>Z ~ffXo computes the (singular) homology of X ([Dol] V. 1.3). It is G-equivariant and it is exact since G is contractible ([BT] 1.2.5.16). In order to motivate later constructions we want to give a more combinatorial description of that complex. By [Dol] V.4.4 and V.6.2 we have the direct sum decomposition H~(X q, Xq-'; Z) = ~D Hq(X', X~\F; Z). FEX8 120 PETER SCHNEIDER AND ULRICH STUHLER Consider, for any F ~ XQ+x and F' ~ X~, the composed map Oq > ~, : Hq+,(X '+~, Xq+~\F; Z) c.~ Hq+~(Xq+l, Xq; Z) H,(X q, X ~- 1; Z) H~(X ~, X"\F'; Z). One has: -- H~(X q, Xq\F; Z) ~ Z for F e X~ (but for q> 0 no canonical such isomorphism exists). -- ~, is an isomorphism if F' _c ~ and is the zero map otherwise. (Using [Dol] V. 6.11 this follows from the fact that in our case the characteristic map ~F of F can be chosen to be injective and hence to be a homeomorphism onto F.) Define now an oriented q-facet to be, in case q > 0, a pair (F, c) where F e X~ and c is a generator of Hq(X ~, X~kF; Z); then (F, -- c) is another oriented q-facet. In case q = 0 an oriented 0-facet simply is a 0-facet F which we sometimes also think of as the pair (F, 1) where 1 is the canonical generator of H0(X ~ X~ Z) = Z. Let X(~) denote the set of all oriented q-facets. Observe that for any (F, c) e X(~+x~ with q >/ 1 and any F' e Xq such that F' ~ F we have (F', The group of oriented cellular q-chains of X by definition is C~ Z) := group of all maps co : X(~) ~ Z such that: -- co has finite support, and, if q >t 1, -- co((F, -- c)) = -- ~((F, c)) for any (F, c) E X(,,. Clearly co-x z) H,(X ", X'-I; Z) o k (q), (F, c) ~ X(q) with ~ = -- 1, resp. 0, in case q > 0, resp. = 0, is an isomorphism which is G-equivariant if G acts on the left hand side by (go) ((F, c)) := o~((g -1 F, g-1 c)). The boundary map 0~ becomes 0q : ~'(X(q+l), Z) ~ C~ Z) o~((F, c))). (F, 9 ~ X(q+l) F'-=~ REPRESENTATION THEORY AND SHEAVES 121 The augmentation map becomes or : Co (X{o}, Z) -~ Z co ~ Y, co(F). F ~ X(o) H.2. Representations as coefficient systems A smooth (or algebraic) representation V of G is a complex vector space V together with a linear action of G such that the stabilizer of each vector is open in G. Let AIg(G) denote the category of those smooth representations. On the other hand a coefficient system (of complex vector spaces) V on the Bruhat-Tits building X consists of -- complex vector spaces V F for each facet F ___ X, and -- linear maps r~, : V F ~ V~, for each pair of facets F' ~ F such that r~ = id and ~,, = ~;, o r[, whenever F" _c ~' and F' _~ F. In an obvious way the coefficient systems form a category which we denote by Coeff(X). We fix now an integer e 1> 0. For any representation V in Alg(G) we then have the coefficient system V := (V v~'}) of subspaces of fixed vectors V~:=V U~'): -{veV:gv=v for all g~U~ '~}; because of U~I, _= for F' _c F the transition maps r~, are the obvious inclusions. Since the U~ '~ are profinite groups the functor y, : Alg(G) -+ Coeff(X) V ~ (VU'{')) F is exact. For any 0 ~< q ~< d the space of oriented (cellular) q-chains ofy,(V) by definition is C~ y,(V)) := C-vector space of all maps o~ : X~} -+ V such that: -- {o has finite support, -- {~((F, c)) ~ V v{/), and, if q >/ 1, -- {o((F, -- c)) = -- o~((F, c)) for any (F, c) e X,~}. The group G acts smoothly on these spaces via (g{~) ((F, c)) := g({o((g -x F, g-~ c))). A straightforward computation shows that the boundary map a : c;'(x,~+., v.(v)) -~ c~ v.(v)) ,~ ~ ((F', c') ~ Z ~((F, c))) {F, c} ff X~q§ F'_cF Or~(C) = C' ~,,TT{'} 122 PETER SCHNEIDER AND ULRICH STUHLER fulfills O o 0 = 0. In this way we obtain the augmented chain complex 0 or v.(v)) L ... c~ (x,0,, v,(v)) A v where the augmentation map is given by : r(x,0 ,, y,(v)) -+ v E X(o) The homology of this chain complex could be called the (cellular) homology of the coefficient system y,(V) on the space X. We will not use this terminology since in the next section it will be shown that these complexes under a rather weak assumption are exact. That assumption has to do with the surjecfivity of the augmentation map. For any open subgroup U _~ G we have the full subcategory AlgV(G) := category of those smooth G-representations V which (as G-representation) are generated by their U-fixed vectors V tr of Alg(G). If the representation V lies in AlgV~')(G) for some vertex x in X then the augmentation map r clearly is surjective. The subsequent two statements are immediate generalizations of the Propositions 1 and 2 in [SS] w 3. Recall that a representation V in AIg(G) is called admissible if the subspace V U, for any open subgroup U ___ G, is finite-dimensional. Proposition II.2.1.- If V in Alg(G) is admissible then the complex C~ y,(V)) consists of finitely generated G-representations. Proof. -- By the admissibility assumption the subspace in C~'(X~ql, y,(V)) of q-chains supported on { (F, + c) }, for a given facet F ~ X~, is finite-dimensional. If F runs over a set of representatives for the finitely many G-orbits in Xq then the corres- ponding subspaces together generate C-~r(x~I, y,(V)) as a G-representation. [] For any continuous character Z : C ~ C � of the connected center C of G we define the full subcategory Alg� :---- category of those smooth G-representations V such that gv = z(g).v for all g E C and v E V of AIg(G). Since C acts trivially on X the complex C~ y,(V)) lies in Algx(G ) if V does. Proposition 1"I.2.2. -- For any representation V in Algx(G ) the complex C~ y,(V)) consists of projective objects in Algx(G ). Proof. -- This relies on the fact that the group P~/CU~ '~ is finite for any facet F _c X. As a consequence of 1.2.9 the group G~ ~ is finite. On the other hand P~/CG~ is finite according to [BT] II.4.6 28. [] REPRESENTATION THEORY AND SHEAVES 123 II.3. Homological resolutions In order to formulate the main result of this chapter let e i> 0 be an integer and let x be a special vertex in A. Theorem lI.a.l. -- For any representation V in AlgV~"~(G) the augmented complex cT(x,.,, v is an exact resolution of V in Alg(G). Proof. -- In the case G = GLa+I(K ) this result was established in [SS]. The proof in the general case in its most parts is a direct generalization of the arguments in [SS]. Insofar we will only indicate the main steps. Step 1. -- Let Cc(T ) denote the space of complex-valued functions with finite (e) support on the G-set T:= G/U, . This is a smooth representation of G which acts by left translations. It lies in AlgV~)(G) and one has the surjective G-homomorphism Co(T) | V V + N v w-~ Y~ +(g) .g(v). g E G[U (*) Bernstein's theorem (I. 3) now ensures the existence of an exact resolution in Alg~")(G) of the form ... -* @ Co(T) -~ (9 Co(T) -~ V -~ 0 11 Io with appropriate index sets Io, Ii, . .. Since the functor 7e is exact a standard double complex argument reduces therefore our assertion in case V to the" universal " case C~(T). Step 2. -- A slightly more sophisticated double complex argument for Co(T) ([SS] w 1) further reduces our assertion to a geometric property of the Bruhat-Tits building X. For any facet F we put T~:= U~'I\T. It follows from I. 2.11.ii) that T~ = T~0II ... lI T,~ T T if{ xo, ..., x, } are the vertices in F. The natural projection T -+ T F is finite and induces an isomorphism C.(Tr) -~ u,(x)u(*)" which we will view as an identification. More generally we have the simplicial set T. ~ : ... , ~T�215 >T� >T > TF Tp ) TF 124 PETER SCHNEIDER AND ULRICA-I STUHLER all face maps of which are finite. There are obvious commutative diagrams /\ T ~ T r T. r' ~ T r, and T. F , T F for F' ___ F. By passing to functions we obtain the double complex (A) 0 0 0 0 ~, @ Co(TF) o o ----- ) ... , @ C,(Tr)--- > C,(T) , 0 FUX d F~Xo 0 , @ C,(T) , ... , @ Co(T) ,. C,(T) ~. 0 l o 0 , @ Co(T � T) > ... > @ C,(T ~ T) , C,(T) :, 0 F~X d TF F~Xo 0 > @ C,(T � T � T) , ... 9 0 C,(T � T � T) , C,(T) , 0 F E Xd TF TF F ~ Xo Te TF l o All its columns are exact. This follows from the fact that each T. ~ is the disjoint union T, r = [J TIt' t ~Tp of the simplicial finite sets T t): ... 9 > :' T, X T t � T t :, T t X T t ~ T t where T t denotes the fiber of the map T ~ T r in t. Simplicial sets of the form TIo tl are well-known to be contractible. The top row in (A) is the complex whose exactness REPRESENTATION THEORY AND SHEAVES 125 we want to establish. Here we have fixed for simplicity an orientation of the building X. Next we have to study, for a fixed m/> 0, the row (A,~) 0~ O Co(T~)~... ~ @ C,(T~)~Co(T)-70 F~X d FEXO from (A). If we view each T~, resp. T, as a subset of T m+l := T � ... x T(m + 1 fac- tors) in the obvious way, resp. diagonally, then the differentials in the above complex are induced by the inclusions T_cT~,'~T~ for F'~F. In order to rewrite this row in a more suitable form we introduce certain subcomplexes of the Bruhat-Tits building. For a fixed (to, ..., t,,) e T "+1 we put X (t0 ..... Ira) :-= union of all facets F __ X such that to, . .., t,~ are not mapped to the same element in U~*)\T. It is easy to see that fl T~=T, ~ex. i.e. that X "~ ..... t,~ is empty if and only if t o ..... t,~. If{ to, ..., t,~ } has cardinality at least 2 than X "~ ..... t,~) is a nonempty closed CW-subspace of X (I.2.11 .i) and [Dol] V.2.7). We now have @ C~(T~)= (~ Cr ..... ~m)) F ~ i. { tO, * 9 -, ~m ) E T m+l and the decomposition on the right hand side is compatible with the differentials. As a result of this discussion we obtain that (A,~) = O (augmented complex of relative chains of the pair (X, X "~ ..... t,)). (tO,..., t m) ~T m+l Since X is contractible this means that we are reduced to show the contractibility of the subspaces X"0 ..... t,) for any { to, ..., t,, } of cardinality at least 2. Step 3. -- The special vertex [l(e) Xo:=goX where t o--go_~ is contained in X "0 ..... t,,) ([SS] w 2 Remark 1). We show that with any pointy e X "0 ..... t,,) the whole geodesic [Xoy ] lies in X "~ ..... t,,). This of course implies the wanted contrac- tibility. Fix a point z e [Xoy ] and let F, resp. F', denote the facet in X which contains z, resp. y. Clearly F' _ X "~ ..... tml SO that there exists I ~<j~< m such that gt~ # t o for allgeU[f). 126 PETER SCHNEIDER AND ULRICH STUHLER If F is not contained in X "~ ..... t,,~ then we find gl ~ U[g ~ with gl tj : t 0. According to 1.3.1 we have U~: U~.~ TTce~ -- Z 0 " ~I~' so that gl E hU~ ) for some h e II (e) i --~,o ~ Put g : = h- ~ gl e U~). Then gtj : h- ~ to : to which is a contradiction. [] Corollary II.3.2. -- For any representation V in AlgV(1)(G) n Algx(G ) the augmented complex C;r(X, ,, v.(v)) -+ v is a projective exact resolution of V in Algx(G ). Proof. -- Combine Theorem 1 and 2.2. [] Corollary H.3.3. -- Let V, resp. V', be an admissible representation in AlgU(1)(G) n Algx(G), resp. Algx(G); then the vector spaces Ext],,x,o,(V , V') are finite- dimensional and vanish for 9 ;> d. Proof. -- Combine Corollary 2 and 2.1 (compare [SS] w 3 Cor. 3). [] Note that because of I. 2.9 any finitely generated smooth G-representation lies in Alg~ if e is chosen large enough. HI. DUALITY THEORY II1. |. Cellular cochaius An element of the space C'~c'(XI,I,y,(V)) also can be viewed as an oriented cellular cochain with finite support on X. This suggests that there is a cohomological differential, too. Indeed, for any pair of facets F, F' ~ X such that F' c ~, we have the projection map p~' : VV~ ;) _+ VO, ") Since the partition of X into facets is locally finite the coboundary map d: c;'(x,.,, v.(v)) -+ c;'(x,.+~,, v.(v)) p~'(~o((r', o~,(c))))) ,. ~ ((F,c) ~ Y~ F' ~F REPRESENTATION THEOI~Y AND SHEAVES 127 is well defined; in case q ---- 0 the summands on the right hand side have to be inter- preted as ~,(c).o)(F'). A standard computation (IDol] VI. 7.11) shows that d d o, e o, x v,(v)) co (X,o,, v,(v)) ... is a complex in Alg(G)--the cochain complex (with finite support) of y~(V). We will see in Chapter IV that this complex computes the cohomology with compact support of a certain sheaf on X. Here we are interested in the relation between the chain and the cochain complex. Again let X : C --~ C � be a continuous character. In Alg� there is the " universal " representation ~ := space of all locally constant functions + : G -+ C such that: -- t~(g-~h) = z(g).+(h) for all g EC, h eG, there is a compact subset Z _c G such that + vanishes outside Z. C where G acts by left translations. This is the x-Hecke algebra of G; its algebra structure will be recalled later on. Note that G also acts smoothly on JV x by right translations. Both actions commute with each other. In the second action the connected center C acts through the character X-i. Fix now a representation V in Algx(G). Its smooth dual V lies in Algz_~(G ). For any 0 ~< q ~< d we have the pairing CO,~ X , v,(V)) x v,(v)) -+ defined by LF~.~(g) := Y, v~((F, c)) [(g-1 c0) ((F, c))]. (F, c) ~ X(q) One easily verifies that ~.ho=hLF~.~ and ~Fh~.~----LF~.~(.h) for any h~G. This means that the above pairing induces a homomorphism ~F: C~ y,(~)) ~ Homa(C~'(X,, ,, y,(V)), ~x) ~ (~o ~ ~/'~,~) which moreover is G-equivariant if the action on the right hand side is the one induced by the right translation action on ~fx" Next one checks that ~F0~. ,, ----- ~. d~ and ~Fa~ ' ~ = ~F~, 0~. In other words ~F is a homomorphism of complexes from the chain, resp. cochain, complex of y,(~) into the Horna(., ~'x)-dual of the cochain, resp. chain, complex 128 PETER SCHNEIDER AND ULRICH STUHLER of y,(V). We claim that )'F is injective. For any (F, c) e X(q~ and any v e V, define an oriented q-chain CO(F,c), ~ of y,(V) by if (F', c') -=-- (F, c), (o(~. ~,.,((F', c')) := prF(v ) if ql> 1 and (F',c') : (F,--c), otherwise; here pr F denotes the projection map pr F : V -+ V U~') v~--~ gv if v E V u for some open subgroup : u] U z We then have V,,o(,,o.0(I ) = 2*.B((F, c)) [prF(v)] ----- 2*.~q((F, c)) Iv] with ,----1, resp. 0, in case q >t 1, resp. = 0. The second identity comes from the observation that any linear form in -~u,(') factorizes through pq,. This clearly implies the injectivity of ~F. Lemma III. 1.1. -- Let V be an admissible representation in Algx(G ) ; then the G-equivariant linear map : C~c~(X(.), 7,('~)) ---%- Homa(C~'(X,.,, 7,(V)), ")fix) is an isomorphism; under this identification we have Hon~(O, 9rt~ = d and Homo(d , ,~~ = O. Proof. -- Only the surjectivity of W remains to be established. Let Wo be an element of the right hand side. Define a map ~ : X(,) -+ V by c)) [v] := .,, o) (1). For a fixed (F, c) the function Wo(CO(F. c,,~) in 9~' z only depends on pr~(v). Therefore the admissibility assumption guarantees the existence of a compact subset Z _~ G such that all functions Wo(O(F ' ~),,) for v ~ V vanish outside E.C. Because of ~(g-'(F, c)) Iv] = ~o(CO,F.c,,~) (g) it follows that ~(g-~(F, c)) ---- 0 for g-' r E.C. REPRESENTATION THEORY AND SHEAVES 129 Since G has only finitely many orbits in XIq) we obtain that the map ~ has finite support. It is now straightforward to see that ~ e C~ y,(~)) is a q-chain such that ~t'(~l) = 2'. ~Fo with the same ~ as above. [] This duality between chain and cochain complexes is perfectly suited to analyze the Ext-groups g'*(V) := Ext~,lgx(V, s~x) in the category Algx(G ). Through the right translation action of G upon s~ffx the space 8*(V) in a natural way is a G-representation which in general might not be smooth. As before we fix a special vertex x in A. Lemma 111.1.2. --For any representation V in AlgU(1)(G) n Algx(G ) we have 6"(V) = h'(HomG(C~ y,(V)), "r H~ "~x))" Proof. -- II.3.2. [] Proposition I11.1.3. ~ For any admissible representation V in AIgU~')(G) n Algx(G ) we have g'(v) = v,(V)), d). Proof. -- Lemmata 1 and 2. [] Remark 1II.1.4. i) The category of finitely generated smooth G-representations is stable with respect to the formation of G-equivariant subquotients ; ii) a smooth G-representation is finitely generated and admissible ~ and only if it is of finite length; iii) let V be an admissible representation in AlgV-('>(G) n Algx(G); then ~ is an admissible representation in AlgV(l>(G) n Algr_l(G ) and we have ~r _= V. Proof. -- i) and ii) [Ber] 3.12. iii) [Cas] 2.1.10 and 2.2.3 together with Bernstein's theorem (I.3). [] In particular it follows from these considerations together with II. 2.1 that the spaces ~"(.) form functors finitely generated and finitely generated o ~" : admissible representations --~ representations in Algx(G ) in Algr_l(G ) If * > d then d'* = 0. For later use we need the following technical consequence of the above results. 17 130 PETER SCHNEIDER AND ULRICH STUHLER Lemma m. 1.5. -- Let V be a representation of finite length in Algx(G ) and assume that there is an integer 0 <~ d(V) <~ d such that g*(V) = 0 for 9 # d(V) ; we then have: i) For e big enough the complex Or (~)) d d or (~)) ) kerd d'v) Co (X~o), T. ) ... ) Co (Xld(v)_~), T, ~I~'(V) is an exact projective resolution of o~v'(v) in AIg~-I(G); ii) #'(gn'v'(V)) = { 0 V otherwise./f * = d(V), HI. 2. Parabolic induction The computation of the spaces oa'(V) in an essential way makes use of the theory of parabolic induction. We fix a decomposition 9 = (D + (I)- of the set of roots (I) into positive and negative roots. Let A ~_ (1) + be the corresponding subset of simple roots. The subsets O G A parametrize the conjugacy classes of parabolic subgroups of G in the following way. First we have the torus S O := connected component of n ker ~Ee of dimension dim S -- $| and the Levi subgroup M e := centralizer of S O in G. Second there is the unipotent subgroup Uo := subgroup of G generated by all root subgroups U, for ~ cO+\< 0 > linear combination of roots in 0 }. The product where (O):={~EO:~isa Po := Me Uo is a parabolic subgroup of G; its unipotent radical is U o. Let ~e : Po -~ Me h [ det(Ad(h) ; Lie Uo) ]-' denote the modulus character of Pc- Because of C ~ M e the category Algx(Me) of all smooth Me-representations on which C acts through the character X is defined. We have the " normalized " induction functor Algx(Me) -+ Algx(G ) E ~ Ind(E) REPRESENTATION THEORY AND SHEAVES 131 where Ind(E) := space of all locally constant functions ~ : G -+ E such that (ghu) = ~2 (h) . h- 1(~0 (g) ) for allgeG, h~M o, and u eU o with G acting by left translations. The reason for introducing the character 8o is the formula Ind(E) ~ ---- Ind(F,) ([Gas] 3.1.2) for the smooth dual E. An irreducible representation V in Algx(G ) is called of type | if there is an irreducible supercuspidal representation E in Algz(Mo) such that V is isomorphic to a subquotient of Ind(E). We put AlgfxZ, o(G) :---- category of all smooth G-representations of finite length all of whose irreducible subquotients are of type | Any irreducible representation in Algx(G ) has a type, i.e. lies in some category Alg~x ~, o(G) ([Gas] 5.1.2). Also Ind(E) lies in Alg~x ~, o(G) if E is irreducible supercuspidal in Algx(Mo) ([Gas] 6.3.7). Technically very important is the fact that Alg~x,o(G ) = Algfxio,(G) if | and 0' are associated ([Gas] 6.3.11). We recall that two subsets | and 0' of A are called associated if S o and S o, are conjugate in G; in this case M o and M o, are conjugate in G, too, and $0 = ~| We actually need a more precise version of that fact. Fix a g E G such that 1 and let E be an irreducible supercuspidal representation in Algx(Mo). M o = gM o, g- Via the map M O, -+ M O h~#g -1 E can be considered as an irreducible supercuspidal representation in Algx(Mo, ) which we denote by gE. Obviously the isomorphism class of ~ does not depend on the choice ofg. Proposition HI. 2.1. -- The representations Ind(E) and Ind(~ have (up to isomorphism) the same irreducible subquotients; moreover given an irreducible subquotient V of Ind(E) there is a subset O' =_ A associated to | such that V is a homomorphic image of Ind(aE). Proof. -- [Gas] 6.3.7 and 6.3.11. [] A key result of this paper which will be established in the Chapter IV (IV.4.18) is the following. Theorem lll.2.2. -- Let E be an irreducible supercuspidal representation in Algx(Mo); there is a subset O' c_ A associated to | such that f Ind(']~) /f 9 = d -- ~O, 8*(Ind(E)) 0 otherwise. 132 PETER SCHNEIDER AND ULRICH STUHLER m.3. The involution Theorem 1II.3.1. -- For any representation V in Alg~xi, o(G) we have: i) o~ = O for 9 4= d -- $| ii) o~-*~ lies in AI~_I,o(G). Proof. -- We prove the vanishing of d"(V) for 0 ~< * < d -- ,| and all V in AlgYx~ o(G) by induction with respect to ,. Fix an integer q such that 0 ,< q < d- ~0 and assume that g'(V) = 0 for all 9 < q and all V. We have to show that g*(V) = 0 for all V. By induction with. respect to a Jordan-H/Sider series we obviously may assume that V is irreducible. Then 2.1 says that there is an exact sequence of G-representations 0 -+V' -+ Ind(E) -+V -+0 where E is an irreducible supercuspidal representation in Algx(Mo, ) for some subset | _~ A associated to | We obtain an exact sequence 6'*-1(V ') -+ oaq(V) -> g'(Ind(E)). Because of Al~xl.e,(G)= Alg~x,o(G ) (and fO'= $| the left term vanishes by the induction hypothesis and the right term by 2.2. For proving ii) we again may assume that V is irreducible. Similarly as above we then obtain an injection gn-~o(V) ~ ~-*~ = Ind(~ where the right hand equality comes from 2.2. This shows that o*~-~*~ hes in Alg~x-, o,(G) = Alg~z~_t,o(G). The remaining vanishing assertion in i) also follows by induction. We already know that ~**(V)= 0 for 9 > d. Since, quite generally, Ind(E)~ = Ind(~,) holds 2.1 can be dualized to the statement that in case V is irreducible we find a mono morphism of G-representations V ~ Ind(E') where again E' is an irreducible super- cuspidal representation in Algx(Me, ) for some subset | _~ A associated to | Therefore an induction argument similar to the above one but downwards from 9 = d + 1 to , -- d + 1 -- ~| is possible. [] This result together with 1.5 ii) implies that @ : Algfx~. o(G) -+ AI~_Le(G ) V ~ o~-~~ is an exact (contravariant) functor such that ~ o d' = id. CorollaTy m.3.2. -- If v is an irreducible representation in Algx(G ) then g(V) is irreducible, too. REPRESENTATION THEORY AND SHEAVES Corollary m.3.3. -- For any representation V in Alg~,e(G) we have: i) V has an exact projective resolution in Algx(G ) of length d- 1~| ii) Ext~agxCol(V , V') -= 0 for 9 > d -- $| and any representation V' in Algx(G ). Proof.- Theorem 1 and 1.5 i). [] Remark m.3.4. -- For any representation V in Al~,zx(G ) we have d"(V) = HomG(V , ~x) = V. Proof. -- We may assume V to be irreducible. Then HomG(V , o~fx) is irreducible, too, by Corollary 2. On the other hand the matrix coefficients of V provide an embedding ~r ([Cas] 5.2.1). [] Fix an invariant measure dg on G/C. Then o~z becomes an associative E-algebra (without unit) via the convolution product q~ (+* q~) (h) :=f +(g) r d~, for +, dtl le Also this algebra o~V� acts from the left on each representation V' in Algx(G ) through d?.v:-=( +(g).gvd~ for ddegffx, v V'. dG /C The antiautomorphism g ~-* g-1 of G induces an algebra antiisomorphism e~a Hence any representation in Algx_l(G ) can be viewed as a right 9f'x-module. In this way the tensor product V" ~x V' and its left derived functors Tor~X(V", V') are defined for any V" e Algx_l(G) and V' e Algx(G). Duality theorem. -- Let V be a representation in Alg~x,o(G); we then have a natural isomorphism of functors Extk~x,o,(V , . ) = Tor~__xt~ e_.(@(v), . ) on Algx(G ). Proof. --- For any V' in Algx(G ) consider the smooth representation ~$"x No V' where G acts only on the first factor. The map a~'x| >> V' +| ~-*+*v is a G-equivariant epimorphism. According to (a slight generalization of) a result in [Bla] (compare also [Ca2] a.4) the representation o~V x and hence o~f' x @c V' is a 134 PETER SCHNEIDER AND ULRICH STUHLER projective object in Algx(G ). We see that the objects in Algx(G ) have a functorial projective resolution by representations of the form ~,~o� ~ V'. It follows from Theorem 1 that Ext~,.z,e,(V, "~x ~ V') = Ext~x,a,(V, ~x) ~ V' = 0 for 9 4= d -- ~@; here the first equality is an immediate consequence of the fact that V has a projective resolution by finitely generated G-representations (II.2.1 and II.3.2). These two properties imply by a standard homological algebra argument ([Har] 1.7.4) that Ext~agz(al(V , .) is the left derived functor of EXffA~:l~l(V, .). In order to establish our assertion it therefore remains to exhibit a natural isomorphism V,x ,(v, v') = "" ~(v) | v' = Ex ea ,l ,(v, a~x) ~x | v'. ,'~gx Using the projective resolution in II. 3.2 (for an e large enough) to compute the Ext's on both sides we see that a natural homomorphism Ext*~a~� a~t~ | V' ~ Ext~a,x,G,(V, V') ,,vg x is induced by the homomorphism of complexes HomQ(C~ I, y,(V)), ~ffx) | V' -+ HomG(C~ ,, y,(V)) V') 9 ,~r 9 , '~" | v' ~ (~ ~ 'v(~), v'). In order to establish that the induced homomorphism in degree d- $| in fact is an isomorphism it suffices, again by applying the above constructed resolution of V', to consider the case V' = ~� But then the map in question simply is ~(v) ~ ~ -. ~(v) w| which, although the ring 9ff x has no unit element, is bijective ([BW] XII.0.3 i)). [] In a more elegant but less precise way these results can be formulated on the level of derived categories. Let D~I(Alg x(G)) denote the bounded derived category of complexes in Algx(G ) whose cohomology objects all are of finite length. Then the functor Dx: D~(Algx(G)) -+ D~(algx_~(G)) V" ~-~ R Homm.� ~x) is well-defined and is an anti-equivalence such that 13� o D x = id. The Duality theorem becomes the statement that R Hom~a~�162 , V") = Dz(V') | V'" for V', V'" in Dh(AIgz(G)). Jf'X These facts constitute a kind of " Gorenstein property" ([Har] V.9.1) for the non- commutative ring ovtax. REPRESENTATION THEORY AND SHEAVES 111.4. Euler-Poincar6 functions In this section we assume that the connected center C of G is anisotropic and hence compact. Then all our previous results hold true without fixing a specific central character Z in advance. Dropping X from a notation has the obvious meaning; e.g. ~ is the Hecke algebra of all locally constant functions with compact support on G. We fix a representation V in Alg(G) of finite length. By II. 3.3 we have, for any other admissible representation V' in Alg(G), the Euler-Poincar6 characteristic EP(V, V') := Y~ (-- l)q.dim Ext~a,(G,(V , V'). ~>~0 It will be a consequence of our theory that this Euler-Poincar6 characteristic is a character value of V'. The character of V' is the linear form tr v, :~ -~C ~-, trace(+ 9 . ; V') which exists since the operator + 9 . on V' has finite rank. In order to see this consequence we fix, for any 0 ~< q ~ d, a set ~ of representatives for the G-orbits in X~. By our assumption on G the stabilizer P~ of a facet F is a compact open subgroup in G. Let Cr : Pt F --> { I } be the unique character such that g((r, c)) ---- (F, r for (F, c) e Xta , and any g 9 P~. We also fix a special vertex x in A and an integer e i> 0 such that V lies in AlgV~')(G). --Fj,~F acts on the finite-dimensional Since U~ *~ is normal in P~ the finite group pttTT~*~ space Vtr~ "). The character of this latter representation will be denoted by "r vF,e : Pt~ --> C. We extend the functions ~-~ and ~v by zero to functions on G. With these notations F, e we define := (- q~O u G ..~"q The volume vol(Ptr) is formed with respect to a fixed Haar measure dg on G. In order to be consistent with our earlier conventions we always let the invariant measure dg on G/C be the quotient measure of dg by the unique Haar measure of total volume one on C. Then the operator ~, . on V', for q~ 9 W, can be written 9 v=fGt~(g).gvdg for v 9 The function f~'p obviously lies in o~. It depends on our choices which, for simplicity, we do not indicate in the notation. We call fvp an Euler-Poincar~ function for the repre- sentation V. If V = C is the trivial representation then fg r is the Euler-Poincar6 function of Kottwitz ([Kot]). Our subsequent results generalize corresponding results in [Kot] w 2. 136 PETER SCHNEIDER AND ULRICH STUHLER Proposition I1"!.4.1. -- For any admissible representation V" in Alg(G) we have: trv,(fVp) ~-- EP(V, V'). Proof. -- According to II. 3.2 we have Ext~,g,o)(V , V') ---- h*(HomQ(C~ yc(V)), V')). But each C~ 7,(V)) decomposes as a G-representation into c~ v.(v)) = 9 C~r(F, v,(v)) F ~ ,,arq where C~ y~(V)) := subspace of all those chains with support in the union of the G-orbits of the oriented facets with underlying facet F. We therefore obtain (-- l)q.dim Homa(e~ y,(V)), V'). EP(V,V') = Z Z q=O F ~ ~q~-q Consider now a single facet F ~ .~'q and fix an oriented facet (F, c). Using the oriented q-chains oIF, c),v introduced in III. 1 we have the isomorphism HornG(C~ y,(V)), V') =-~ ,, ,v(/' V,U~"),, Home~v , ,v ~ (~ ~ 'v(~0,F.,,,o)) where the exponent e F on the right hand side stands for the er-eigenspace of P~. It is a standard fact from the representation theory of finite groups ([CR] (32.8)) that dim Home(V U(~'), V'U~')) " ---- vol(Ptr)-1.jp~ ~-~v 9 ~, ,. ~'~. dg. Therefore our assertion finally comes down to the following general observation: For any function d/~ ~ which is supported on P*~ and is constant on the cosets modulo U~ *) one has " F, e dg trv,( ) = fp, +._v ([Car] p. 120). [] The real number (-- 1) ~. vol (PtF) - 1. dim V t-(') f~(~) = z y. REPRESENTATION THEORY AND SHEAVES 137 is independent of the choice of the sets a~-. Moreover the invariant measure dV g := f p(1).dg on G does not depend on the choice of the Haar measure dg. We call it an Euler-Poincard measure for V. The corresponding volume function is denoted by vol v. In case of the trivial representation V-----C the measure dCg is the canonical measure of G in take sense ofSerre ([Ser] 3.3). We recall that d c g is nonzero with sign (-- 1) a ([Ser] Prop. 28). Serre's " Euler-Poincar6 property " of dCg has a counterpart for any V. Proposition hi. 4.2. -- For any cocompact and torsionfree discrete subgroup P in G we have volv(P\G ) = Y,, (-- 1)*.dim He(P , V) ---- Y, (-- 1)*.dim Ext~tr~(V , C). e~o ,>/o Proof. -- Since V is admissible the spaces V re,') are finite-dimensional. Moreover 1-' being torsion free and cocompact acts freely on X, with finitely many orbits. Hence G-~'(Xcr y,(V)) is a finitely generated free C[P]-module. We therefore can use the resolution in II.3.2 in order to compute the homology groups H.(P, V) and we see that those groups are finite-dimensional and vanish in degrees > d. The second identity is also clear from that. We obtain (compare [Ser] p. 140) Y~ (-- 1)*.dim n~(P, V) = 5] (-- 1)*. Y,, dim V v(,') ~...~> 0 a~ > 0 P ff I"~Xq v(,~) = 1~ (-- 1)*. ]~ dim V . ~r\G/P~ ,>.~ o F E.~r -----vol(l'\G). ]~ (--1)*. E vol(P~)-l.dimV U(') ,~o 1~ Esrq = vol (r\G). [] If K has characteristic 0 then a discrete subgroup P as in Proposition 2 always exists ([BH] Thin. A and Remark 2.3). Hence in this case the measure dVg is uniquely determined by the representation V (and does not depend on the choice of U.). I.i We also introduce the rational number fVp(1) dEp(V) := ; it fulfills d v g = dEp(V ) . d c g. The denominator of d~v(V ) is bounded independently of V. If K has characteristic 0 then as a consequence of Proposition 2 the number dEp(V ) only depends on V; in the rare case that V is finite-dimensional we have dEv(V ) = dim V ([Ser] p. 85). We f0p(1------~ 138 PETER SCHNEIDER AND ULRICH STUHLER therefore call dsp(V ) the formal dimension of V. In a moment it will be seen that this is compatible with the notion of the formal dimension (or degree) of a square-integrable representation. Remark 1II.4.3. -- If V lies in Alge~(G) then we have: i) EP(o~(V), V') = (-- 1)a-~~ V')for any admissible representation V'; ii) ]~ (-- 1)q.dim Ha(P , o~ = (-- 1) a-~~ ~ (-- 1)*.dim Ha(P , V) for any cocom- q~>o ~>~o pact and torsionfree discrete subgroup I' in G; iii) d~,(@(~)) = (-- 1)a-~~ ) /f K has characteristic O. Proof. -- In the proofs of the above two Propositions we used the chain complex (C~ -(,(V)), 0) whose only nonvanishing homology is V in degree 0. On the other hand we know from 1.5 i) and 3.1 that the only nonvanishing homology of the cochain complex :r'~ ~, ~ c.I,Y,(V)),d) is @(V) in degree d ~| [] Associated with any irreducible square-integrable representation V is a unique Haar measure d v g on G which makes the Schur orthogonality relations hold (compare [Car] p. 122 and p. 131); dvg is called the formal degree of V. Proposition 11-1.4.4. -- If either V is irreducible supercuspidal or V is irreducible square- integrable and K has characteristic 0 then we have d v g = d v g. Let us draw immediately the following consequence which reproves results of Harish-Chandra, Howe, and Vigneras. Corollary 1II.4.5. -- If either V is supercuspidal or V is square-integrable and K has characteristic 0 then the rational number dEp(V ) only depends on V and has sign (-- 1)a; its denominator is bounded independently of V. Proof. -- The only additional observation which we have to make is that dEp (.) is additive in short exact sequences. [] The proof of Proposition 4 requires the abstract Plancherel formula. Let G denote the unitary dual of G, i.e. ([Car] p. 133), the set (of isomorphism classes) of preunitary irreducible representations in AIg(G). For any ~ e~Y the Fourier transform ~ is the function on (~ defined by ~(V') := try,(+ ). Since the group G is of type I([Car] p. 133) the abstract Plancherel formula ([Dix] 18.8) is available; it says: -- d/(1) = Ia~(~) dff for any + ~ where d~ denotes the Plancherel measure (cor- ~Lt responding to dg) on (~; -- a V" in G is square-integrable if and only if vola~({ V' }) > 0 in which case we have g = vol (( V' }).dg. REPRESENTATION THEORY AND SHEAVES 139 Let us now first consider the case of an irreducible supercuspidal representation V. Since V is a projective object in Alg(G) ([Gas] 5.4.1) Proposition 1 implies that 1 if V' ~ V, (fVP)^(V')---- 0 otherwise. Inserting the function fv e into the abstract Plancherel formula therefore gives d v g =fEvp(1).dg ---- vol~({ V }).dg = d v g which proves Proposition 4 in the special case under consideration. The argument in the other case is the same once we use the following two additional facts. Firsdy the support of the Plancherel measure is contained in the tempered irreducible represen- tations ([Be2] Example 4.3.1). Secondly we have the following result. Theorem Ill. 4.6. -- Assume that K has characteristic O. If V is irreducible square-integrable and V' is irreducible tempered then 1 ,f v' ___- v, EP(V,V') = 0 otherwise. Proof. -- The vanishing assertion is a consequence of the subsequent Theorem 21 and [Kal] Cor. on p. 29. In the case V' ---- V we have to show, again by Theorem 21, that f 0v(c-1).0v(C) de ----1; here C 'u denotes the set of regular elliptic conjugacy ell dc classes in G and de the natural measure on it ([Kal] w 3 Lemma 1). But this can easily be deduced from [Kal] Thin. F and [Kal] w 5 Prop. 3 [] There are two more consequences of this type of arguments. Corollary HI. 4.7. -- Assume that K has characteristic O. If V is irreducible tempered but not square-integrable then d~.p(V) = 0. Proof. -- More generally Theorem 21 and [Kal] Cor. on p. 29 imply that, for V and V' irreducible tempered, we have EP(V, V') = 0 unless V and V' are relatives in the sense of [Kal] p. 10; the latter means that there is a representation which is parabolically induced from a square-integrable represen- tation of a Levi subgroup and of which V and V' both are constituents. None of these finitely many relatives is square-integrable ([Kal] Lemma 1.4). Using Proposition I we see that the Fourier transform (fVp)^ has support in a set of Plancherel measure 0. Hence the abstract Plancherel formula says that fVp(1) = 0. 140 PETER SCHNEIDER AND ULRICH STUHLER Recall that if V is irreducible square-integrable then a function + E ~ is called a pseudo-coefficient for V if 0 for V' irreducible tempered but V' g V, try'(+) = 1 for V'~-V. Corollary III.4.8. -- Assume that K has characteristic O. If V is irreducible square- integrable and lies in AI~o(G ) thenfVi , and (-- 1)d-#e.fff~ g~ are pseudo-coefficients for V. Proof. -- Proposition 1, Remark 3 i), and Theorem 6. [] It is very likely that the restriction to characteristic 0 is unnecessary in all of the above statements. Actually we strongly believe that Ext~og~o~(V, V') = 0 for any * t> 0 holds true whenever V and V' are two irreducible tempered representations which are not relatives in the sense of [Kal] p. 10. A possible strategy to prove this would be the following. Define an appropriate category Temp(G) of tempered representations and show that the Ext-groups in Alg(G) and in Temp(G) of any two admissible tempered representations (which naturally belong to both categories) coincide. Next we will discuss the orbital integrals of the Euler-Poincar6 functions fvp and their relation to the character trv as a locally constant function on the regular elliptic set. Recall that an element of G is regular elliptic if its connected centralizer in G is a compact torus. For the sake of completeness let us include the following well-known fact. Lemma HI. 4.9. -- If h ~ G is regular elliptic then the map G~G g ~g-lhg is proper. Proof. -- It suffices to show that the preimage of a compact subset of the form Ug0 U where U _c G is a compact open subgroup is compact. This preimage equals [.] GhgU g E oh\o/tr a-zhgC Ugo U where G h denotes the centralizer of h in G. The element h being regular elliptic its centralizer is compact. Hence the double cosets G h gU are compact. On the other hand it is a particular case of Lemma 19 in [HCD] that the set over which the above union is taken is finite. [] We denote by G ell the open subset of all regular elliptic elements in G. The above Lemma says that for each h ~ G ~ and each + ~ ~ff the integral + (h) = +(g-1 hg) dg REPRESENTATION THEORY AND SHEAVES 141 exists. As another consequence of that Lemma the set X h of fixed points of a given element h e G en in the Bruhat-Tits building X is compact (compare also [Rog] Lemma 1). To see this fix a facet F e #'.. The Lemma says that the set {g ~ G :g-' hg e pt }/pt = {g e G: h e Pt F }/Pry ={gEO:X nngF+ O}/P~, is finite. It follows that X n is covered by finitely many facets and hence is compact. We have hF=F for a facet F~X, if and only if F(h):=FnXh+0; moreover, as explained in [Kot], F(h) then is a polysimplex whose dimension fulfills ~(h) = (-- 1) ~-"'m~'~'. In this way X h is a finite polysimplicial complex; we denote its set of q-dimensional facets by (Xh),. Lemma Ill. 4.1O. -- For any h e G *n we have (fvx,)V(h) = ~ N (-- 1)'. v--~-, ,(h). Proof. -- We first quite generally consider a function d/~ ~f' whose restriction to P~, for some facet F ~ X,, is a class function and which is zero outside of pt. Let (X~) h denote the set of fixed points of h in X~ and let (G.F) h be the intersection of (X~) h with the G-orbit G. F of F in X~. We compute d?(g-X hg) dg = Y~ d?(g -1 hg).vol(G~ gP t) fG 9 e c,\G/P~ ~r- l hg ff P~ = Y~ +(g-' hg). vol(P;). [Gh: Gh c~ P,tF] oF E Gh\CG. 7# = vol(P;). X; +(g-' hg). Applying this to each summand of our function fv we obtain = ('r~,,. ~,) (g-' hg) q=O FG,~q cF ~ (O.F)h = N 2~ 2~ (-- 1)".%F(h).-rvF,,(h) q=0 FG..~q gFff(tl.F) h ---- X ~ (-- 1)q.~F(h).v[,(h) q = 0 F f3 (Xq) h = Z ~] (-- 1)a.'r~,,(h). [] An element in G is called noncompact if it is not contained in any compact subgroup. 142 PETER SCHNEIDER AND ULRICH STUHLER Remark m.4.11 (The Selberg principle for Euler-Poincar6 functions). -- For any noncompact semisimple element h ~ G we have fo,\o f p(g-1 hg) = 0 where dg' is any Haar measure on the centralizer Gh of h in G. Proof. -- Note that since semisimple orbits are closed their orbital integrals exist in any characteristic. The Euler-Poincar6 function fvp vanishes on noncompact elements. [] Lemma III.4.12. -- The function (fvp)v on G ell is locally constant. Proof. -- This is a consequence of Lemma 10 once we know that for any given h e G *u there is an open subgroup U __c_ G such that X ~'--X n for any h'EhU nG on. First note that X h is never empty ([Tit] 2.3.1). We choose a point y e X h. Since X n is compact we find a constant r > 0 such that x { z ex:d(y, z) < r}. Consider now the open subgroup U:={geG:gz-=z for any zeX with d(y,z)~ r}. Clearly, for any h' E hU, we have Xh~ X h' and Xh'\X h _c { z e X : d(y, z) > r }. Since with any z e X h' the whole geodesic [yz] is contained in X h' the latter inclusion forces Xh'\X h to be empty. [] We define an equivalence relation on G ~u by h,~h' ifXh=X h'. In the preceding proof we have seen that the corresponding equivalence classes are open. Hence any function ~ E o~ ~ with support in G en can in a unique way be written as += 5: h ~ Gellf ~ where d& e ,~' has support in the equivalence class of h. Also the function q-- 0 F(h) E {Xhlq REPRESENTATION THEORY AND SHEAVES 143 in o-~ only depends on the equivalence class of h; here, for any compact open subgroup U _c G, cv denotes the idempotent ~tr(g) := { v~ if g ~ U, 0 otherwise inaff. Lemma 111.4.13. -- For any ~? ~,~ with support in G en we have +,, try( fot~(g) (fvv)V(g-~) dg = Y~ h E Gell/~ Proof. -- It is clear ([Car] p. 120) that 9 ~..(h) = tr~(h(~.~)) holds true for any facet F in X such that F c~ X h + 0. Using this together with Lemma 10 we obtain (fvp) v(h-1) = trv(h(eh)) -= trace (v ~--~ foch(h-~ g) gv dg) = trace (v ~ fo~dg) hgv dg) = trace (v ~ h(eh * v)). Therefore the left hand side in our assertion becomes fo qb(g).trace(v ~-~g(%, v))dg = trace (v ~-'fa ~b(g).g(%, v)dg). If t~ has support in the equivalence class of some h E G en the last expression obviously is equal to ) dg)=trace(v~--,+, (ah,v))= trv(+,~h). [] Lemma HI. 4.14. -- For any h e G eu and any F0(h ) e (Xh)o we have ~;~ , ~ = ~;~. Proof. -- Let F0(h ) = {y } be the given vertex of X h. We introduce a relation between facets F and F' in X as follows: We write F -~ F' if -- V(h), ~. V'(h) + ~, -- F' ~ F (equivalently F'(h) _ F(h)), F' 4: F, and -- there are points z e F(h) and z'e F'(h) such that z e [yz']; trace(v~fod?(g).g(an*v 144 PETER SCHNEIDER AND ULRICH STUHLER moreover in this situation F, resp. F', is called y-large, resp. y-small. These notions have the following elementary geometric properties: I. Any facet F 4:F0 with F(h) 4= 0 is either y-large or y-small. 9. A y-large facet is not y-small and vice versa. 3. For any y-small facet F' there is a unique y-large facet F such that F-~ F'. 4. If F is y-large we have (- 1)"lm~ + X (- 1).~m~,~h~ = 0. In order to see these properties consider an arbitrary point y' E X ^ different from y. The whole geodesic [yy'] then belongs to X h. In addition one has: 5. There are only finitely many facets F0, ..., F,~ in X such that F, n [yy'] 4: O. Indeed, X is locally finite. 6. Each intersection F~ ~ [yy'] either consists of one point or is an open convex subset of [yy']. Choose an apartment A'_ X which contains [yy'] and therefore each F~. Let ( F( ) denote the affine subspace of A' generated by F d note that F( is open in ( F, ). If the intersection F( n [yy'] consists of more than one point then [.yy'] ~ ( F~ ). The enumeration F0, ..., F,. obviously can be made in such a way that d(y, z,) < d(y, z,+l) whenever z, e F, n [yy'] and z,_,, E F,+ 1 n [yy']. Then the intersections consisting of one point precisely are the F2, t~ [yy'] for 0~< i~< -~ and we have F 0c_ F1- F2-~ F3=- F4~- --- It is clear that F,._ 1 -~ F,. if F,. n [yy'] =: {y' }. The converse holds in the following stronger form. 7. If F ~ F,. then F = F,._ 1 and hence F,. n [yy'] = {y' }. Let z E F(h) and z'e F,.(h) be points such that z e [yz']. Choose A' as above so that F= _~ A' and hence [yz'] u F _c A'. Applying the preceding discussion to z' we obtain that (F) contains [yz'] u F., and hence [yy'] and F.,_ 1. We also obtain that F,. n [yz'] -- { z' } so that [yy'] cannot belong to (F,.); this means that F,. n [yy'] = {y' } or in other words that F,.-a - F,.. As a consequence ( F,._~ ) contains [yy'] u F,. and hence [yz'] and F. Therefore F and F=_~ must be facets of the same dimension both having F,. in their boundary. Assuming F and F,._ I to be different there would exist an affine root for A' which is 0 on F,. but has different signs on F and F,._ 1. On the other hand because of F n [yz'] 4: O, reap. F=_ 1 n [yy'] 4: 0, this affine root has the same sign on y and on F, reap. F,._ 1, which is a contradiction. REPRESENTATION THEORY AND SHEAVES 145 This implies 3. together with the following characterization. 8. A facet F' 4: F 0 in X such that F'(h) 4:0 isy-small if and only if F' c~ [yz'] == { z' } for some (or any) z' r F'(h). It also implies the direct implication in the following analogous characterization of y-large facets. 9. A facet F # F 0 in X such that F(h) # 0 is y-large if and only if F n [yz] is open in [yz] for some (or any) z ~ F(h). For the reverse implication let A'c X be an apartment which contains ~z] and let L ___ A' denote the affine line generated by [yz]. It follows from I. 1.5 that rn L ~ X h. The intersection (F\F)c~ L consists of exactly two points and those belong to X h. Taking as y' that one of bigger distance to y we obtain, with the previous notations, that F --~ F,. Clearly 6., 8., and 9. imply 1. and 2. It remains to discuss the property 4. Fix a y-large facet F. In particular F ~ F 0. _-'F'_ "F" 10. Let F" and F' + F be facets such that F" c F, c ~, and F'(h) ~: 0; if F-+ then F -L F'. We choose points z e F(h) and z" e F"(h) such that z e [yz"]. We also choose an apartment A'c_ X containing y and F and therefore also F' and F". Fix a point ~" e F'(h) and consider the euclidean triangle in A' with verticesy, z", and ~'. It follows that (z~') ::= [z~']\{ z, 7' } is nonempty and is contained in F(h). Fix a point ~e (z~"). The two affine lines in A' through y and ~ and through z" and 7' intersect in a point z' ~ F'(h). Then~ e [yz'] and hence F -~ F'. The last statement means that Y:=union of all F'(h) where F'___F, F'~e F, and not F~F' is a subcomplex of F n X h. Since the latter is contractible we have Z (-- 1)d'~ r'(h) = 1. F'--q~ ]~"(h) ~ o The property 4. therefore is equivalent to Z (-- 1)aJmv'(a) = 1. , F'ff~) _r Y The latter certainly holds if we show Y to be contractible. We may assume thaty and F are contained in the standard apartment A. Let < F > denote the affine subspace of A generated by F. Since F is y-large we necessarily have y e < F >. Let H~, ..., H, _r A be affine root hyperplanes such that the Fx, ..., F, defined by F, : H~ n 19 146 PETER SCHNEIDER AND ULRICH STUHLER precisely are the codimension 1 facets ofF. For cacti H, fix a defining affine root cq(. ) + t, in such a way that F ={xc(F):~,(x) +t, 2>0 for any 1~< i~< s}. Since F 4= F 0 the set I :={ 1 ~< i< s : g,(y) +t,.< 0} is nonempty. Using [Bou] V. 3.9.8 ii) one sees that the ~, for i e I restricted to the linear subspace parallel to ( F ) are linearly independent. Hence there is a facet F' _c such that fl- ,~zF, 9 Since h fixesy and F it permutes the F, with i c I and fixes F'. This means that F'(h) 4 ~ 0. Once we show that Y = 0 n X h iEI it is then clear that Y can be contracted to any point in F'(h). Consider first a point xcF~nX h for some ieI. Then oq(x) + g, -= 0 and ~,(y) + l, ~< 0. This implies (~,(.) +gi) l[yx]<<. 0 and hence [yx] nF=0. If F_c ~ is the facet containing x then it follows that not F ~ F. We conclude that x e ~'(h) _~ Y. Now let x be a point in Y. Then [yx] n F = 0. Moreover we have (~,(-) +t,)] [yx]\{x}>0 for any ir The case x =y is clear since, in that case x 9 F~ for all i 9 I. Assume therefore that x 4: y and that x is not contained in the right hand side of our claimed equality. Then =,(x) +t,> 0 for any i e I. Hence we would find a x'e [_yx], x'+ x, such that ,ti(x' ) + g, > 0 for any 1 .< i ~< s. The latter means that x' 9 F which is a contradiction. This finishes the proof of the properties 1.-4. It follows from 1.-3. that eh = ~u~, "~. + Y' ((-- 1) "'=''~' ~-~,,) + Z (-- 1) ~='''~' e~:~). F y--lart~ F~F' Because of 4. it is therefore sufficient to show that e,c,~. * eo~,~ = so~;) * ~;~, whenever F -~ F'. Fixing a pair F ~ F' we have to check that U~,~ U~ U ~.~ 11',' F0' ~ ~- Fo'~F' REPRESENTATION THEORY AND SHEAVES 147 holds true. The inclusion U~', ) _c U~, ~) is clear from I. 2.11. Hence it remains to establish the other inclusion U,., c U~.~ U~' This is a variant of I.3.1. We may and will assume that the facets Fo, F and F' lie in the basic apartment A. Let z ~ F(h) and z'~ F'(h) be points such that z ~ [yz']. It is trivial to see that, for , ~ @~a, we have f~(~) >~f~,(~) and hence U~*' c~ U~ _= U(~? except possibly in the case fF(~) =fF,(~), :c [F not constant, but a iF' constant. In that case we have -- e(y) < -- .(z) < -- ~(z') -----fi(e). If fFo(~) ~< -- e(z') then f;(e) /> fx~o(~) and hence U(v e, n U~ _c Otherwise there are two consecutive values g < t' in P~ such that t < -- ~(y) < -- ~(z') < t'. Using 1.2.10 we obtain TT{e) A U~ = Ug t,.U2~,~t,+ -- - U~ ~ n U~ [] ~Fo , r 9 To get further we need the subsequent result which doubtlessly holds true in general but which we can establish, at present, only under some additional assumption. Let [ogf, og'] be the additive subgroup of .gf generated by all commutators ~b, q) -- r for % tp ~ o~f and put oVf "b : ---- o~f/[o~f, ogf]. Proposition 111.4.15. -- If G is split or if K has characteristic 0 then the class of fVv in ~b and hence the function (fv)v is uniquely determined by the representation V (and the Haar measure dg) and does not depend on the choice of U(~ ~. Proof. -- Proposition 1, [Kal] Thm. O, and [Ka2] Thm. B. [] Theorem III.4.16. -- If G is split or if K has characteristic 0 then we have try(+) = fa +(g) (fvv) V (g-~) dg for any qJ ~ ~ with support in G% Proof. -- By Proposition 15 we may choose the number e as large as we want. Let hi,..., h,~ e G ~a be representatives of those equivalence classes which meet the ~Fo'TT'e) 148 PETER SCHNEIDER AND ULRICH STUHLER support of + and, for each 1 ~< i ~< m, fix a F~(h~) e (X~)0. We now choose e large enough so that +hi is U~]-right invariant for any 1 ~< i ~< m. This means that +~i * *v~]) = +hi and hence, by Lemma 14, that +h~* ah~ = +h~. The statement follows then from Lemma 13. [] Of course we know from Harish-Chandra that the distribution trv is given by a locally constant function on the regular semisimple subset G ~ in G; let 0 v denote the restriction of that function to G en. The last Theorem can then be rephrased by saying that under the assumption made there we have 0v(h) = (f~)V(h-X). This might be viewed as a kind of explicit formula for the character values on the regular elliptic set; compare also the Hopf-Lefschetz type formula in IV. 1.5. Corollary HI. 4.17. -- If G is split or K has characteristic 0 then we have: i) Ov(h -1) = Ov(h ) for h e G~n; ii) 0,,~)= (-- 1)a-~~ /f V lies in Alg~o(G). Proof. -- i) We obviously have fVp(g-1) =fV(g) for any g e G. ii) It follows from Proposition 1 and Remark 3 i) that trv,~j~. P _ (_ l)d-~o.fvp) = 0 for any admissible V'. Hence that function is contained in [~,~] by [Kal] Thm. 0 and [Ka2] Thm. B, respectively. [] J~mma HI.4.18. i) EP(V, V') = EP(V', V)for V and V' of finite length in Alg(G); ii) let | C A be a proper subset and let E be a representation of finite length in Alg(Mo) ; then EP(V, Ind(E)) = 0. Proof. -- i) The symmetry is obvious from the expression = 1 e 1 $-x v v' EP(V, V') ~ ~ (--) .vo (PF) . ag which was given in the proof of Proposition 1. Of course e >/ 0 here should be chosen u(e) C. in such a way that both V and V' lie in Alg 9 (_). ii) The subsequent argument is due to Kazhdan. The set of unramified characters of M o is a complex algebraic torus of dimension d -- $O (compare [Car] 3.2). The function REPRESENTATION THEORY AND SHEAVES 149 on this torus is regular according to [BDK] w 1.2. Using Proposition 1 we see that the function ~-~ EP(V, Ind(E | ~)) is regular and integral valued; therefore it is constant. On the other hand it is shown in [Ca2] A. 12 that Ext~,o~(V , Ind(E | ~)) = Ext~.~(~o,(V~o , E | ~) holds true. But for any character ~0 such that the central torus So in M o acts on the Jacquet module V~o and on E | ~0 by different characters we have Ext~,,,o)(V~o , E | ~0) = 0 ([BW] IX.1.9). [] Lemma 1II.4.19. --- If K has characteristk 0 then we have dg h\o ~g, = 0 for any h e fo fVp(g-1 hg) G~g\G 'n. Proof. -- Proposition I, Lemma 18, and [Kal] Thm. A. [] Following [Kal] we put A(G) :: { qj eoCf :fo +(g_Xhg ) dg = 0 for any h e G~'\G "n } h \G ~g' and A(G) := A(G)/[.~,.,~V]. Let R(G) := Grothendieck group of representations of finite length in AIg(G) (w.r.t. exact sequences) tensorized by C. The induction functor Ind(.) induces a homomorphism R(Mo) -+ R(G) for any subset 0 ~ A. We put R~(G) := E image of R(Mo) and R(G) := R(G)/R~(G). OCA If K has characteristic 0 then it follows from Proposition 15 and Lemma 19 that R(G) -+ ~,(G) class of V ~ class offVl, is a well-defined homomorphism; as a consequence of Proposition 1, Lemma 18, and [Kal] Thin. 0 this map is trivial on RI(G ). 150 PETER SCHNEIDER AND ULRICH STUHLER Proposition 111.4.20. -- If K has characteristic 0 then the map R(G) ~-~ A(G) class of V v--> class of fVp is an isomorphism. Pro@ -- It follows from Theorem 16 that, up to the substitution h ~h -1, the map in question is the inverse of the isomorphism in [Kal] Thm. E. [] Our approach allows to establish a kind of orthogonality formula for characters which was conjectured by Kazhdan and which generalizes [Kal] Cor. on p. 29. Let C ~n denote the set of all regular elliptic conjugacy classes in G; then +, for any + e ~, as well as 0 v can be viewed as functions on C en. According to [Kal] w 3 Lemma 1 there is a unique measure dc on C eu such that d? dg = climb(c) dc for any t~ e ~ with support in G "n. Theorem 111.4.21 (Orthogonality). -- If K has characteristic 0 then, for any two representations V and V' of finite length in Alg(G), we have fO Or(C--l).Ov,(C )dc = EP(V, Vt). ell Proof. -- According to [Kal] Thm. F we have the identity try'(+) = ,n 0v'(c)" +(c) dc for any function + e A(G). The Lemma 19 allows to apply this identity to the function d/ =f~p. Using Theorem 16 we obtain trv(fNVp) = fGel I 0V(C-- 1). 0V ,(C) rig. It remains to apply Proposition 1 to the left hand side. [] By Lemma 18 the Euler-Poincar~ characteristic induces a symmetric bilinear form EP(., .) : R(G) x R(G) -+ G. Because of Theorem 21 this form coincides with the form [., . ] considered on p. 5 in [Kal] provided K has characteristic 0; hence it is nondegenerate in this case. As is pointed out in [Clo] w 5 a better undcrstanding of this form is tied up with the study of the L-packets. Finally we want to relate our concepts to the notion of the rank of V introduced by Vigneras ([Vig]). She extends the formalism of the Hattori-Stallings trace to the REPRESENTATION THEORY AND SHEAVES 151 context of smooth representations. The technical difficulty which arises is that the Hecke algebra .OF has no unit element in general. For us it is most natural to work with the subalgebras o,~(e) := ~ui,~ , ~, ~,) for e t> 0 in which ~v~*~ is the unit element (recall that x is a fixed special vertex); by I. 2.9 we have 9~' = U ~,~(e). e~>0 The point is that UI, *1 fulfills the assumptions of [Ber] 3.9 as we noted already in I. 3 so that the functor AIgU~')(G) _Z_> category of unital left ~,Vg(e)-modules V' ~ V'(e) := (V') of*) is an equivalence of categories which in addition ([Ber] 3.3) respects the property of being finitely generated. First let V' be a finitely generated projective representation in AlgV~')(G); then V'(e) is a finitely generated projective ~ff(e)-module and we have the obvious isomorphism Homae,,l(V'(e),.,Cd'(e)) Q V'(e).~ Endae,,,(V'(e)). .~(e) If E v~ | v, is the element in the left hand side which corresponds to the identity endo- morphism in the right hand side then the rank of V' is defined as r v, := class of Z v~ (v~) in ~,b. Now consider an arbitrary finitely generated representation V' in Alg(G). Bernstein has shown ([Vig] Prop. 37; alternatively we can use II.3.2) that V' has a resolution 0 ~V~ -+ ... -+V~ -+V' -+0 by finitely generated projective representations V~ in Alg(G). Choosing e large enough so that the whole resolution lies in AlgV~')(G) the rank of V' then is defined to be r v, := 5", (-- 1)i.rv ,. t~0 It is shown in [Vig] Prop. 39 that if G is split or K has characteristic 0 then the class r v, E w,b only depends on V' and is characterized by the property that trw(rv, ) = EP(V', V") holds true for any irreducible representation V" in Alg(G). Combining this with our Proposition 1 we see that the Euler-Poincar6 functions fvp of our representation of finite length V are representatives of its rank r v. 152 PETER SCHNEIDER AND ULRICH STUHLER Proposition Ill. 4. '2"2. -- If G is split or if K has characteristic 0 then we have r v = class offVv in ,~b. Actually a more precise result holds true. Each individual summand offVv is the rank of a corresponding direct summand of our projective resolution in II. 3.2: As already used in the proof of Proposition 1 there is the decomposition C~r(X(ql ~'e(V))= O c~ , o, , v,(v)). PG.~q Fix a facet F e ~'~ and put V' := C~ ~ , y~(V)). Since V' is projective its rank is characterized by the property that trv,,(rv, ) = dim HomG(V' , V") for any irreducible representation V". But in the proof of Proposition 1 it was shown that the latter dimension is equal to trv,,(vol(P~)-I xv ~r). 9 F, r Hence we obtain that rank of C~ yr = class of vol(ptv)-l.xvv~.~-r in ~t '~. Propositions 2 and 22 together give a different proof, for a representation of finite length, of the dimension formula in the Main Theorem 36 in [Vig]; the positivity statement in loc. cit. is, by the above discussion, trivial for the projective representations appearing in our resolution II.3.2. More importantly we obtain a relation between the rank r v and the trace 0 v which is entirely similar to the case of a finite group ([Hat]). Note that the function + on G en only depends on the class of + in ~ff~b. Theorem HI. 4.23. -- If G is split or if K has characteristic 0 then we have Ov(h ) = (rv)V(h -x) for heG ell. Proof. -- Theorem 16 and Proposition 22. [] IV. REPRESENTATIONS AS SHEAVES ON THE BOREL-SERRE COMPACTIFICATION IV. 1. Representations as sheaves on the Bruhat-Tits building Let V be a smooth representation of G. For any open subgroup U_ G we have the space V v := maximal quotient of V on which the U-action is trivial of U-coinvariants of V. We write v mod U for the image in V v of a vector v ~ V. Fix an integer e >/ 0. In order to simplify the notation we sometimes will suppress indicating REPRESENTATION THEORY AND SHEAVES 153 the dependence on e in the notions to be introduced. Let F be a facet of X. Since l;~*~ ~F is profinite the projection map prr : V ~> V U~(') in III. 1 induces an isomorphism ~ VU?) Vr{,) =, Whenever F' is another facet such that F' _ F then we have the commutative square where p~, is the other projection map from III. 1 and pr is the obvious quotient map (coming from the fact that U~,}~ U~*}). The representation V gives rise to a sheaf V on the Bruhat-Tits building X in ev the following way: For any open subset fl ~ X put V(~) := C-vector space of all maps s: ~ -+ 0 Vu~, ~ such that z~fl -- s(z) c Vu~,) for any z c t, -- there is an open covering fl = [J fl~ and vectors v~ ~ V with ~I s(z) = v~modU~- *) for any z~"/i and icI. The stalks of the sheaf V are the expected ones as we will see in a moment. The star of a facet F' in X is the subset of X defined by St(F') := union of all facets F ~ X such that F' __q F. These stars form a locally finite open covering of X. Lemma IV.I.1. i) (V), = Vv(/) for any z ~ X; ii) the restriction of V to any facet F of X is the constant sheaf with value Vv6,). Proof. -- There is the obvious map (v), vo{,, germ of s ~ s(z). 20 154 PETER SCHNEIDER AND ULRICH STUHLER It is an isomorphism since if z lies in the facet F then St(F) is an open neighbourhood of z with the property that U~/~_c _,,II ~"~ for any z' e St(F) (by 1.2.11 i). The same argument shows more generally that, for any nonempty subset 2; open in F, we have lira> V(f2) = locally constant Vv~,)-valued functions on Y~. [] -~ St(F) open ~F-~ Lemma IV. 1.2. -- Let F be any facet in X; then Vu~,) f* ~0~ H*(St(F), V [ St(F)) -= H*(F, V I F) = ( /f, > 0. Proof. -- This is (a polysimplicial version of) [KS] 8.1.4. [] It follows from Lemma 1 that the functor Alg(G) -+ sheaves on X V~V is exact. Our aim in this Chapter is to compute the cohomology with compact support H~(X, V) of the sheaf V.~ The interest in this comes from the fact that this cohomology can be calculated from the cochain complex of y,(V) considered in III. 1. * h,/Corfx Proposition IV.1.3. -- We have the equality H,(X, V) = ,.~ ,~ ~.,, y,(V)), a). t~J Proof. -- The filtration X = ~0~ f~l~_ ... ~ Oa of X by the open subsets ~2" := X\X "-1 induces the filtration ro(x, .) _= ro( l, .) =_ ... r4n', .). Because of [God] II.4.10.1 the spectral sequence of this filtration reads El"" := (~ H:+'~(F, V [ F) ~ H:+'~(X, V). F~Xn 9 ,~ According to Lemma 1 ii) we have ifFeX, and * =n, H;(F, V I F) ---- ( H~(F,o Z) | Vu~,) otherwise. Inserting this into the spectral sequence we obtain dd- l, O 0 Hd(F, Z) | Vu~,)]. H:(X, V) = h'[F2x ~ S~ Z) | Vv#, ... F E X d The description of the cellular coboundary in [Dol] V. 6 and VI. 7.11 shows that the complex on the right hand side coincides with (C~ y,(V)), d). [] REPRESENTATION THEORY AND SHEAVES Corollary IV.1.4. -- Let V be a representation of finite length in Alg� ~e is chosen big enough then we have oP'(V) = H:(X, ~). Proof. -- Proposition 3 and III. 1.3. [3 The sheaf V at hand, we can reformulate III.4.16 as a trace formula. Proposition IV. 1.5 (Hopf-Lefschetz trace formula). -- We assume that the connected center C of G is anisotropic and that either G is split or K has characteristic O. Let V be a repre- sentation of finite length in Alg(G) and choose e big enough. For any h e G ~1I we have 0v(h ) ----= Y, (-- 1)a.trace(h; H*(X h, V)). q~O Proof. -- First of all note that since X h is compact and V is admissible the coho- mology H*(X h, V) has finite dimension. By III.4.16 (as explained in the paragraph after that Theorem) we have 0v(h ) = (fvp)V(h-'). Moreover III.4.10 says that (fVp) V (h- 1 ) = Y, Y~ (-- 1)*.trace(h; VOW')). Proposition 3 of course is completely formal and applies to the finite polysimplicial complex X h as well. Therefore the right hand side in the last identity is equaI to Y~ (-- 1)Ltrace(h; C~ y,(V))) ~=0 ---- Y, (-- 1)q.trace(h; Hq(X h, V)). [] q=O IV. 2. Extension to the boundary In [BS] Borel and Serre have constructed a compactification X of the Bruhat-Tits building X with the help of which they could determine the cohomology with compact support of a constant sheaf on X. Our strategy for computing the cohomology with compact support of our sheaves V will be similar. In this section we will defne an appro- priate " smooth" extension j, V of V to a sheaf on X. The boundary cohomology of that extension will be discussed in the next section. Finally in the section after the next one it wilt be shown, for V of finite length at least, that j,, o~ V is cohomologically trivial on X. The result about the cohomology with compact support of V will then be obtained from the long exact cohomology sequence. 156 PETER SCHNEIDER AND ULRICH STUHLER We first give a description of the Borel-Serre compactification X which is adapted to our purposes. Let A denote the compactification of the basic apartment A by " the directions of half-lines " ([BS] 5.1). As an explicit model one can take A:={x~A:d(O,x)<~ 1} together with the embedding j:A~A x~ t 1 -- d(0, e -d(~ x) .x if x+ 0, 0 ifx =0. A boundary point x e A| := A\j(A) then corresponds to the half-line [0x) := { rx : r >>. 0 } in A. The N-action on A extends uniquely to a continuous action of N on A. Note that Z acts trivially on the boundary A~. For any boundary point x e A~ we have the parabolic subgroup P, := subgroup generated by Z and all U, for ~ e 9 such that ~(x) >/ 0 in G; its unipotent radical is U, := subgroup generated by all U~ for ~ e@ such that ~(x)> 0; clearly nP, n-l=P,, for heN holds. Moreover [BoT] 5.17 and 5.20 imply that P, nN=N~:={n~N:nx=x} for any xeAoo. These two properties allow to formally imitate the defnition of X by setting X:= G x A/,-~ with the equivalence relation ~ on G x P. defined by (g, x) ~ (h, y) if there is an heN such that nx=y and g-ihneP,. The group G acts on X through left multiplication on the first factor. The map Y~->X x ~ class of (1, x) is injective and N-equivariant. There is an obvious G-equivariant map j:X -~X REPRESENTATION THEORY AaND SHEAVES 157 which is injective. The latter fact follows from the observation that, because P. = U.. N. for x ~ A, we could have used the groups Px instead of U x in the definition of X. On the other hand the boundary X~ := X\X is X~ = G � Ar Hence X~ as a G-set coincides with the Tits building of parabolic subgroups in G (compare [GLT] 6.1). We see that at least as a G-set X is the Borel-Serre compactifi- cation of X. We equip .~ with the quotient topology of the product topology on A given by the natural topology on A and the ::-adic topology on G. Zemm~ IV. 2.1. -- The space X is the Borel-Serre compactification of X. Proof. -- Without loss of generality we may assume that the origin 0 in A is a spccial vertex. Also fix a decomposition (I) = 9 + w (I)- into positive and negative roots; this corresponds to fLxing a fundamental Wcyl chamber D:----{x~A:~(x) t> 0 for any ~O +} in A. Let D denote the closure of D in A. Then the obvious map P0 � b -~ X is surjective. The Borel-Serre topology on X is the quotient topology with respect to this map if the left hand term is equipped with the product topology of the n-adic topology on P0 and the natural topology on D (see [BS] 5.4.1). Therefore the Borel-Serre topology is finer than our topology. But it is also shown in loc. cit. that the former one induces on D its natural topology and that the G-action on X is continuous in the Borel-Serre topology. This implies that the two topologies under consideration actually coincide. [] We have ([BS] 5.4): -- X is compact and contractible; -- X is open in X and the topology induced by X on X is the metric topology of X; -- X~o with the topology induced by X is the :z-adic Tits building of G ([BS] w 1) ; -- the topology induced by X on A is the natural topology of A; -- the G-action on X is continuous. In the following we keep the assumptions and notations introduced in the proof of Lemma 1. One advantage of viewing X as the quotient of P0 � ~j is that since is a fundamental domain for Po in X ([BS] 4.9 iii)) the equivalence relation ~ for (g, x) and (h,y) ~ P0 � ~) simplifies to (g, x) ~ (h, y) if and only ifx=y and g-lheP0np,. For later purposes it is necessary to explicitly construct appropriate neighbourhoods in X of any point in the boundary X,. Since D~, := D\D is a fundamental domain 158 PETER SCHNEIDER AND ULRICH S'I'UHLER for Po in X,o it suffices to consider a point x e D,o which is fixed throughout the following. The set St.(x) :={x' eDoo :P., _c p=} is an open neighbourhood of x in Do. Put r := { ~ 9 r : ~(x) > o } _ r We also fix an open normal subgroup U in P0 and a real number r >/ 0 such that U nU~_D U=,, for any ~eq)(x). Lemma IV.2.~,. -- Let f~ c_ Stw(X) u{y eD: a(y) > r for any o: er } be any subset; then the subset U(P0 n P=) x ~ is ,,~-saturated in Po � i). Proof. -- Consider a point y 9 and elements g e P0 and h 9 U(P o n P=) such that g-lh e P0 n P~. We have to prove that then necessarily g 9 U(Po n P,). Since by assumption g c h(Po n P,) c_ U(Po n P=) (Po n P,) it suffices to show that Po n Pv -~ U(Po n P=) holds true. In case y ~ StD(x) we even have Py _c p=. Therefore we may assume that y 9 with ct(y) > r for any a 9 According to 1.1.2 and 1.1.4 we have Po n P~ = Pio~j = l-I U= Stoyj,~" Nf~l r ~ ~red for an appropriate ordering of the factors on the right hand side. Since N acts on A by affine automorphisms Nc0,~ is contained in Nw=, j where x' e Do is such that y lies on the half-line [0x'). But x'e Stw(X); this follows from ~(x')> 0 for any 9 e q)(x) which amounts to U=_ U=,. We obtain Nt~ c_ NLo~, ~ _c Po n P=, _q Po n P=. For a root 0t eO ~ we distinguish two cases: ~ eO\(--O(x)) or at ~--O(x). In the first case we have ~(x) >/ 0 which means Us __c_ P= and hence U~,,Ico~c~,~ _c U,,.o c_ Po n P=. In the second case we have -- e(y) > r and hence U,~. ftoylr ~ U=,_ =~v~ c U=., __c U. [] REPRESENTATION THEORY AND SHEAVES 159 For any subset f~o c StD(x), we now define the subset C,(~o0) := ~o u { y 9 D : y 9 [0x') for some x' 9 f~oo and a(y) > r for any ~ 9 ~(x) } in D. For any x' 9 f~o there is a unique point y' 9 [0x') such that [0x') n C,(~) = [0x')\[0y']. Lemma IV.B.a. -- Let f~o---Stw(x) be an open neighbourhood of x in D~; then U(P 0 c~ Px).C,(f~) is an open neighbourhood of x in X. Proof. -- It is easy to see that C,(f~oo) is open in D. Hence U(Po n P,) � Cl,(f~oo ) is open and w-saturated in P0 x D. [] These neighbourhoods have the disadvantage not to reflect the cellular structure of X. We therefore define Cl',(~o) := f~ u [J { St(y) n D :y 9 C,(fl~o) n D a vertex such that St(y) c~ D _ Cl,(f~oo) }. /,emma IV.2.4. -- Let ~ _c StD(x ) be an open neighbourhood of x in Do~ and put := U(P 0 n P,).C',(fl~) ; we then have: i) G',(fl~o ) is an open neighbourhood of x in I); ii) f~ is an open neighbourhood of x in X; iii) f~ nX-- [.J St(y). V 9 a vertex Proof. -- The set U(Po n P,) � C',(f~o) is open in P0 � D if we assume i) and is w-saturated by Lemma 2. Hence ii) is a consequence of i). Moreover iii) follows from ii). Indeed, by construction fl n X is a union of facets. But any open subset of X which is a union of facets contains with any facet F the whole star St(F). This in particular shows that the right hand side in iii) lies in f~ n X. To see the reverse inclusion first note that the right hand side is invariant under U(P0 c~ P,). It therefore suffices to consider a point z EC',(~)~)riD. Then by definition there is a vertex y 9 C~(f~) n D c f~ n X such that z 9 St(y). The crucial assertion to establish is i). Since St(y) n D is open in D for any vertex y 9 D it remains to ensure that any point x' 9 f~o~ has an open neighbourhood in which is contained in C',(f~,~). For this it is convenient to use certain standard neigh- bourhoods of x' in Doo. Thinking of A~ as being the unit sphere in A (as we do in our explicit model) we have, for any 0 < ~ < 1, the open neighbourhood f~, := { x" eD= "d(x',x")< ~} of x' in Doo. We may choose r small enough so that ~2, _~ fl~. Then C,(~,) is an open neighbourhood of x' in D which lies in C,(~o). Let now c > 0 be a fixed real constant. 160 PETER SCHNEIDER AND ULRICH STUHLER It is an elementary computation to show that by decreasing c and increasing r appro- priately we obtain a C,,(f~,,) _c C,(f2,) with the property that { z' e D : d(z, z') < c } C,(f2.) for any z e C,,(f~,,) n D. We choose the constant c in such a way that d(z, z')< c whenever z E D and z' e U { St(y) n D: y any vertex of the facet containing z }; this is possible by I. 2.10. It follows easily that then C,,(f~,,) _c C',(f~,). [] Lemma IV. 2.5. -- Let f~ c__ X be an open neigkbourhood of x; then we can choose U and r in suck a way that U(P0 n P,).C',(f~) ___ f~ for some open neighbourhood f~ c_ StD(x ) of x in D o . Proof. -- Consider the quotient map ~ : Po � b ~ X. The subset ~t-x(f~) is open in P0 X D and contains (P0 n P~) x { x }. We therefore find, for any h e Po n P~, an open normal subgroup U(h) ___ P0 and an open neighbourhood f~o(h) of x in D such that u(h) k x no(k) _ By the compactness of (P0 n P~)/C we have U(h~) hx C u ... w U(h,.) h,. C _~ Pon P. for finitely many appropriate elements hi, 9 9 h,, e P0 n P,. Now put f~o := f~o(hl) n ... n f~o(h,,) and U := U(hx) n ... n U(h=). We then obtain U(Po n P,) � f~o c V.-~(~) and hence U(Po n P~).f~o c_ fL It remains to observe the elementary geometric fact that for any open neighbourhood f~o ofx in D we find an open neighbourhood f~g StD(x ) ofx in D~o and a r/> 0 such that C,(f~) _c Do. p, Lemma IV.2.6. -- Any boundary point in X~ has a fundamental system of open neigh- bourhoods D in X such that f~ n X = [.J St(y). w~t~x & vertex Proof. -- Lemmata 4 and 5. [] Lemma IV.2.7. -- Let c _ U. be a compact subset; then there is an open neighbourhood D of x in ~, such that r =_ U~ ~) for any facet F c_ f~ n X. __%_ REPRESENTATION THEORY AND SHEAVES Proof. -- (Recall that e >/ 0 is fixed throughout this Chapter.) Fixing an enume- ration of the roots in ~ c~ ~(x) any element g e c can be written in a unique way as g = l-I g~ where g~ ~ U~. The compactness of r implies that for any such root ~ the set t,:={t(g,):gec such thatg~4= 1} is bounded below; we put r : = min { t : t e G }- Define now f~o:={yeA:~(y)>e+ 1--/, for any ~e~ar3~(x)). Clearly there is an open neighbourhood f~ of x in ,~ such that f~ n A = f~0. It therefore suffices to show that tg Uk "~ for any facet F_~D0. Fix a root =t ~ ~ c~ ~(x), an element g E c, and a facet F __q f~o- We actually check that g~ e U,,~ +, __q U~ 'J c~ U, holds true. The case g, = 1 is trivial. Otherwise we have f~(~,) ~< -- inf a(y) ~< -- (e q- 1 --l,) ~< -- e -- 1 q-t(g,,) w Et'l, and hence f;(~) + e < t(g.). [] The sheaf V has the two obvious extensions j, V _ j. V to sheaves on X. We will work with a third " intermediate" or " smooth " extension j,V---~j.,| ,j.V which is constructed as follows. Let i : X~ -+ X denote the inclusion of the boundary. The stabilizer P, of any boundary point z ~ X~ is a parabolic subgroup of G; let U, denote the unipotent radical of P,. For any z e X~o we may form the Jacquet module Vtr ~ of U,-coinvariants of V; similarly as before we write v mod U, for the image in Vtr * of a vector ~ ~ V. Analogously to V we can define a sheaf _V_V on X| in the following way: For any open ~ _~ X~ put V(t2) := G-vector space of all maps s : l) -~ 6 Vv, such that -- s~t'l --s(z) e Vv, for any z eO, -- there is an open covering f~ = L) f~ and vectors v~ 9 V with (EI s(z) =v~modU, for any zet2, and ieI. 21 162 PETER SCHNEIDER AND ULRICH STUHLER It was mentioned already that X~ is a simplicial complex but equipped with a topology which is coarser than the simplicial one. For any point z ~ X~ we put St(z) :={ z' ~X~ :P,, _~ P,}. Remark IV.2.8. -- Let z ~ X~ be a boundary point; for any open subgroup U ~_ G the subset U.St(z) is an open neighbourhood of z in X~. Proof. -- Because of St(z') _~ St(z) for any z' E St(z) it suffices to prove that there is a subset fl _ St(z) containing z such that U .fl is open in X~. We may assume that z c D~o. Choose an open neighbourhood ~ ___ StD(z ) = St(z)nDo of z in D,. According to Lemma 3 the subset U(P0 n P,).~ is open in X,. Therefore f~ := (P0 n P,).fl~ meets the requirement. [] Lemma IV.2.9. -- We have V, = Vv, for any z E X| Proof. -- There is the obvious map Zz ~ VU z germ of s ~-* s(z) which clearly is surjective. In order to see the injectivity let s be a section of V in a neighbourhood f~ _~ X~ of z such that s(z) = 0. By shrinking the neighbourhood we may assume that s is represented by a single vector v ~ V, i.e. s(z') = v mod U,, for any z' e ~. For z' ~ St(z) we have U, _ U,, and hence v rood U,, = 0. Let U be the stabilizer of v in G. It easily follows that actually v mod U,, = 0 for any z' e U. St(z). We obtain s]U.St(z) n~ = O. [] Since the formation of Jacquet modules is exact ([Car] p. 128) Lemma 9 implies that the functor AIg(G) -+ sheaves on X~ V~V_V_ is exact. By construction the sheaf V, resp. V, is a quotient V r-~V, resp. V-~-V, of the constant sheaf with value V on X, resp. X~. The first arrow induces by adjunction and restriction a not necessarily surjective homomorphism V ~i*j.V. REPRESENTATION THEORY AND SHEAVES 163 /,emma IV.2.10. -- We have the commutative triangle /\ v , i.s.v where the upper term is the constant sheaf with value V on X= and the oblique arrows are the natural sheaf homomorphisms. Proof. -- We have to show that, for any point z c X,~, the natural map v (j. v). contains in its kernel all vectors of the form gv -- v for some g e U, and some v ~ V. By the G-equivariance of this assertion we may assume that z c Doo. Choose an open subgroup U ~ G such that v, gv c V v. By Lemma 7 and the fact that (P0 n P,)/C is compact we find an open neighbourhood ~o of z in )k such that { h- 1 gh : h c P0 n P, } ~ U TM for any vertex Yo ~ ~o n X. -- it 0 Consider now a vertex y ~ U(Po n P,) .~0 n X, say, y-~uhyo with ucU, hcPonP,, andyo cY~onX. We then have U~*~ = uhU~*~ h-X u -1 and g' :=- h-l gh cU ~*) and hence gv -- v = u(gu -I v - u -iv) ---- uhg' h-a u -av-v=OmodU~*'. It is quite clear that ~0 contains a subset of the form C',(~oo) as considered above. Using Lemma 4 we therefore see that z has an open neighbourhood ~ in X such that n X = U St(y) ~G~nx & vertex and gv--v=OmodU --y I*l for any vertexycf~nX. If y' ~ fln X is an arbitrary point, say, y' e St(y) for some vertex y e fln X then UI~I ~, = -- li~,) ~ by 1.2 9 11. We obtain that gv--v = 0modU I*1 for anyy'efl nX. --I/" This means that the image of gv -- v in (2". V) (f~) and a forfiori its image in (j, V), is zero. [] 164 PETER SCHNEIDER AND ULRICH STUHLER We now define j.,~o V to be that sheaf on X which makes the diagram j..oo V ,j.V i,V__ - > i.i*j.V cartesian; the right perpendicular arrow hereby is given by adjunction and the lower horizontal arrow is the direct image of the arrow in Lemma 10. By construction we have 9 , 9 3 d.,| V = V and z "* J,,~o " g =- V. In particular the functor AIg(G) -+ sheaves on X V~j., V co~ is exact; of course this functor depends on the choice of the number e whereas the sheaf V does not. We also obtain the short exact sequence of sheaves O~j,V -+j.,| V ~i.V -+0. Since H*(X,j~ V) ----- H:(X, V) ([KS] 2.5.4 i) and (2.6.6)) the associated long exact cohomology sequence reads ... -+ H'(X,j.,~ V) -+ H'(X=, V) -> H',+'(X, V) 1 X 9 ... -+H '+ ( ,),,~oV)-+ (e) Later on it is technically important that for the representation V := C~(G/U= ), x a special vertex in A, our " smooth" extension has a simpler description. Proposition IV.2 11. For the representation V ~ special vertex in A, 9 -- = Co(G/U~ ), x a We haY8 j.,~ V = image(V --+j. V). Proof. -- By Lemma 10 we quite generally have natural homomorphisms V >>j.,~V---~j.V the first one of which is surjective. We therefore have to show that in case of our parti- cular V the second one is injective, i.e. that for any boundary point z 9 X.~ the natural map between stalks Vv, -+ (j. V). REPRESENTATION THEORY AND SHEAVES 165 is injective. Let us first make this map more explicit. Put l~(z) := system of all open neighbourhoods fl of z in X with the property that n n X = O St(y). & vertex Because of Lemma 6 we have (J. v), = lim V(n n X). The sheaf axiom says that, for any ~ e lI(z), the restriction map v(n n x) II v(st(y)) vertex is injective. According to 1.2 the terms on the right hand side are V(St(y)) = Vv~,,. Putting this together we see that what we have to show is the injectivity of the natural map Vu~-+ lim 1-[ Vv~. fl ~ 11(~) ~t~X ~ett*x -~- G (e) In other words fix a function + ~V C,( /U, ) and a neighbourhood ~ e lI(z) such that + = 0 mod II TM for any vertex y e ~) n X. --y We have to cheek that then necessarily + = 0 mod U,. We write ~b as a sum += Z +4 h e u~\G/ui,~ of functions ~b n ~ Co(G/U! ')) in such a way that ~b h has support in U, hU(~I/U~ ~ and we will show that each summand fulfills +h = 0 mod U,. Consider an individual element h e G. There is an apartment which contains both points x and h-Xz in its closure ([Bro] Thin. VI.8). By I. 1.6 we find an element g e P, such that x, gh -~ z e A. Using the G-equivariance of our problem and replacing z and + by gh-~ z and gh-a ~, respec- tively, we see that it suffices to deal with the case z e A and h e P,. We choose ele- ments u~,..., u,, e U, such that hit ~e) , u~ .__= }. hU~e~ supp(+n) = { u 1,.~, , .., Note that u~hll _~ TM = u~ _~ IT (~ h since P~ normalizes _~1I c*~. According to Lemma 7 we find an open neighbourhood ~' of z in ~, such that ~' ___ ~ n A and ux, .,u m~U (~) for any vertexye~' AA. 9 . ~Lr 166 PETER SCHNEIDER AND ULRICH STUHLER In particular we have = 0 mod IT TM for any vertex y ~ f~' n A. On the other hand it was shown in I. 3.2 that U(e) ~ TT IT (e) for any x' [xz). Choosing x' ~ [xz) close enough to z in such a way that the closure of the facet F which contains x' still lies in f~' and choosing a vertex Y0 ~ F we obtain -- TT(~) 1. 2L 1~ . . .~ 2L m ~ UVO 2. ~ = 0 mod U TM and YO ~ 3. IT ('1 c U(F ~ IF ~) ~ U,.TT ~> The properties 1. and 3. imply that 4, IT (~) hi? ~) (TT (~) hTT TM __,0 ___, n supp(~b) = U, ___,hit 'e) n supp(~b) = ,--,o n U,) .__, ~ supp(~b). It follows from 2. and 4. that IT (e) n U, [] ~b h = 0 mod IT (`) and even ~b h = 0 mod --~0 --VO Appendix. -- Geodesics in X. Implicitly in our thinking about the compactification X is the existence of a unique " half-line " in X between any given point in X and any given boundary point. In the section after the next one we will have to make explicit use of this fact. Also it is the link between [BS] and [Bro] VI. 9. Since we could not find an appropriate reference this will be justified in the following. We fix a point x e X and a boundary point z ~ X~. According to [BT] I. 7.4.18 ii) or [Bro] Thm. VI.8 there is an apartment A' _ X such that x ~ A' and z e (A')~. Let [xz)a. denote the half-line in A' in direction z emanating from x and put [xz]A, := v ( z }. The subsequent result allows to simply write [xz) and to view the latter as the geodesic between x and z in X. Proposition. -- The half-line [xz),' does not depend on the choice of the apartment A'. Proof -- Let A" _~ X be a second apartment such that x ~ A" and z e (A")~. We have to show that [xz)A, = [xz)A,. We first treat in several steps special cases where additional assumptions about A' and A" are made. Step 1. -- Here we assume that x is a special vertex. For notational simplicity we may assume by G-equivariance that A" = A is our standard apartment and that REPRESENTATION THEORY AND SHEAVES 167 z z D~. By I. 1.6 we find an dement h e P, such that A' = hA. Write z = hz o with z0 z A~o. Since x is a special vertex we find an n z N, such that nzo ~ D~o. We then must have hn -a E P, because D o is a fundamental domain for Pz in X~o ([BS] 4.9 iii)). Therefore replacing k by hn-1 we may assume that A'=hA with hEP, nP,. Again since x is a special vertex we obtain from [BS] 4.10 that h fixes the half-line [xz)a pointwise. We now conclude that = [hx, hz)h,, = h[xz), = [xz)A. Step 2. -- Here we assume that the intersection A' n A" contains a special vertex Xo. From the first step we know that [x o z)~, = Ix 0 z)x,,. Hence [xz)A, and [xz)A,, are parallel rays in the sense of [Bro] VI. 9A and therefore have to coincide by the Lemma 1 in loc. cit. Step 3. -- Here we assume that the intersection A' n A" contains a sector D' such that z e (D')~. Choose any x 0 e D'. Then clearly Ix o z)a,--Ix 0 z)a,,. Hence [xz)a, and [xz)a,, again are parallel and therefore equal. In order to establish the general case we will show that there is an apartment A ~ X such that x ~ A, z e A| A' n A contains a special vertex, and A" n A contains a sector D with z ~ ~=. Using steps 2 and 3 we then obtain = [x )x = In order to find A we choose an h e G with A' = hA", a facet of maximal dimension F ~ A" with h-1 x z F, a special vertex x 1 z F ([BT] I. 1.3.7), and a sector D" _c A" such that z z (D")| By [BT] I. 7.4.18 ii) or [Bro] Thm. VI. 8 there exists an apartment A_ X which contains kF and an appropriate suhsector D c D". Then x z hF ~ ~,, z z D~ ___ A~, ~A"nA, and x o:=hxxzA'nA. [] Corollary. -- Any element in Pz n P, fixes [xz] pointwise. IV.3. Cohomology on the boundary In this section we explicitly compute the boundary cohomology H*(Xoo, V) in the case of an induced representation. Throughout the notation introduced in III.2 168 PETER SCHNEIDER AND ULRICH STUHLER will be in order. In particular A _~ r is a fixed choice of simple roots. Corresponding to A we had defined in IV. 2 the subset D~o ~ Xoo, it is a (d -- 1)-dimensional simplex whose simplicial structure is given by the subsets D ~ := { x e Do~ : U, = Uo } for any proper subset | C A. The closure D ~ of D ~ in Do| (equivalently in Xoo) is a (d- 1 -- $| simplex. Since Do| is a fundamental domain for the G-action on X~o ([BS] w 1) we have an obvious projection map "r : X~o -~" D~ = G\X~o ; it is proper and has totally disconnected fibers. The proper base change theorem ([God] 11.4.17.1) therefore implies H'(X=, V) = H'(D| v. V). For any subset | _~ A we introduce the space Ind~,o(Vvo ) := space of all locally constant functions ~:G-* Vvo such that e?(ghu) -= h- l(~?(g) ) for allgeG, heMo, and ueUo on which G acts smoothly by left translations. (This is unnormalized induction!) Lemma IV.3.1. -- For any proper subset | C A the restriction of% V_ to D ~ is the constant sheaf with value Ind~o(Vvo ). Proof. -- The map G/P o x D~ --% ~-~ D~ (gPo, x) ~ gx is a homeomorphism and x corresponds to the second projection map on the left hand side. The inverse image of V on the left hand space can be computed as follows. For any x e D ~ and any g e G we have the obvious map Vvo = Vv~ -~ Voo ~. Also let p:=pr � x D ~ o � D ~ 9 REPRESENTATION THEORY AND SHEAVES 169 An argument as in the proof of 2.9 shows that the inverse image in question can be identified with the sheaf on G/P o � D ~ whose space of sections in an open subset f~ is the C-vector space of all locally constant maps r : p- 1 ~ ~ Vu ~ such that ~(gh, x) =h -lq~(g,x) for any (g,x) eiz -1~ and hePo. On the level of sections this identification is given by s(gx) --- g~(g, x). It is quite clear that the direct image of this latter sheaf under the projection map to D ~ is the constant sheaf with value Indr~ Our assertion follows now by an application of the proper base change theorem. [] Specializing to the case of an induced representation we once and for all fix a subset 19o-~ A and an irreducible supercuspidal representation E of Moo and we put V := Ind(E). For any @_ A we need the subgroup W o:= (s=:x~O) of W; moreover let [W/Wo0], resp. [Wo\W/Wo0], denote the subset in W of representatives of minimal length for the cosets in W/Wo0 , resp. the double cosets in Wo\W/Woo. The Weyl group W acts on the set of roots ~. According to [Ca.s] I. 3.4 two subsets 19 and | in A are associated if and only if | = w| for some w e W. Lemma IV. 8.2. -- For any proper subset | C A the following assertions are equivalent: i) Vuo + O; ii) woo ~ | for some w ~ [Wo\W/Woo]; iii) O contains a subset which is associated to @o. Proof. -- The equivalence of i) and ii) follows from [Cas] 6.3.5. The third assertion is a trivial consequence of the second one. To see the reverse implication assume that WOo~ t9 for some wEW. We then have wE[W,,oo\W/Wo0 ] by [Cas] 1.1.3 and hence Vv, o, 4: 0. But Uo -~ U~oo so that Vtr o 4: 0, too. [] Corollary IV.3.3. -- The support of the sheaf v, V is equal to the (d- I -- $| dimensional simplicial subcoraplex D| := U{D~ wO0-c 0C A for some w ~W} of O~. Let ~< on W denote the Bruhat order. We now fix an enumeration /W/Woo/ = { 1 = Wo, wl, W 2 .'. } 22 170 PETER SCHNEIDER AND ULRICH STUHLER in such a way that m~< n ifw~< w.. For any proper subset | C A, this allows us to define a decreasing filtration F]V:={~elnd(E):c?iPow,~Poo=0 for any m<n} of V by Po-invariant subspaces. It induces corresponding filtrations (F~ V)v o of Voo (forming the Jacquet module is exact!) and F" IndP~ ) :---- Ind~ V)vo) of Ind~ ). The latter clearly is G-equivariant. Most importantly we obtain a G-equivariant filtration F" ~. V of the sheaf ~, V defined by F" - V :--~ subsheaf of all sections s such that s(z) ~ F" Ind~ for any z~D ~ and any | Our further computation is based on the associated G-equivariant spectral sequence E~" := n "+"(D~o, gr~ ~, V) :~ H ~+ "(D~, ~, V) = H "+ "(X| V). First of all we have (gr~.V), =Ind~ for any z~D ~ and | A. (The functor Ind~ is exact by [Car] I. 1.8.) By construction gr~oV=0 ifw. r ]. For any w, e [W/W| let us fix a lifting g, e N. If w, e [WokW / Woo] then the compu- tation in [Cas] 6.3.1 and 6.3.4 shows that (gr~o V)v o = 0 if and only if w, G 0 $ | and 9 . g-Z1 [normalized mductaon of . E] = 8o v~ @ [ from g, Poo g~- 1 c~ M| to Mo] if w, | _c | Since, by [Cas] 1.3.3, we have the Levi decomposition g, Poo g-- 1 c~ M o = M~, Oo. (g, Uoo g,- ~ c~ Mo) , the last formula simplifies to [normalized parabolic induction] if w, O| -~ O. (gr~) V)u O = ~1[2 ~ I_ Of ~nXE from M,,,o0 to M o ] REPRESENTATION THEORY AND SHEAV'ES 171 Using the transitivity of parabolic induction ([BZ] 1.9 (c)) we therefore obtain that if w, e [Wo\W/Wo 0] and w. O0 --= O, (gr~, v, Y)'- = { Ind(g~'E)o otherwise for any zeD ~ and OCA. Put D~(n):= u{D ~174 OCA and w, [wo\w/Woo] } r }; = w{D~:| and W. 0o_~ | w. the equality is a consequence of [Cas] 1.1.3. This set is empty if w, | is not properly contained in A; otherwise it is an open subset in the closure of D~ .~176 which contains D~-o0. Altogether this establishes the following fact. Lemma IV.3.4.- The sheaf gr~v.V is the constant sheaf with value Ind(~E) on D| extended by zero to all of D| For the corresponding cohomology groups this has the consequence that H'(Doo, gr~, % V) = H;(D| Z) | Ind(~lE) H'-a(D~"~176 Z) | Ind(r"lE) if * 1> 2, if* ----- 1, = coker(Z -+ H~176176 Z)) | Ind(~ if * =0 ker(Z -+ H~176174 (n), Z)) | Ind(O~'E) provided D| + 0. Lemma IV. 3.5. -- Assume that w, Oo is a proper subset of A and that n 4= O, ~tW/Woo; then D~,~176 is contractible. Proof. -- The set D~ .~176 is the geometric realization of the abstract simplex given by the poset (w.r.t. inclusion) of all nonempty subsets of A\w. O o (note that by assumption A\w. O0 4= 0). The set Do~.~176 is the geometric realization of the subcomplex given by the subposet of all those subsets which do not contain A\w, ~+. Our assumption that n 4= 0, resp. 4= SW/Wo,, implies that A\w. ~+ 4= 0, resp. + A\w. | The first implication is a consequence of [Bor] 21.3. In order to see the second implication assume n + SW/Wo, and put w := w.. Let wa, resp. Woo , be the unique maximal (w.r.t. the Bruhat order) element in W, resp. Woo; then w + wa woo so that WWoo + w,x. Hence there is an e0 e A such that s~, WWoo 7> WWeo and a fortiori s~, w > w 172 PETER SCHNEIDER AND ULRICH STUH.LER which means ~o e w~ +. On the other hand denoting by t(. ) the length function on W w.r.t. A we have, for any 0~ e | wwo0) = t(ws wo.) = t(w) + t(s wo~ = e(w) + t(w..) - I = t(wwoo ) -- 1. It follows that ~0 r woo. What we have to convince ourselves of therefore is the following. Let C be a non- empty finite set, let 0 C C O C C be a nonempty proper subset, and denote by II the poser (w.r.t. inclusion) of all nonempty subsets of C which do not contain (3 o. Then the geometric realization ]II [ of the abstract simplicial complex given by II is contractible. But this is clear: Fix an element c e (3\C 0. The subset { c } corresponds to a vertex in [ H [. Since for any C' e II also (3' u { c } e H, it follows that [ H I can be contracted onto that vertex. [] Lerama IV.3.6. -- i) D| = D~ o/f Oo C A. ii) D~,(n) = D~, Oo /f Oo C A and n = SW/Woo. Proof. -- i) Obvious. ii) This follows from the fact that w. | = w. 9 + n A if w. is the unique maximal element in [W/Wo0 ] ([(:]as] 1.1.4). Theorem IV.3.7. -- Assume that V = Ind(E) for some irreducible supercuspidal repre- sentation E 0f Moo; let g-a eN be a lifting of the unique maximal element in [W/Weo]; we then have V@Ind('E) /f. =0, ~| 1, V /f, =0, $O0<d--1, H*(X~, V) -~ Ind(gE) /f * = d -- 1 -- $0 0 > 0, 0 otherwise. Proof. -- In case Oo = A we have V = 0. In the following we therefore assume that Oo C A. According to the previous results the only nonzero El-terms in our spectral sequence then are V ifn=m=O, E~"'~ ~ Ind(~E) ifn=~W/Woo, n+m=d-- 1 --$0o. Moreover since V is a quotient of the constant sheaf with value V on X| we have a natural augmentation map V--~ H~ V) which splits the edge homomorphism H~ V) = H~ % V) -+ H~ gr ~ v, V) = E ~176 = V. Hence the spectral sequence degenerates and the assertion follows. [] REPRESENTATION THEORY AND SHEAVES 173 IV.4. Cohomology with compact support Our main aim in this section is to establish the following result. In the proof we will follow the strategy developed for II. 3. I; but the tools used there have to be analyzed in more depth. Theorem IV.4.1. -- Let x be a special vertex in A and let e >>. 0 be an integer;for any representation V in AlgU"~')(G) we have: i) the natural map V _2~ H0(X,j,,oo V) is an isomorphism; ii) H,(X,j,,= V) = 0 for * > O. In the proof we will make use twice of homological resolutions. This is made possible by the following observation. Lemma IV.4.2. i) cd(X) = d; ii) cd(X| /fd~> 1; iii) cd(X) = d. Proof. -- (Here cd(.) refers to the cohomology with compact support.) iii) is a consequence of i) and ii) by the additivity of the cohomological dimension. Concerning i) it is a standard fact that a d-dimensional locally finite polysimplicial complex has cohomo- logical dimension d. Finally ii) follows from the existence of a proper map from X~ onto a (d -- 1)-simplex whose fibers are compact and totally disconnected ([BS] 3. I). [] The functor V ~j,,~ V is exact and commutes with arbitrary direct sums as can be seen most easily from the description of the stalks. Moreover X being compact the cohomology functor H*(X, .) commutes with arbitrary direct sums. In the proof of II. 3.1 we had seen that any V as in Theorem 1 has an exact homological resolution in AlgV~')(G) by representations which are direct sums of the " universal " representation C~(T) with T := G/U~ "1. Using Lemma 2 and the facts given in the above paragraph we conclude by standard arguments of homological algebra that in order to prove Theorem 1 it suffices to treat the case V = Co(T). For the rest of the proof V always denotes the representation C~(T). We begin by constructing a very convenient simplicial resolution of the sheaf V on X. Fix an integer m >/ 0 and consider the (m + 1)-fold product T~:=T~+I:=T x ... x T. As in the proof of II. 3.1 we put, for any facet F in X, TF:= V~')\T and T~:= T X ... � T (m -t- 1 factors). TF TF 174 PETER SCHNEIDER AND ULRICH STUHLER The latter is a subset in T m+l. For facets F'_c F we have T~' _c T~. Extending functions by zero therefore induces inclusions Co(T ) _= Co( +1) in such a way that Co(T~') ~ Co(T~) if F'__q I ~. We recall that C~(. ) stands for the space of complex valued functions with finite support. If the point z e X is contained in the facet F we write T, := T r and T~, := T~. A sheaf $',. on X can now be defined in the following way: For any open subset f~ _ X put g'..(f~) := C-vector space of all maps s : f2 ~ LJ C~(T~,) such that sell -- s(z) ~ Co(T~) for any z ~ f2, --there is an open covering f~ = U f~i and functions +~ eC~(T "+1) with s(z) -= qb i for any z ef~ and i eI. /.emma IV.4.3. i) (W=), = C~(T~) for any z e X; ii) the restriction of g',~ to any facet F of X is the constant sheaf with value Cr iii) for any facet F in X we have C~(T~) 0, H'(St(F), St(F)) = H*(F, g'. I F) = { 0 /f, > 0. Proof. -- Entirely analogous to I. I and 1.2. [] In order to distinguish various constant sheaves in the following it is convenient to follow the convention that M/y , for any abelian group M and any topological space Y, denotes the constant sheaf with value M on Y. Obviously ). T.:... ~ >TxT� >TxT ~T as wcll as T F : >TxTxT ~TxT ~T 9 TF TF > TF REPRESENTATION THEORY AND SHEAVES |7b for any facet F are simplicial sets in a natural way. The push-forward of functions with finite support with respect to these face maps commutes with extension by zero. In this way we obtain simplicial sheaves ). ( ) C+.T../x : ... ) ) C~(T2),x ~ Co(T1)/x ~ Cr and W. : " ~'2 , --'gO on X together with an inclusion .g" _c Cr which in degree 0 is an equality W 0 = Co(T)/x. The obvious surjection Co(T)/x ~V defines an augmentation ~" -7 V. Applying j, we obtain the augmented simplicial sheaf j.r.-+j.v on X, Lemma IV.4.4, -- For any abelian group M we have j,(M/x ) = M/~. Proof. -- We will establish a slightly stronger fact. Fix a boundary point z E X| We will show that z has a fundamental system of open neighbourhoods ~ in X such that both ~ and ~ n X are path-connected. The tool to construct such neighbourhoods is the notion of the angle between two intersecting geodesics in X ([Bro] VI. 7 Ex. 1). We first need some notation. Let x ~ X be any point. For any other pointy E X different from x we put [xy) := ( [xy]\{y} ify ~ X, half-line emanating from x in direction y if y ~ X~o ([Bro] VI.9 A). In either case [xy] := [xy) ~3 {y } is a path from x toy in X. We also put (xy] := [xy]\{ x } and (xy) :--- [xy]\{ x,y }. There is a unique facet F(x;y) in X such that x ~ F(x;y) and (xy) n FIx;y ) 0; 176 PETER SCHNEIDER AND ULRICH STUHLER clearly the latter intersection is of the form (xy) n F(x;y) = (xy,) for some y~ e (xy]. -- t Given now two points y,y' e X different from x the two geodesics [xy,] and [xy,] lie in a common apartment (which is euclidean) so that the angle 0 ~< "r(x;y,y') ~ 3, 14... between them is defined (and, in fact, is independent of the chosen apartment). For any real number 0 < ~- < 1 we consider the subset n(x; z; .-) := {y ~ x\{ x }: v(x;y, z) < ~ } of X which contains z. We will successively prove: 1. ~(x; z; ,) and a(x; z; ~) n X are path-connected. 2. The function x\{ x } --, R+ y ~ 7(x;y, z) is continuous. 3. There is a constant 0 < ,(x; z) < 1 such that F(x; z) ~ F(x;y) for any y ca(x; z; ~(x; z)). 4. For any z' e f~(x; z; ,) n Xo~ and any d< min(a -- 7(x; z', z), a(x; z')) we have ~(x; z " , ~') _ c a(x; z; ~) 5. f~(x; z; ~) is open in X. 6. For x' e [xz) and 0 < d ~< ~ we have a(x'; z; ~') _q ~(x; z; q. 7. Let a(.) : R+ -+ (0, 1) be a decreasing function; then fl n(~'; ~; ~(a(x, x'))) = { ~ }. 8. The f~(x; z; ~) for varying x and ~ form a fundamental system of neighbourhoods of z in X. The assertions 1., 5., and 8. contain what we wanted to establish. Ad 1. -- Lety andy' be points in fl(x; z; ,). Then obviously [y,,y] u [y~y'] ~_ f~(x; z; ~). By looking at an apartment which contains F(x;y) and F(x; z) and hence the convex hull of{ x,y,, z, }one sees that [y, z,] _c f~(x; z; ~). We similarly have [y'_ z,] __c_ ~)(x; z; ~). Ad 2. -- The sets F\{ x } with F running through all the facets of X form a locally finite closed covering of X\{ x }. It therefore suffices to check the continuity of the function REPRESENTATION THEORY AND SHEAVES restricted to each such set. But the latter is clear again by looking at an apartment which contains F(x; z) and F. Ad 3. -- This follows from [BT] I. 2.5.11 and the elementary geometry of an apartment containing F(x; z) and F(x;y). Ad 4. -- Lety be a point in ~(x; z'; ~'). Looking at an apartment which contains F(x; z) and F(x;y) and hence, by the assumption on ~', also F(x; z') we find v(x;y, z) .< + v(x; z', z) < Ad 8. -- As a consequence of 2, we know already that fl(x; z; r n X lies in the interior of~(x; z; ~). It remains to consider a boundary point. Because of 4. it is actually sufficient to show that ~(x; z; ~) is a neighbourhood of z where in addition we may assume that z ~ Doo. Suppose this would not be true. Then we find a sequence (z,),E M of points in X\(fl(x; z; ~) u { x }) which converges to z. This means in particular that y(x; z,, z) >/ r for all n e N. On the other hand we will show below that lirn y(x;z.,z) =0 whenever limz,= z. This gives a contradiction and proves our claim. By the construction of X we have z, = tt,,y,, with h, ~ P0 and y, e D. Since P0/C and D are compact we may assume by passing to a subsequence that the h, C converge to hC for some h ~ P0 and that the y, converge to some y E D. It follows that hy = z and hence even y ---- z so that h ~ P0 n P,. Passing again to a subsequence we further assume that all tth-~ 1 fix F(x; z) pointwise. Using that the G-action on X respects angles ([BT] 1.7.4.11) we conclude that v(x; z., z) = v(hh; -1 x; by., hh. z) = v(x; by., z). This argument shows that it suffices (replace D by hD) to consider a sequence z, contained in D. Now using [BT] 1.7.4.18 ii) or [Bro] VI.8 Thm. we find a subsector D' := x' + D ___ D such that x and D' are contained in a common apartment A'. In particular z e D' because of (D'), = D| moreover (x' + z.).E! is a sequence in D' which converges to z (note that x' + z. = z. ff z, ~ D~). It is clear from the definition of the topology of A' that lim y(x; x' + z., z) = 0. m ,--~ O0 From the cosine inequality in [Bro] VI. 7 Ex. 2 it is also clear that lim.. T(x; z., x' + z.) = O. We finally pass for a last time to subsequences of (z,) and (x' + z,) in such a way that the facet F := F(x; z,), resp. F' := F(x; x' + z,), is independent of n ~N. Then 23 17B PETER SCHNEIDER AND ULRICH STUHLER necessarily F(x; z) _~ F'. In this situation we see by working in an apartment which contains F and F' that "~(x; z,, z) ~< ~,(x; z,, x' + z,) + ~'(x; x' + z,, z). Combining the last three formulas we obtain lirn y(x; z,, z) = 0. Ad 6. -- Lety be any point in f~(x'; z; d). Assume that y r [xz) because otherwise y lies in f~(x; z; r for trivial reasons. Then the function x' ~ V(x';y, z) on [xz) is defined. We show that it is increasing as a function of d(x, x'). Choose an apartment A' which contains F(x'; z) u {y }. Looking at the convex hull of { x', z,,,y } in A' it is clear that T(x";y, z) >>. "~(x';y, z) for any x" ~ [x' z,,]. Ad 7. -- Let y be any point in the intersection on the left hand side. Then y r [xz) and T(x';y, z) <~ ,(d(x, x')) for any x' ~ [xz). The left hand side is increasing with d(x, x') by the previous argument whereas the right hand side is decreasing by assumption. It follows that T(x';y, z) = 0 for any x'~ [xz). Would y and z be different then we would have [xz] n [xy] = [xx'] for some x' e [xz). On the other hand one easily deduces from y(x ,y, z) 0 that (x'y.,) n (x' z..) + r Hence we would obtain a contradiction. Ad 8. -- Choose a sequence of points xl, x2,,., in [xz) such that d(x,x,) is increasing and tends to oo and choose a decreasing sequence of real numbers 0 < E 1, ~2, 9 9 9 < 1 which converges to 0. ]'hen, by 5. and 6., the subsets f~(x,; z; ~) form a decreasing sequence of open neighbourhoods of z such that the intersection of their closures is, by 6. and 7., equal to { z }. It is a general fact about compact Hausdorff spaces that such a sequence has to be a fundamental system of neighbourhoocks. [] This has two consequences: First of all we obtain an inclusion j. ~. _~ Co(T.)~ of simplicial sheaves on X. Secondly because of j. W 0 ---- Vt2 it follows from 2.11 that j,, ~ V is the image of the augmentation map. Therefore we actually have an augmented simplicial sheaf j.g'.-~j., V with surjective augmentation map. REPRESENTATION THEORY AND SHEAVES 179 Proposition IV. 4.5. -- The associated complex of sheaves ....~j,~'~j.~-~.L, V~O is exact. Proof. -- This is shown stalkwise. First let z be a point in X. By 1.1 i) and Lemma 3 i) the sequence of stalks in z is the complex of functions with finite support associated with the augmented simplicial set T; -+ T,. It is exact since the fibers of that augmen- tat-ion are contractible simplicial sets (compare [SS] p. 22). The same reasoning works for a boundary point z e X~ where we use the unipotent subgroup U, in order to analogously define T,:=U,\T and T~,:=T x ... � T(mq- l factors) T~ T z once we show that the augmented simplicial vector spaces (3", ~,), -, Co(T)~ h and Co(T*. ) --~ Co(T,) coincide. Both simplicial vector spaces are contained in Cc(T. ) so that the comparison can be done termwise. We need the subsequent two facts. Lemma IV.4.6. -- Let z e X~ be a boundary point;fir any open ndghbourhood Q of z in X we have fl T~,~ ~_ T~. wE~t~X Proof. -- Assume that (to, . .., t,,) is a tuple which is contained in the left hand side but not in the right hand side. Then there is 1 ~< j ~< m such that gt o + tj for anygeU,. Let go e G be a coset representative of to =-- go--, IP '~ 9 Choose a point y e Q n [g0(x) z). By assumption we have ht o=t~ for some h~U c') In 1.3.2 it was shown that UCel c U,.U~o~ * or equivalently U(e) iT(e) ij(e) holds true. (Observe that by [Bro] Thin. VI. 8 there is an apartment which contains the special point go x and the boundary point z in its closure.) But this implies UC,~ t c U,.t 0 which is a contradiction. [] V " 0-- 180 PETER SCHNEIDER AND ULRICH STUHLER /.emma IV.4,.7. -- Let z e X~ be a boundary point; for any tuple (to, . .., t,,) E T~ there is an open neighbourhood f~ of z in X such that (to, ..., t,,) e TV,, for any y e f~ n X. Proof. -- By G-equivariance we may assume that z ~ D~o where D is the funda- mental Weyl chamber introduced in 2.1. Let U' G G be an open subgroup such that gt~-=t t for all geU' and 0,<j,< m. By assumption there are elements u t e U, such that t t=u~t o for l~<j,< m. The subset c:= U {hu~h-l:hePonP,} X~</~<m of U, is compact. Hence we find, by 2.7, an open neighbourhood f2 o of z in A such that c __c_ U~f ) for any facet F ___ f/o n X. According to 2.5 there is now an open neighbourhood f~ of z in X of the form = U(P0 n P,).f21 where -- U __G U' is an open subgroup, -- ~1 is an open neighbourhood of z in D such that f~x n X is a union of facets, and -- ~1 _c ~o. In particular we have r U c') for anyy'ef2 lnx. Consider an arbitrary point y=ghy" with geU, heP0nP,, andy'ef~x nX in f~ n X. By construction we obtain tj = gt t = gu t t o =- gu t g-1 to ~T'r(e) h--1 g-a T'T(e) 9 = gh(h -1 u~ h) h-lg -1 t o egnur t o = for any 1 ~< j ~< m which means that (to, ..., t,.) ~ T~,.. [] Returning to the proof of Proposition 5 let first ~ be a function in (j. $',.), ~ Co(T,. ). This means that there is an appropriate open neighbourhood ~ of z in X such that ~_ Co(T~.,) for anyy ~ f~ n X. In other words the support of + is contained in ~ T*., and hence in T~. by Lemma 6. ~Etanx REPRESENTATION THEORY AND SHEAVES 181 Conversely let d? be a function in Co(T~). Then it follows from Lemma 7 that + can be viewed as a section in g',~(f2 ra X) ----j. g',,(~) for some neighbourhood of z in X. Finally in order to compare the two augmentation maps we have to check that the push-forward of functions with respect to the projection map T ~ T, induces an isomorphism Co(T)v, ~ C~(T,). The surjectivity is trivial. For the injcctivity it suffices to consider a function ~ e Co(T) whose image in Co(T,) vanishes and which is supported on a single U,-orbit in T. Wc then find a compact opcn subgroup U in U, such that ~ even is supported on a single U-orbit in T. Hence the image of~ in C~(U\T) vanishes which implies that ~/= 0 mod U. This finishes the proof of Proposition 5. [] The computation of the cohomology of a sheaf j. g;, will be based on the obser- vation that this sheaf has a natural direct sum decomposition. For each tuplc (to, ..., t,~) e T,, we introduce the subset X.0 ..... t.) := { z e X : (to, ..., t,~) r T~ } of X and we put X,o ..... t~.) : -= X(to ..... t..) n X and X"~ ..... ~.,) := ~,..o ..... '") ~ X~. Clearly ~,o ..... ")=0 if t o ..... t,. Assuming therefore that {to, ...,t,,} has cardinality at least 2 let us first collect a number of properties of the subspace X"* ..... ~,,) which will be of use later on. IV.4.8. X "~ ..... t,.) is a nonempty closed geodesically contractible CW-subspace of X. (This was already noted in the proof of II. 3.1.) IV.4.9. For any open subset ~__c X such that Y2 n X"J ..... t~)+ 0 we have f2 n X "~ ..... t,.~ 4: 0. (This follows from Lemma 6.) IV.4.10. Any boundary point z e X~\X~ ~ ..... t~,~ has an open neighbourhood -Q such that ~ n X "0 ..... t.,) = 0. in X (This follows from Lemma 7.) 11(,). for any z e X"o ..... e,) we have IV.4.11. Choose go e G such that t o ---go--~ , --~ [g~ z) ___ X"~ ..... '-'. (This is proved literally in the same way as Lemma 6.) 182 PETER SCHNEIDER AND ULRICH STUHLER IV. 4.12. We have X(to ..... tm) z X(t0 ..... tm); in particular ~(to ..... tin' is closed in X. (This is a consequence of 10. and 11.) For an arbitrary tuple we now define the sheaf C,o ..... ,=) on X to be the constant sheaf with value C on the open subset X\X It~ ..... tin) extended by zero to all of X. Lemma IV.4.13. -- We have $',, = C(to .... , ira). (to, ..., trn) E T m Proof. -- Straightforward. [] Lemma IV.4.14. -- The sheaf j, C(t ~ ..... t,,) is the constant sheaf with value C on the open subset X\X "~ ..... t,.) extended by zero to all of X. Proof. -- This is a consequence of 12 and Lemma 4. [] By comparing stalks we obtain j. g'= = 9 j. C,,o ..... ,=, (tO, ..., tm)ET m and hence , -- o H (X,3 , #',.) = 0 H (X,j. C.o ..... t.,)) ( tO,..., tm)U. T m = | H*(X, X"~ ..... '"; C). (tO,..., tm)~T m Proposition IV. 4.15. -- We have /f* -----0, n (x,j. ~-,~) = Co(T) 0 /f.> 0. Proof. -- The space X is contractible by [BS] 5.4.5. The same argument together with 8. and 11. shows that X "~ ..... t,,), for {to,..., t,.} of cardinality at least 2, is contractible as well. [] Because of Lemma 2 the resolution in Proposition 5 gives rise to the hypercoho- mology spectral sequence , H,+,,X 9 v~ I, = u,(x,j. ~,) ~ v ,a.,. v). It degenerates by Proposition 15 and exhibits H'(X,j,,~o V) as the homology of the complex ...~176 This proves Theorem I. REPRESENTATION THEORY AND SHEAVES 183 Corollary IV. 4.16. -- Let V be a representation of finite length in Alg(G); /f e is chosen large enough then we have (ker(V /f 9 = 0, H:(X, V) = ~ coker(V ~ V(X.~)) /f 9 = 1, kH'-I(X~o,V) /f* /> 2. We fix a subset @ __c A. As in the proof of 3.5 let wa, rasp. wo, denote the unique maximal (w.r.t. the Bruhat order) element in W, resp. W o. We also fix a lifting g e N of w o w~; note that (w o wA) -x is the unique maximal element in [W/W| Theorem IV.4.17. -- Assume that V = Ind(E)for some irreducible supercuspidal representation E of Mo; /f e is chosen large enough then we have , (Ind(OE) /f, =d-- $| He(X, V)~ - [ 0 otherwise. Proof. -- Corollary 16 and 3.7. [] Corollary IV.4.18. -- Let E be an irreducible supercuspidal representation in Alga(M| we then have [Ind(~ /f* =d--~| o"*(Ind(E)) 0 otherwise. Proof. -- Theorem 17 and 1.4. [] Corollary IV.4.19. -- Let V be a representation in AlgSx~,o(G) ; /re is chosen large enough then we have HI(X, V) ---- 0 for 9 oe d -- ~| Proof. -- This follows from Theorem 17 by an induction argument as in the proof ofIII. 3.1. (Because of 1.4 the present assertion and III. 3.1 i) actually are equivalent.) [] IV. 5. The Zelevinsky involution We fix a central character )~ and let Rz(G; )~) be the Grothendieck group of representations of finite length in Algx(G ) (w.r.t. exact sequences); the class in Rz(G; ?() of a representation V is denoted by [V]. It follows from III. 3.1 that ~: Rz(G; Z) -~ Rz(G; Z) [V] ~ Y,, (-- 1)'. [~(~)] is a well-defined homomorphism such that t([V]) = (-- 1)a-lt~ for any V in Algexl, o0(G). 184 PETER SCHNEIDER AND ULRICH STUHLER Proposition IV. 5.1. -- The homomorphism ~ respects up to sign the classes of irreducible representations. Proof. -- This is III. 3.2. [] Consider a representation V in Alg~x, o0(G). It is a consequence of 3.1 that there is an augmented complex Ind~o (Vvo) =V O~A ~O=d @ Ind~ (V) ~ ... , @ Ind~ (V) 0 _~ A 0 UO 0 ~ A 0 UO ~o=~-a ~o~o which computes the cohomology H*(X~, V). By combining 1.4 and 4.16 we see that the only nonvanishing homology group of that complex is ~(~); it sits in degree d -- 1 -- $00 if the complex is put in degree -- 1 up to d -- 1. Since the formation of Jacquet modules as well as the parabolic induction respect representations of finite length ([Ber] 3.1) each term in the above complex has a well-defined class in Rz(G; Z). Proposition IV.5.2. -- For any representation V of finite length in Alggxl(G) we have t([V]) = E (-- 1) a-~~176 O_=A Proof. -- Obvious from the preceding discussion, rq For any O c A let P_ o be the parabolic subgroup of G which contains M o and is opposite to Po; then P_ o n Po = Mo. Let U_ o denote the unipotent radical of P o. The modulus character of P-o is pr P-o ,~ Mo 8~ R~ (leas] 1.6). Lemma lWV.fi.3. -- Let E be a representation of finite length in Algx(Mo) for some | ~_ A; we then have [Ind~ = [Ind(~o u~ | E)]. Proof. -- If g ~ N lifts wzx then g- i p_ o g = P~ ~e o. We obtain Ind~ ) _~ Indpwaw e o(~ = Ind(8~ we o | ~ = Ind(a(8o v' | E)). Because of g-1 Mo g = M~ A ~o o we may apply III. 2.1 and we see that the latter representation has the same irreducible constituents as Ind(8ol~| E). [] REPRESENTATION THEORY AND SHEAVES Proposition IV.5.4. -- For any representation V in AlgSxl, o0(G) we have [~(~)] = [d(V)~]. Proof. -- Dualizing the discussion preceding Proposition 2 we obtain [e(V)~] = E (-I)*~176176 ] = E (-- 1)~~176174 t~_cA where the second equality even holds termwise by [Ca.s] 4.2.5. On the other hand Proposition 2 holds true, of course, for any choice of simple roots, e.g. --A. Hence we have &,o E (-- 1) ~~176176 [Ind~_o(Vu_o)]. O_cA Apply now Lemma 3. [] Corollary IV.5.5. -- The map ~ is an involution, i.e. to t= id. Proof. -- Combine Proposition 4 and III. I. 5. [] The Proposition 2 shows that (-- 1) a. t coincides with the involution D o studied in [Aul] 5.24. It therefore follows from [Aul] 5.36 that in case G = GLa+I(K ) the Zelevinsky involution i considered in [Zel] 9.16 is equal to -- t. Hence Proposition 1 proves the Duality Conjecture 9.17 in [Zel]. It also follows that the orthogonality property discussed in [Aul] 5.D holds true. Assume K to have characteristic 0 and the center of G to be compact. Then the above results hold without having specified a central character. In III. 4 after Thin. 21 we had seen that the Euler-Poincar~ characteristic EP(., .) induces a nondegenerate symmetric bilinear form on the quotient R(G) = R(G)/R~(G) of the Grothendieck group R(G)-Rz(G)| It follows from Proposition 2 or from III.4.3 i) that EP(t., .) = EP(., .). Hence the involution t respects the subgroup RI(G ) and induces the identity on the quotient i~(G) or, equivalently, (id -- ~) (R(G)) _ Rz(G ). V. THE FUNCTOR FROM EQUIVARIANT COEFFICIENT SYSTEMS TO REPRESENTATIONS In this final Chapter we want to examine more closely the relation between representations and coefficient systems. The category Coeff(X) of coefficient systems (of complex vector spaces) on X was introduced in II.2. We say the group G acts on the coefficient system (VF) v if, for any g E G and any facet F_ X, there is given a linear map gv : V F ~ Vg~ 24 186 PETER SCHNEIDER AND ULRICH STUHLER in such a way that -- ghr o hF = (gh)F for any g, h 9 G and any F, -- le = idv~ for any F, and -- the diagram gp V F > Vg F Vr o,,> Vo r is commutative for any g 9 G and any pair of facets F' _~ F. In particular the stabilizer PtF, for any facet F, acts linearly on V F. Definition. -- An equivariant coefficient system on X is a coefficient system (V~) F on X together with a G-action on it which has the property that, for any facet F, the stabilizer P~ acts on V F through a discrete quotient. Let Coeffo(X ) denote the category of all equivariant coefficient systems on X. This is an abelian category. Fix an object r = (VF) F in Coeffo(X ). By definition, for any 0 ~< q ~< d the space of oriented q-chains of r is ". C~j(X~,), r := C-vector space of all maps to : X(,) --+ U V F such that FEXq to has finite support, -- to((F, c)) 9 VF, and, if q >/ 1, to((F, -- c)) = -- r c)) for any (F, c) 9 X(,,. The group G acts smoothly on these spaces via (gto) ((F, c)) := gc_~r(to((g-~ F, g-X c))). The boundary map o~ ~) -, ~r(X,q, ~) 0 " ~'Jc k (~z+l), r~,(to((F, c)))) to ((F', c') Z (F, e)~ X(q+x ) F'~ O~,(c) : c' is G-equivariant. Hence we obtain the chain complex cor(x, ), & ... & co,(X,o,, REPRESENTATION THEORY AND SHEAVES 187 in AIg(G). Its homology is denoted by H.(X, "//). It is not difficult to see that the above complex as well as its homology actually lies in the full subcategory AlgC(G) :== category of those smooth G-representations V which are generated by V ~ for some open subgroup U ~ G. As a consequence of Bernstein's theorem (I.3) the category AlgC(G) is stable with respect to the formation of G-equivariant subquotients; moreover it is closed under extensions. In the following only the right exact functor Ho(X , .): CoeffQ(X) ~ Alg"(G) will be of importance for us. Let E be the class of morphisms s in Coeffa(X ) such that H0(X , s) is an isomorphism. We then have a unique commutative diagram of functors ~(X,. ) Coeff~, (X) , AlgC(G) where Q is the canonical functor into the category of fractions with respect to Z. Theorem V. 1. -- The functor p : Coeffo(X)[Y~-' ] -~ Alga(G) is an equivalence of categories. Proof. -- The following properties are a consequence of the right exactness of Ho(X , .) ([GZ] 1.3): 1. 2; admits a calculus of left fractions. 2. Coeffo(X) [Z-~] is additive and has finite direct limits. 3. The functors O and p are additive and respect finite direct limits. 4. The functor p detects isomorphisms. The latter two properties imply: 5. The functor ~ is faithful. Namely, let a and b be two morphisms in Coeffo,(X ) [~-~] such that p(a) ---- p(b). Using 3. we have that ~(coker(a -- b)) = coker(p(a -- b)) ---- coker(p(a) -- p(b)) = coker(0) is an isomorphism. By 4. then coker(a -- b) is an isomorphism, too; hence a = b. Fixing a special vertex x in A we have Alga(G) :: O Alg~'~(G). e~>0 188 PETER SCHNEIDER AND ULRICH STUHLER In II. 2 we have constructed, for any e >1 0, an exact functor (e) y, : Alg~x (G) ~ Coeffo(X) ; moreover there is an obvious natural transformation Ye ~T,+I I AlgU~')(G) 9 It follows from II. 3.1 that the latter induces a natural isomorphism in homology H,(X, V,(-)) -~ H,(X, y,+,(.)) on AlgV))(G). After composing with the functor Q the above natural transformation therefore becomes a natural isomorphism = uic) Qo~tr QO~e+l IAlg , (G). Hence we obtain in the direct limit the functor 3' := lira Qo y, : Alg"(G) -+ CoeffQ(X) [Z-a]. Again by II.3.1 we have 6. p o 3' _-_ id~cla ~. It is an immediate consequence of 5. and 6. that P and ~, are quasi-inverse to each other. [3 Because of their practical importance let us state separately the following facts which were established in the course of the previous proof. Lemma V . 2. i) Z admits a calculus of left fractions; ii) thefunctor ~, : AlgC(G) -+ Coeffc(X ) [Z -a] is quasi-inverse to O. INDEX OF NOTATIONS a nonarchimedean locally compact field o the ring of integers in K a fixed prime element in o r the discrete valuation of K normalized by c0(n) = I the residue class field of o the base change to K of some object X over o G a connected reductive group over K the group of K-rational points of G the center of G a maximal K-split torus in G W the Weyl group of G the set of roots of G ~:)red the set of reduced roots of G q)+, ~)- the set of positive resp. negative roots in REPRESENTATION THEORY AND SHEAVES 189 A the set of simple roots in ~+ U~ the root subgroup corresponding to the root a subset of the set A of simple roots <| the subset { c~ E 9 : cc is a integral linear combination of roots from | } the connected component of f] ker(~) So ctEo M| the centralizer of S o in G, i.e. the Levi subgroup corresponding to O U| the unipotent subgroup of G generated by all root subgroups U~, for ~ E ~+\( | ) P| = M0 U0 the parabolic subgroup of type | with respect to the choices S, ~+ the modulus character of the parabolic subgroup P| 8o X the Bruhat-Tits building corresponding to G d the distance function on the metric space X X(q~(Xr the set of oriented (nonoriented) q-dimensional polysimplices of X X(q) the q-skeleton of X A a fixed (basic) apartment of X, a d-dimensional affine space D a fundamental Weyl chamber in the apartment A F a polysimplex of X St(F) the star of the facet F the Borel-Serre compactification of X x~ the boundary of X in Pfl the pointwise stabilizer in G of a subset fl _ X the stabilizer in G of a subset fl c X uT for any integer e >I 0, the e-th filtration subgroup of P~ subgroup of U~ of t-value >/ r Uot, r T = C,/U~ ) for a special vertex x ~ A, the basic homogeneous G-set the subset of all regular elliptic elements of G Gell the set of conjugacy classes of regular elliptic elements of G Cell V a smooth representation of G Alg(O) the category of smooth G-modules AlgU(G) for U a compact open subgroup, the subcategory of Alg(G) of tho~e G-modules which are gene- rated by the subspaee of their U-fixed vectors V ~ for Z a character on the connected center C of G the full subcategory of Alg(G) of those Algx(G) G-modules on which C acts by the character X the full subcategory of Algx(G ) of those G-modules which are of finite length and whose irre- Alga, o (G) ducible subquotients are all of type | R(G) the Grothendieck group of representations of finite length in Alg(G) tensorized by r the (urmormalized) induction to G of a smooth Po-module e Ind~o(e) lad(e) the normalized induction to G of a smooth Po-module e for a character X of the connected center C, the Hecke algebra of locally constant functions on G, ~x compactly supported modulo C and transforming with respect to the action of C by the character Z Ee(V, V') the euler-Poincard characteristic of the smooth G-modules V, V', V finite length, V' admissible d v g the euler-Poincard measure for V volv the volume corresponding to dVg ca - ~o the involution functor on Alg~x ' o(G) Coeff(X) the category of coefficient systems on X y~(V) the coefficient system associated to a smooth G-module V for a fixed integer e >/ 0 ye the corresponding flmctor from Alg(G) to Coeff(X) ~r(x(.),-r,(v)) the oriented chain complex associated to a coefficient system y,(V) v the sheaf on the Bruhat-Tits building associated to the smooth representation V the smooth extension of V to the compactification Z,~ j,,=o v V the sheaf on the boundary Xco of X in "X corresponding to the smooth representation V 190 PETER SCHNEIDER AND ULRICH STUHLER REFERENCES [Aul] A.-M. 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K~TO, Duality for representations of a Hecke algebra, Pror AMS 119 (1993), 941-946. HA~RIsH-CHANDRA, REPRESENTATION THEORY AND SHEAVES 191 [Kal] D. KAZHDAN, Cuspidal geometry of p-adic groups, J. d'analyse math. 47 (1986), 1-36. [Ka2] D. K~ZnDAN, Representations of groups over close local fields, J. d'analyse math. 47 (1986), 175-179. R. KOTTWrrz, Tamagawa numbers, Ann. Math. 12e'/ (1988), 629-646. [Kot] [MP] A. MoY, G. PR.aSAD, Unrefined minimal K-types for p-adic groups, Invent. math. 116 (1994), 393-408. G. PRASAD, M. S. RAOaUNAT~a~, Topological central extensions of semi-simple groups over local fields, [PR] Ann. Math. 119 (1984), 143-201. [Pro] K. Plto~rzR, The Zelevinshy Duality Conjecture for GLN, Thesis, King's College, London, 1994. [Rog] J. ROGAWSXL An application of the building to orbital integrals, Compositio Math. 42 (1981), 417-423. [ss] P. SCHNEXDZR, U. STUHr~R, Resolutions for smooth representations of the general linear group over a local field, J. reine angew. Math. 4.86 (1993), 19-32. [Ser] J.-P. SERR~, Cohomologie des groupes discrets, in Prospects in Mathematics, Ann. Math. Studies 70, 77-169, Princeton Univ. Press, 1971. A. SILBEROmt, Introduction to harmonic analysis on reduetive p-adic groups, Princeton Univ. Press, 1979. [Sill [Tit] J. Trrs, Reductive groups over local fields, in Automorphic Forms, Representations and L-Futvtions, Pror Symp. Pure Math. 38 (1), 29-69, American Math. Soc., 1979. M.-F. VXQNZRAS, On formal dimensions for reductive p-adlc groups, in Festschrlft in honor ofL L Piatetshi- [Vig] Shapiro (Eds. Gelbart, Howe, Samak), Part I, 225-266, Israel Math. Conf. Proc. 9., Jerusalem, Weizmann Science Press, 1990. A. ZZLEVINSXY, Induced representations of reductive p-adic groups II, Ann. Sci. ENS 13 (1980), 165-210. CZel] P. Schneider Mathematisches Institut der Universit~it Miinster Einsteinstr. 62 48149 Miinster U. Stuhler Mathematisches Institut der Universit~it G6ttingen Bunsenstr. 3-5 37073 G6tfingen Manuscrit re~u le 28 juin 1995. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals http://www.deepdyve.com/lp/springer-journals/representation-theory-and-sheaves-on-the-bruhat-tits-building-cJF0E3c1Gf

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Publisher: Springer Journals
Copyright: Copyright © 1997 by Publications Mathématiques de L’I.H.É.S.
Subject: Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN: 0073-8301
eISSN: 1618-1913
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Abstract

REPRESENTATION THEORY AND SHEAVES ON THE BRUHAT-TITS BUILDING by PETER SCHNEIDER and ULvacri STUHLER The Bruhat-Tits building X of a connected reductive group G over a nonarchi- medean local field K is a rather intriguing G-space. It displays in a geometric way the inner structure of the locally compact group G like the classification of maximal compact subgroups or the theory of the Iwahori subgroup. One might consider X not quite as a full analogue of a real symmetric space but as a kind of skeleton of such an analogue. As such it immediately turned out to be an important technical device in the smooth representation theory of the group G. As a reminder let us mention that the irreducible smooth representations of G lie at the core of the local Langlands program which aims at understanding the absolute Galois group of the local field K. In this paper we develop a systematic and conceptional theory which allows to pass in a functorial way from smooth representations of G to equivariant objects on X. There actually will be two such constructions--a homological and a cohomological one. Since the building carries a natural C W-structure the notion of a coefficient system (or cosheaf) on X makes sense. In the homological theory we will construct functors from smooth representations to G-equivariant coefficient systems on X. It should be stressed that the definition of the coefficient system only involves the original G-repre- sentation as far as the action of certain compact open subgroups of G is concerned. One therefore might consider the whole construction as a kind of localization process. Our main result will be that the cellular chain complex naturally associated with a coefficient system provides (under mild assumptions) a functorial projective resolution of the G-representation we started with. In the cohomological theory we will associate, again functorially, G-equivariant sheaves on X with smooth G-representations. The main task which we will achieve then is the computation of the cohomology with compact support of the sheaves coming from an irreducible smooth G-representation. The result can best be formulated in terms of a certain duality functor on the category of finite length smooth G-represen- tations. As a major application we will prove Zelevinsky's conjecture in [Zel] that his duality map on the level of Grothendieck groups preserves irreducibility. For carrying 13 98 PETER SCHNEIDER AND ULRICH STUHLER out our computation we have to extend the sheaves under consideration in such a way to the Borel-Serre compactification X of X that the cohomology at the boundary becomes computable. Since the stabilizers of boundary points are parabolic subgroups it might not surprise that this can be achieved by using the Jacquet modules of the representation as the stalks at the boundary points. The cohomology at the boundary then is computed by adapting a strategy of Deligne and Lusztig ([DL]) for reductive groups over finite fields to our purposes. Apart from the theory of buildings we will very much rely on the beautiful results of Bernstein on the category of smooth representations in [Ber]. All the known homo- logical finiteness properties of this category follow already from his work. The point of our paper is rather that we construct nice projective resolutions in that category in a functorial as well as explicit manner. Exactly this explicitness enables us to apply our theory to the harmonic analysis of the group G. It turns out that Kottwitz' Euler- Poincard function in [Kot] is a special case of a general theory of Euler-Poincar6 functions for finite length smooth G-representations. Besides being pseudo-coefficients their main property is that their elliptic orbital integrals coincide with the Harish-Chandra character of the given representation. This leads to a Hopf-Lefschetz type trace formula for the Harish-Chandra character at an elliptic element. Combined with powerful results of Kazhdan in [Kal] it also leads to a proof of the general orthogonality formula for Harish- Chandra characters as conjectured by Kazhdan. Let us now describe the contents of the paper in some more detail. The first chapter contains most of the input which we need from the theory of buildings. The ceils of the natural C N-structure of X actually are polysimplices and are called facets. For any such facet F of X let PF denote its pointwise stabilizer in G. The technical heart of our theory is the construction of certain decreasing filtrations PF -~ U~ ) ~ 9 9 9 ~- U~, ~ 2 -- - of PF by compact open subgroups U~ '1. This is done in I. 2 where also the more basic properties of these filtrations are established. Since we work with an arbitrary connected reductive group G that construction involves more or less all of the finer aspects of the theory developed in the volumes [BT]. This unfortunately makes numerous references to [-BT] unavoidable so that any reader without an expert knowledge of the work [-BT] might find this section hard to read. We apologize for that. In order to make it a little easier we give in I. 1 a brief overview over the theory of buildings for reductive groups. In the section I. 3 we give those properties of the groups U~ '1 which later on are needed for the computation of the (co)homology. Notably we study how the groups U~ "~ behave if the facet F is moved along a geodesic in the building X. In case G is absolutely quasi- simple and simply connected similar filtrations appear in [PR] and [MP]. The second chapter contains the homological theory. In II. 1 we briefly recall the formalism of cellular chains. The section II. 2 contains the definition of the functor y, from smooth G-representations to equivariant coefficient systems on X. Here e/> 0 is a fixed " level ". The coefficient system y,(V) corresponding to a representation V is formed by associating with a facet F the subspace of U~'~-invariant vectors in V. In 99 REPRESENTATION THEORY AND SHEAVES addition properties of finite generation and projectivity of the chain complex of u are discussed. The main result is shown in II.3. It says that at least for any finitely generated smooth G-representation V we can choose the level e large enough so that the chain complex of y,(V) is an exact resolution of V in the category Alg(G) of all smooth G-representations. In the case that G is the general linear group we proved this already in [SS]. The strategy of the proof for arbitrary G is the same once the necessary properties of the groups U~ ~ are known. In the third chapter we develop that part of the duality theory which uses the chain complex of y,(V). Since the polysimplicial structure of the building X is locally finite we actually can associate with the coefficient system y,(V) also a complex of cochains with finite support. Let us fix a character X of the connected center of G, let Algx(G ) denote the category of all those smooth G-representations on which the connected center acts through X, and let ovg x denote the ?(-Hecke algebra of G. If V is an admissible representation in Algx(G ) then the functor HomQ(., 9fix) transforms the chain complex of y,(V) into the cochain complex of y,(V) where V is the smooth dual of V. Assume now that V even is of finite length and choose e large enough. Then we know that the chain complex of y,(V) is a projective resolution of V in Alg� It follows that the cochain complex of y,('~) computes the Ext-groups o"*(V) := Ext~lgx~Q~(V , Yt~ All this is shown in III. 1. Later on in IV. 1 wc will see that the same cochain complex computes the cohomology with compact support of a certain sheaf on X associated with the representation V. This fact will enable us in chapter IV to compute the groups d~'(V) in the case that V is a representation which is parabolically induced from an irreducible supercuspidal representation of a Levi subgroup. In III. 2 we briefly recall the theory of parabolic induction following [Gas]. Then in III.3 taking the computation of $r for induced V for granted we deduce the following result for an arbitrary irreducible smooth representation V: The groups g"(V) vanish except in a single degree d(V), 8(V):----- ~a~v~(v) again is an irreducible smooth representation, and moreover g'($*(V))= V. Standard techniques of homological algebra now allow to establish a general duality formalism which relates the Ext- and Tor-functors on the category Algx(G ). Loosely speaking one might say that o~r x is a " Gorenstein ring " Since our applications to harmonic analysis all come from the exactness of the chain complex of ye(V) we include them here as the section III.4 before we turn to the sheaf theory on X. For reasons of convenience we assume that the center of G is compact. Since in the paper [Kal] the field K is assumed to be of characteristic 0 we have to make the same assumption in most of our results of this section. We obtain: A general notion of Euler-Poinear6 functions, a formula for the formal degree, the existence of explicit pseudo-coefficients, the Harish-Ghandra character on the elliptic set as an explicit orbital integral, the general orthogonality relation for Harish-Chandra characters, and the 0-th Chern character on the Grothendieck group of finite length representations. 100 PETER SCHNEIDER AND ULRICH STUHLER In the fourth chapter we present the sheaf theory on X. In IV. 1 we functorially associate a sheaf V on X with any representation V in Alg(G). Of course this construction again depends on the choice of a level e >i 0 which is fixed once and for all and which, for simplicity, is dropped from the notation. The sheaf V is constant on each facet F having the U~e'-coinvariants of V as stalks. As promised earlier we show that the coho- mology with compact support of V is computable from the complex of cochains with tq~ finite support of the coefficient system %(V). We also rewrite our earlier formula for the Harish-Chandra character of a finite length representation V at an elliptic element h ~ G as a Hopf-Lefschetz trace formula: The character value at h is equal to the trace (in the sense of linear algebra) of h on the cohomology of the sheaf V restricted to the fixed point set X h. In IV. 2 we construct a " smooth" extension of V to a sheaf j., ~ V on the Borel-Serre compactification X of X. This extension is in some sense intermediate between the extension by zero j: V and the full direct image j. V. It requires a rather detailed and technical investigation of the geometry of X. Let X~o := X\X be the boundary. In IV.3 we compute the cohomology of j.,~ V restricted to X~ in the case where the representation V is parabolically induced from a supercuspidal representation. Then in IV. 4 we show that, for any finitely generated V and any e large enough, the sheaf j., ~ V in fact has no higher cohomology. The combination of these two results immediately leads to the computation of the cohomology with compact support of the original sheaf V provided V is parabolically induced as above. This is the fact which we had taken for granted in chapter III. So the duality theory, i.e. the investigation of the Ext-groups o~(V), now is complete. As an application we prove in IV. 5 Zele- vinsky's conjecture. At this point we want to mention that Bernstein has a completely different proof (unpublished) of this conjecture along with the fact that the Zelevinsky involution comes from the functor d'*(. ). The last chapter complements the discussion of coefficient systems. We show that a rather big subcategory of AIg(G) is a localization of the category of equivariant coefficient systems on X. As will be explained in a forthcoming paper of the first author the latter objects constitute something which one might call perverse sheaves on the building X. From this point of view our constructions bear a certain resemblance to the Beilinson-Bernstein localization theory from Lie algebra representations to perverse sheaves on the flag manifold. During this work we have profited from conversations with M. Harris, G. Henniart, M. Rapoport, M. Tadic, J. Tits, and M.-F. Vigneras for which we are grateful. We especially want to thank E. Landvogt whose expert knowledge of the building has helped us a lot. The support which we have received at various stages from the MSRI at Berkeley, the Newton Institute, the Tata Institute, and the Universit~ Paris 7 is gratefully acknowledged. REPRESENTATION THEORY AND SHEAVES 101 Added in proof. -- As the referee has pointed out, special cases of the Zelevinsky conjecture are treated in the papers [Kat] and [Pro]. Their methods are completely different from ours. Also in the meantime Aubert has given in [Au2] a proof of the Zelevinsky conjecture in the general case by studying on the Grothendieck group a certain involution wtfich is defined in terms of parabolic induction (compare our IV. 5.2). CONTENTS I. The groups U~ I ........................................................................ 102 I. 1. Review of the Bruhat-Tits building ................................................ 102 1.2. Definition of the groups U~ ) ...................................................... 105 1.3. Properties of the groups U~ I ...................................................... 117 II. The homologicai theory ................................................................. 119 II.1. Cellular chains ................................................................... 119 II.2. Representations as coefficient systems ............................................... 121 II.3. Homological resolutions ............................................................ 123 III. Duality theory ......................................................................... 126 III.l. Cellular cochalns ................................................................. 126 III.2. Parabolic induction ............................................................... 130 III.3. The involution ................................................................... 132 III.4. Euler-Poincar~ functions ........................................................... 135 IV. Representations as sheav~ on the Borel-Serre compactification ................................ 152 IV. 1. Representations as sheaves on the Bruhat-Tits building ............................... 152 IV.2. Extension to the boundary ......................................................... 155 Appendix: Geodesics in .X ........................................................ 166 IV.3. Cohomology on the boundary ..................................................... 167 IV.4. Cohomology with compact support ................................................. 173 IV.5. The Zelevinsk'y involution ......................................................... 183 V. The functor from equivariant coefficient systems to representations ............................. 185 LIST ov NOTATIONS ........................................................................... 188 REFE RENC-.ES ................................................................................. 190 NOTATIONS K a nonarchimedean locally compact field, o the ring of integers in K, a fixed prime element in o, ca : K � ~ Z the discrete valuation normalized by co(n) = 1, := o]~o the residue clas~ field of o, for any object X over o for which it makes sense to speak about its base change to K we denote this base change by X, G a connected reductive group over K, G := G(K). 102 PETER SCHNEIDER AND ULRICH STUHLER I. THE GROUPS U[g I 1.1. Review of the Bruhat-Tits building The (semisimple) Bruhat-Tits building X of the group G is the central object in this paper. Since the monumental treatise [BT] is not so easily accessible for the nonexpert we believe it to be necessary to briefly review the construction and basic properties of X. Most of the notations to be introduced for tiffs purpose will be needed later on anyway. We fix a maximal K-split torus S in G. (Strictly speaking S is the group of K-rational points of that torus. This kind of abuse of language will usually be made.) Let X*(S), resp. X.(S), denote the group of algebraic characters, resp. cocharacters, of S. Similarly let X.(C) denote the group of K-algebraic cocharactcrs of the connected center C of G. The real vector space A := (X,(S)/X.(C)) | R is called the basic apartment. Let Z, resp. N, be the centralizer, resp. normalizer, of S in G. The Weyl group W := N/Z acts by conjugation on S; this induces a faithful linear action of W on A. On the other hand let (,): X,(S) � X*(S) -~ Z be the obvious pairing; its R-linear extension also is denoted by (,). There is a unique homomorphism : Z --~ X.(S) | R such that z l s ) = - o(z(g)) for any g e Z and any K-algebraic character ) of Z. We let g e Z act on A by the translation gx:----x+image ofv(g) in A for xeA. The first important observation in this theory is that this translation action of Z on A can be extended to an action of N on A by affine automorphisms ([Tit] 1.2). We fix one such extension and simply denote it by x ~-, nx for n E N and x e A. One has: -- All possible other such extensions are given by x ~-, n(x + xo) --xo where x0 e A is a fixed but arbitrary point. -- IfweWis the image ofneN then (x ~ wx) = linear part of (x ~-, nx). In order to equip A with an additional structure we need the set 9 _ X*(S) of roots of G with respect to S. Any root e obviously induces a linear form e : A ~ R. Also REPRESENTATION THEORY AND SHEAVES 103 corresponding to any a ~ ~ we have the coroot ~ ~ A and the involution s, E W whose action on A is given by s,x=x--~(x).~ for xeA. For us the most important object associated to an ~ e 9 is its root subgroup U, ~_ G ([Bor] 21.9 where the notation Uc, ~ is used); in particular it is a unipotent subgroup normalized by Z. Let Om := {, e 9 : ~/2 r (I) } be the subset of reduced roots. Crucial is the following fact ([BoT] w 5): For 0c eel) ~a and each u e U,\{ 1 } the intersection U_, uU_ ~ n N = { m(u) } consists of a single element called m(u); moreover the image of m(u) in W is s~. A central assertion in the Bruhat-Tits theory now is the fact that the translation part of the affine automorphism of A corresponding to m(u) is given by -- g(u). ~ for some real number t(u), i.e. we have m(u) x = s~x --t(u).~ = x -- (~(x) +t(u)).~ for any xcA. We may view m(u) as the " reflection " at the affine hyperplane { x e A : ,(x) = -- t(u) } ([Tit] 1.4). Put F~ := {t(u) : u e U~\{ 1 } } _ R; this is a discrete subset in R unbounded in both directions and --F, = F_~ ([BT] 1.6.2.16). The affine functions ~(.)+g on A for ~ e @,,a and g e P, are called affine roots. Two points x and y in A are called equivalent if each affine root is either positive or zero or negative at both points; the corresponding equivalence classes are called facets. This imposes an additional geometric structure on the apartment A which is respected by the action of N. Parallel to this structure the root subgroup U~ for ~ e (I) "~ possesses the filtration U,.,:={ueU,k{1}:t(u)>~r}u{1} for reR. This is an exhaustive and separated discrete filtration of U, by subgroups ([BT] 1.6.2.12b)); put U,.~o := { 1 }. For any nonempty subset fl c-_ A we define fn:*~R u{oo} e~-- inf e(x) and U n : = subgroup of G generated by all U,. tn~ for e e ~,,a. This group has various important properties ([BT] 1.6.2.10, 6.4.9, and 7.1.3): 1. nUan -1 = U,n for any n eN; in particular Nn:={n eN:nx = x for any x etl} normalizes Un. 2. UnnN ___N n. 3. U n n U, ----- U~. ~'ac~l for any 0c e ~. 4. Let q) = q~+u q)- be any decomposition into positive and negative roots and put U e := subgroup of G generated by all U~ for c~ e ~+ n q)~a; then Un = (Ua n U-) (Ua n U +) (Un n N); 104 PETER SCHNEIDER AND ULRICH STUHLER moreover the product map induces bijections II U~, Ial~J -% Ua r~ U whatever ordering of the factors on the left hand side we choose. Define Pa :----- Na. Ua which contains Un as a normal subgroup by 1. By 2, we have Pan N = N n. (Warning: In [BT] our groups N a and Pa are denoted by iq a and Pa and our symbols have a different meaning.) In case ~ = { x } we write f,, U,, N,, and P, instead off~,~, ... We now are ready to define the Bruhat-Tits building X. Consider the relation on G x A defined by (g, x) --~ (h,y) if there is a n ~ N such that nx = y and g- 1 hn E U.; it is easily checked that tiffs is an equivalence relation. We put X:=G xA/,-~. It is straightforward to see that G acts on X via g.class of (h,y) := class of (gh,y) for g~G and (h,y) ~G x A and that the map A~X x ~ class of (1, x) is injective and N-equivariant. Viewing the latter map as an inclusion we can write gx for the class of (g, x). A first basic fact ([BT] I. 7.4.4) is that, for f~ _~ A nonempty, Pa={geG:gx-=x for any x~f~} holds true. The relation between the facet structure of A and the subgroup filtration in U~, for ~ e ~a is given by the fact that for u e U~\{ 1 } we have {xcA:ux=x}={x~A:~(x) +g(u)>/ 0} ([BT] 17.4.5). The subsets of X of the form gA with g ~ G are called apartments. A very important technical property of the G-action on X is the following: 5. For any g e G there exists a n EN such that gx = nx for any x ~A ng-lA ([BT] 1.7.4.8). REPRESENTATION THEORY AND SHEAVES 105 For example it implies that the partition into facets can be extended from A to all of X in the following way: A subset F' c X is called a facet if it is of the form F' = gF for some g 9 G and some facet F _c A. It also implies: 6. For any nonempty ~ ___ A the group Uo acts transitively on the set of all apartments which contain fL From the Bruhat decomposition G=-U, NU v for x,y 9 one concludes: 7. Any two points and even any two facets in X are contained in a common apartment ([BT] 1.7.4.18). For any nonempty subset ~2 _c X we define Pn:={geG:gz=z for any z 9 and P~:={g 9 and we abbreviate P, := P{,) = P~,} for any z 9 X. Finally we fix once and for all a W-invariant euclidean metric d on A. The action of N on A then automatically is isometric. As a simple consequence of 5-7, this metric extends in a unique G-invariant way to a metric d on all of X. The metric space (X, d) together with its isometric G-action and its partition into facets is called the Bruhat-Tits building of G. Further properties of this very rich structure will be recalled when they are needed. 1.2. DeAnition of the groups U~ ~ For any facet F in A let G ~ be the smooth affine o-group scheme with general fiber G constructed in [BT] II.5.1.30. By [BT] II.5.2.4 the group G~ is the subgroup of G generated by U F and .~o'~ where ~0 is the connected component of the " canonical " extension ~ of Z to a smooth affine o-group scheme ([BT] II. 5.2.1). Put H:={neN:nx=x for all xeA} and H 1 := { n 9 H : co(z(n)) = 0 for any K-algebraic character )( of G }. According to [BT] II.5.2.1 we have .~(0) = H 1. Therefore L~~ is of finite index in H 1. Since H ~ N F we see that u, G~ = PF. 14 106 PETER SCHNEIDER AND ULRICH STUHLER It follows from [BT] I1.4.6.17 that any inner automorphism g ~-~ ngn -1 of G with n e N extends to an isomorphism of o-group schemes G0 0 The closed fiber ~,o is a connected smooth algebraic group over I(; let R~ denote its unipotent radical. Put R F := { g E t3~ : (g mod ~) e RI,(K ) }; this is a compact open subgroup of G. Because of nR~ n-1 = R, F for n e N, it is nor- realized by Nt~:={neN:nF=F}. The property 1.5 implies that P; = N; P, = N; U.. Hence R F is a normal subgroup ofPtF. In the sequel we will construct a specific decreasing filtration 13~ ~ R, -----: U~'_~ U~'_~ ... by subgroups U~ *~ which are normal in Pt F and compact open in G. For doing this we need the concept of a concave function in [BT] 1.6.4.1-5. First we have to introduce the totally ordered commutative monoid R:=Ru{r+:r~R}u{oo}. Its total order is given by the usual total order on R and by r~< r+~<s~< co ifr<s; its monoid structure extends the addition on R and is given by r+ (s+) = (r+) + (s+) =- (r+s) + and r+co= (r+) +oo =co+co=co. We put 89 (89 + and 89 := oo. A function f: q) --~R is called concave if for any a,~,a+~q~, f(a) +f(~) >--f(a + ~) for any a e q~ and f(a) +f(-- a) >/ 0 hold. For a E ~i and r ~ R we define U,,,+ := O U~,,. s~R,.>r REPRESENTATION THEORY AND SHEAVES 107 Then, for any concave function f, the group U t :-----subgroup of G generated by all U,. tl-I for a e ~t and all U2~ n U~. tI~2~ for 0c, 2~ e has properties completely analogous to 1.1-1.4 ([BT] I. 6.4.9). Observe that Un = Urn. Starting from the concave function fF, for a facet F in A, we define a new function f;:r -+R by /f~(~)+ if ~IF is constant, f~(~) := ~ fF(~) otherwise; it is concave, too, by [BT] 1.6.4.23. In case ~, 2~ er we have f;(2~) -~ 2f~(~) so that UI; is the subgroup generated by all U~. ~,~1 for a e (I) '~i. Lemm I. 2.1. -- We have R~, n U~, I,~ = U~. g~j for any ~ e (I) ~. Proof. -- We have to introduce further notations. In case 2~ e ~) put := { 2t(u) : 1 } } (recall that U~, _~ U~) and F'~ := { g(u) ~ F, : u 9 U,\U2~ and g(u) = sup/(uU2, ) }; one has P~ ----- P', u 89 ([BT] 1.6.2.2) and F', 4 ~ ~ ([BT] II 4 2.21). In case 2~ r put F'~ := F~. The " optimization " gr : ~ ~ R of the function ft, is de ined by g,(~) := inf{ l e F; : t >t fF(~) }. Moreover we put (g,(~)+ ifgF(~) +gF(--~) =0, := { [ gF(~) otherwise. (The functions g~ and g~, in general are no longer concave but only quasi-concave in the sense of [BT].) In [BT] II.4.6.10 (compare in particular the third paragraph on p. 321) and 5.1.31 it is proved that U~.~ if 2~ r ~, R r c~ U~, tr = (U~ c~ holds true. Let us first consider the case 2a r (I). If g,(~) + g~(- ~) = 0 then clearly alsof~(a) +fl,(-- a) = 0, i.e. a IF is constant and gF(a) ----f~(a); we obtain U,.a~ =- U,.gr = U,.,,,). 108 PETER SCHNEIDER AND ULRICH STUHLER Assume now that gF(e) + gF(-- ~) ~e 0. If ~ [ F is not constant then by the definitions we have U~,.;~) = U~, ~,(~) = U~, ~.(~ = U~, ~(~1; the same holds if ~ [ F is constant since, in that case, fF(~ ) r I'~, which implies the last identity. We turn to the case 2~ ~ q). There are the following four possibilities: 1) g~(00 = g~(~)+ and g~(2~) = g~(200§ 2) g;(~) = gF(~) + and g;(2e) = gx,(200, 3) g~(~) = gF(e) and g~,(2e) = gr(2e), 4) g~(e) = gF(o0 and g;(2e) = gt(2e)+. In case 1) we have 1 , ] F is constant, ft(e) = g~(e) e r', n ~ r~, and gi,(2~) = 2ft(~) and hence U.. ~.(~. (U2. n U., toI(2~)) = U., t;.(~. (U2. n U., #(~) = U., ~.~. In case 2) we have 0c [ F is constant, fF(e) = gF(~) E r',, and ~ gr(2e) >fF(~) and hence In case 3) we have g)(~) = inf{ t e : t ~> fF(00 } 1 g~,(2~)=inflte~r;.:t>~f~(~) t. and Let us first assume that 89 ) 1> g~(~); then g~(e) ---- inf { t e r~ : t/> fF(e) }. This implies U=. 4(~," (U2~ n U=, to;~2=1) = U=, o;~=) = U=, I,(=~" Now assume that 89 < g~(a); then x g~(2~) = inf{t e P~ :e >~ fF(~) }. Z REPRESENTATION THEORY AND SHEAVES 109 We are going to use the following general fact which is a straightforward consequence of the definition of the set P'~: If r < s are values in F, such that rCF'~ and s=inf{g 9 r} then (,) U~,, _= U.,,. (Uz~ c~ U~,,). Applied to our situation this leads to U~, g;(~. (U2~ ca U~, i,;~) = U~, I~;~ = U~, ,,(~. Moreover in case 3) we always have U=,t.t=> = Ue, t;,~, since if el F is constant then both g~(e) and 89 are strictly bigger thanf.(e). Finally in case 4) we have 1 1 [ F is constant, f~(~) = ~ g~,(2~) e ~ P~, and gv(~) >f~(~) and hence U.,,lc.,. (U2. c~ U.,t,~,...~ ) = U.,~.I.~. (U2. n U.,,,I.,+) = U.,~ where the second identity again is a consequence of (*) sincefF(a ) r F',. [] There is a scheme-theoretic version of 1.4 ([BT] II.5.2.2-4): G ~ possesses smooth closed o-subgroup schemes o//,, F for e e ~ and o//~ for any fixed decomposition = ~+ t3 ~- into positive and negative roots such that ~'~,F(o) = U~,I.,. ~ and ~(o) = U~ n U (We have simplified the notation a little by writing q/=,r for ~., ~&l~l, &12.)~ in loc. cit.) Moreover the product map induces an isomorphism of o-schemes (whatever ordering of the factors on the left hand side we choose) as well as an open immersion of o-group schemes qG x ~rOx ~'~6 ~ We put Zr176 := { g e ~L~~ : (g mod r~) 9 R.(.~ ~ (K) } where R.(.~ ~ denotes the unipotent radical of .~o Proposition 1.2.2. -- The product map induces a bijection II U~.,~,~,) � Z '~ x ( II U~ ,:c~) _v~ RF. ~ ~. ~-t-h q) red r ~ ~" t'5 ~, red ' 110 PETER SCHNEIDER AND ULRICH STUHLER Proof. -- We recall from [BT] II. 4.6.4 and 5.1.31: If oq '~ denotes the connected component of the N6ron model over o of S then 5oo is a maximal K-split torus in ~o and .o~ ~ is its centralizer. Also the q/,,r are the root subgroups in G ~ By [BT] II. 1.1.11 the above open immersion therefore induces an isomorphism ( I/ ~,~ n ~) x ~(~) x ( II ~, n ~,) -~ ~,, ~ ~- ~ ~red ~ ~ ~* t~ ~red ' and hence also an isomorphism between formal completions (I/~,) x ~ x (II~,)-% G ~ where ?R denotes the formal completion of ? along ? n RF. It remains to observe that qz~ F(o),, = R~ n U,,,,,,, = U,,a,, ,, Lr~ = Z '~ and G~R(o) = R F. [] Corollary 1.2.3. -- We have R F = Uj~ Z ~~ With fx; also the functions f~ + e, for any integer e i> 0, are concave. Hence we have the descending sequence of subgroups Ut~ -~ UI~+I ~_ UI~+,, ~ ... We also need a corresponding filtration Z ~~ __ Z ~a~ ~ ... __ Z ~'~ ~ ... The subgroups we are looking for then will be defined to be U~ "~ := UI~+,.Z~'~; note that H normalizes U 1 for any concave function f ([BT] 1.6.2.10 (iii)). The properties which we want the subgroups U~ '1 to have impose certain conditions on the possible shape of the filtration Z ~'~. These conditions are axiomatized in [BT] I. 6.ff in the following way. First of all it is notationally convenient to define U2~:={1}in case ~e(l) but 2~r and U2~,k := U2~ c~ U~,t~ for any ~ e~ and any k ~ R. For any k eR put H~k I := set of all h e H such that (h, U~.,) := { (h, u) : u e U~., } ~ U~.,+k.U~.~,+~ for any ~ e(I) and any r e R. REPRESENTATION THEORY AND SHEAVES Ill The H~k ) form a decreasing family of subgroups in H which are normal in N; obviously H~k~ = H for k ~< 0. Another such family denoted by H[k ~ is given as follows: For k ~< 0 put H[,j := subgroup generated by all Hc~<U~,,wU_~_,) for 0~e~ and reR. In case 0<k<oo the commutator (u,u') for u eU=,,, u'eU_~.o with r+s-----k or u' e U_2~ ' , with 2r -t- s = k, r, s e It, and any a e r lies in a double coset U~ h~,., U_~ with a uniquely determined element h~,r e H ([BT] 1.6.3.9); we put H~l :-= subgroup generated by all those h,,~,. Finally we set Ht~l:= [7 H[~ 1. k<oo The H[~ 1 again form a decreasing family of subgroups of H which are normal in N. One has H[,+~ = [J H[0~ for any r e R. The key property of this latter family is the S>r following. Let us call a function f: 9 u { 0 } -+ R concave if f] 9 is concave and if f(~) +f(-- ~) >if(0)/> 0 for any ~ e~ holds. In this situation we have H c~ Utt o _ HEs(0)] ([BT] 1.6.4.17). A good filtration of H now by definition is a family of subgroups H, _ H for r ~ R such that -- H,-~H for r~ 0, -- H,_H. ifr>ls, -- H m_H,~H(,) for any r~R, and -- (H,, H,)_~ H,+, for any r,s~R. These properties together with [BT] I. 6.4.33 imply that the H, are normal in N. A necessary and sufficient condition for the existence of a good filtration is, according to [BT] 1.6.4.39, that H m ~ Hlk) holds true for any k e R. In [BT] I. 6.4.15 it is stated that this condition actually is fulfilled--see Proposition 6 below. For the moment we assume that a good filtration is given. We pose H,+:---- [J H, for reR and H~:= ~ H,. a>r r6R 112 PETER SCHNEIDER AND ULRICH STUI-ILER Also for any concave function f: 9 u { 0 } -+ R we define the subgroup U I := U11 | Ht<o~- Lemma 1.2.4. ---Let f,g:~ u{ 0}-+R be two concave functions such that g(p~ + q~) <<. pg(a) qf(~)foranya, ~ eOu{O}andp, q~Nsuchthatpoc + q~er u{ 0}; then U t normalizes Ug. Proof.--- [BT] 1.6.4.43. ~] Lemma 1.2.5. i) Hto4] ~ Z ~~ ~ H~o+); ii) Z c~ is normal in N. Proof. -- i) The subgroup Hco+j is generated by the h~, u, for u ~ U~.,+ and u'~U_~,_, with ~'~a and rER. Fix such u and u' and choose a vertex x in A (i.e. { x } is a facet in A) such that U.,, = U.,_~.,. Then u e Rt.~ by Lemma 1 and u' e U. __q P.. Since R~| I is normal in P. the commutator (u, u') lies in RI. ~ . It follows now from Proposition 2 that h~,.. e Z ~~ On the other hand we have to show that (Z ~~ _ U~,,! for any eeO~a and r e R. But choosing the vertex x as before we have (Z '~ U~, ,) ~ (Z ~~ U~, t,~) -~ Rr n U~. t,c~, = U~. t~,4 c_ U~. ,+. ii) By the very construction of the o-group scheme ~Lr ~ any automorphism g ~-, ngn-x of Z with n e N extends to an automorphism of the o-group scheme :Lr0. [] Proposition 1.9~.6. -- There exists a good filtration (tt,),~l~ of H such that: i) Ho+ =Z ~~ ii) H~ ={l}, and iii) H,+ is open in H for any r ~ R. Proof. - - As mentioned already this is a slightly sharpened version of [BT] I. 6.4.15. We are indebted to J. Tits for explaining to us the proof which is missing in [BT] and which we briefly sketch in the following. First of all we note that it suffices to find a good filtration H', which fulfills ii), iii), and the weaker condition i') Zl~ Ho+ , because then H for r~< 0, H,:= ZI~ for r>0 is a good filtration satisfying i)-iii). This follows from Lemma 5 i) and the fact that Z ~~ is open in Z. REPRESENTATION THEORY AND SHEAVES 113 Step 1: The split case. ~ IfG is split then we have S = Z -~ (K� and ~e ~ ~ G~,/o. We define H, := ker(.o~e~ -+ .o~~ n+x o)) if m < r ~< m 4- 1 with m ~ N u{ 0 }. The only thing which has to be checked is H~, 1~ H,~ H~,~ for r>0. The left inclusion, by [BT] II.3.2.1, can be checked in SL,(K) where it is straight- forward. For the right inclusion we use the following two identities. Let ~ e 9 be a root. -- ([BT] II.3.2.1) (h, u) = (0~(h) -- 1) u for h e Z and u e U~. -- ([BT] 1.6.1.3 b) and 6.2.3 b)) t(au) = ~o(a) + l(u) for a e K and u e U~ (here l(1) := oo). By definition any h 9 H, satisfies ~(a(h) -- 1) >/m + 1. For u 9 U~.0 we therefore obtain t((h,u)) =r l) +l(u)/> m4- 1 4-s>/r+s, i.e. (h,u)~U~.,48. This shows that h e H~,~. To deduce the general case we observe that Bruhat and Tits proceed by applying the descent theory in [BT] I. 9 in two steps: First from the split to the quasi-split case ([BT] II.4.2.3) and then from the latter to the general case ([BT] 11.5.1.20). Hence we may use [BT] 1.9.1.15 in order to see that our assertion descends as well. In each of the two steps we have to check that the assumption (DP) in loc. cit. is fulfilled and that the descent preserves the properties i'), ii), and iii). Step 9: From the split to the quasi-split case. ~ If G is quasi-split then Z is a maximal torus in G and ~e is that part of the N6ron model of Z over o which in the closed fibre consists of the connected components of finite order. The condition (DP) follows from the explicit computations in [BT] II.4.3.5. The filtration of H by construction is the intersection of H with a corresponding filtration over a splitting field of Z. From this it is obvious that the properties i'), ii), and iii) are preserved. Step 3: From the quasi-split to the general case. -- Note that the descent is along an unramified extension L]K. The condition (DP) (with t = 0) holds by [BT] II. 5.2.2. The descent of the properties i)-iii) is deduced from the following fact: Applying the argument in the proof of Proposition 2 to the group scheme ~r0 (compare [BT] II. 5.2.1) we obtain the decomposition Z'~ II U~o+ x 7-~x H U~,o+); here Z~ ~ denotes the analog of Z ~~ for G(L), ~ is the root system of G(L) with respect to some mammal L-split torus which contains S, and O0 is the subset of those reduced roots which restrict to 0 on S. [] 15 114 PETER SCHNEIDER AND ULRICH STUHLER We fix once and for all a good filtration of H as in Proposition 6. Define U (e) ZC'l:= He+ and U~; I:= ~+,. for e>~ 0. In other words we have U~ *1 = U~,+, where the concave function h r : 9 u { 0 } ~ R is defined by hr [~ :=jr; and hg(0) := 0-4-. The functions h r and fr extended by fF(0) := 0 fulfill the assumption of Lemma 4: This is straightforward if one of the ~, 9, P~ -4- q~ is equal to 0; otherwise it is shown in the proof of [BT] 1.6.4.23. Therefore U~ normalizes U~ '~ for any e >/ 0. Since N normalizes Z TM and N~ normalizes U~+, ([BT] 1.6.2.10 iii)) we obtain that U~ '~, for any e/> 0, is normal in P~. The same argument shows that nU~ n-1 = for n e N and e >i 0. Proposition 1.2.7. -- For any e >1 0 the product map induces a bijection ( II Ua+ . n U~) x z `~ x ( II Ur.+. n U,) --~ U~"; moreover we have U,~+, c~ U, = U,,t~(~+,.U~.~c,~+, for any at e~ ~a. Proof. -- Proposition 2 and [BT] I. 6.9 i). [] Corollary I. 9. .8. -- The following equality holds for any e >1 0: W' = (W' n u-) (W I n z) (W' n u+). Corollary 1.9..9. -- The U~r "' for e >1 0 (and F fixed) form a fundamental system of compact open nzighbourhoods of 1 in G. Proof. -- Since ~ x -~ x ~l ,+ ~ G~ is an open immersion the subset ( H u~.,,,~,) x .~(o) x ( H u,.,,,~,) ~t ~ ~- ~ ~red tt 6 4) + ~ ~red is compact open in G. Qlearly the U,+, n U= form a fundamental system of compact open neighbourhoods of I in U~ for any ~ e ~m. Similarly Proposition 6 implies that the Z ~*~ form a fundamental system of compact open neighbourhoods of 1 in 2L~'~ [] --',~zTc"~ REPRESENTATION THEORY AND SHEAVES 115 Using 1.5 we may define, for any facet F' in X and any e/> 0, a compact open subgroup U~):=gU~ '~g-~ if F'-=gF with g 9 and F a facet in A in G. By construction we have gU~, g-1 -- ~gF'lI~ for any g ~ G. If x is a vertex of X, i.e. { x } is a facet, then we replace similarly as before { x } by x in all our notations; e.g. we write U ~ instead of 11~ Lemma I. 9,. 10. -- There is a point Yo E A such that we have vet = (y0) + - z for any rta where n~, 9 N is a natural number which moreover is even in case 2o~ 9 ~. Proof'. Step 1. -- According to [BT] 1.6.2.23 the statement at least holds with some real number ~et > 0 instead of --.1 (The point --Y0 has to be a special point; in net loc. cit. it is assumed to be the origin and therefore does not appear. Also note that 9 '=~ by [BT] II.4.2.21 and 5.1.19.) Step ~. -- It suffices to show that Fet contains a subset of the form cet 4- n-- Z with some n" 9 N which is even in case 20t ~ 9 and some c~ ~ 11. Because then there has to be amap~:Z~Zsuchthat c~4-n~m=~(y0) 4-~et~(m) for any m 9 If m = 0 this means that r = ~(Y0) 4- s~ ~(0) which inserted back implies ~, v(0) 4- n', m set ,~(m) for any m 9 Z. We obtain so that our assertion holds with net := n'~. (v(1) -- ~(0)). Step 3. -- Let K ~ be the strict Henselization of K. The quasi-split group Gm,h possesses a maximal K'h-split torus T which is defined over K and contains S ([BT] II. 5.1.12) ; let ~h be the set of roots of G/K,h with respect to T. Restricting characters defines a surjecfive map ~Bhu{0}-~U{0}. By [BT] II.5.1.19 we have 116 PETER SCHNEIDER AND ULRICH STUHLER F~ c_ P~, _c F~, whenever ~ c ~'~ restricts to ~ e r The sets r~ are explicitly computed in [BT] II. 4.2.21 and 4.3.4: For any ~ c ~ one has r~ = c~ + -- z nff with appropriate constants c~ ~ R and n~ e N. Let now an a e ~,,d be given. If 2a r (I) then we choose an ~ e (1) ~ restricting to a and we obtain F~ ~_ P~ =c~ +--Z. n~ If 2e e q) then we choose a ~ e ~'~ restricting to 2e and we obtain 1 1 1 1 F~ = ' = c~ + Z. [] Proposition 1.2.11. -- On~ has: i) U~',' G U(;' for any two facets F, F' in X such that F' ~_ F; ii) U(~ '~ = II U(~ '~ for any facet F in X and any ordering of the factors on the right hand side. 9 vex'~x Proof. ~ We may assume that F __c A. First we consider the case that F' = { x } is a vertex. Then U"~ n U~ = U~,c_~+,~+.U~,~_~,~+,~+ for ~ e~ ~. If a(x) = inf ,t(y) then clearly (-- e(x)) + >>.f;(o:) and hence I1 ~*1 n U, c U[g' n U,. If > inf e(y) then el F is not constant so that vql* -- =(x) <fF(~) =f;(~). Furthermore, by the definition of facets, we then have --a(y) eP~ for anyyeF. Hence inf{ t ~ P~ : g > -- ~(x) } >i fF(~) and because of Lemma 10 also inf{ger~:g>--e(x) +e}/>fF(=) +e e. inf teP~:t>--c((x) +~ l>fF(~) +~ m case 2~e~. and This implies that again UI; I n U, _ U~ e~ n U,. Using Proposition 7 we obtain U TM ? a IT~'~ = the subgroup generated by all _, for x c vertex. REPRESENTATION THEORY AND SHEAVES 117 To get the reverse inclusion we fix an ~ e ~ and consider U~ "~ c~ U,~. Let x e F be a vertex such that a(x) = sup a(y). Then IrEF )~< -- l~ey inf,(y) =f~(~) if ~l F is not constant, (-- ~(x)) + = (-- infer(y))+ if ~[ F is constant; ~eF hence U~ '~ c~ U, _~ U~ ~ n U,. Again using Proposition 7 we see that Uk '~ = the subgroup generated by aU U~; ~ for x ~ F a vertex. This implies in particular the assertion i) for an arbitrary facet F' _ F. For ii) it remains to show that for any two vertices x,y ~ F the subgroup --~ II TM normalizes the subgroup --v II TM " But by i) we have UIs~ c U~ ') : PF = Pv z -- -- -- and _~N ~'~ is normal in P~. [] Finally we define, for any z e X, UC;~ := U~ '~ if z lies in the facet F of X. Note that U, = U F in this situation if F _~ A ([BT] 1.7.1.2). 1.3. Properties of the groups U~ > Here we will establish those properties of the groups U~ '~ which are responsible for our later results about the cohomology of the Bruhat-Tits building. We recommend the reader to skip this section at first reading and only come back to it when the results are needed. Fix an e/> 0. A first technical clue is the observation that the following representation-theoretic fact is at our disposal. The notion of a smooth G-representation will be recalled in II. 2. A vertex x in A is called special if ~(x) ~ -- r~ for any ~ e @~a. There always exists a special vertex ([BT] 1.6.2.15). Theorem (Bernstein). -- Let x be a special vertex in A. The category of smooth G-repre- sentations V which are generated (as a G-representation) by their U~'~-fixed vectors V~Z~ (') is stable with respect to the formation of G-equivariant subquotients. Proof. -- This is [Ber] 3.9 i). We only have to check that our group U~ ~ fulfills the assumptions made there. Since the vertex x is special the Iwasawa decomposition G = U, P = G~~ P for any parabolic subgroup P G G holds true ([BT] 1.7.3.2 ii)). Moreover the decomposition property (3.5.1) in [Ber] is a consequence of 2.7. 118 PETER SCHNEIDER AND ULRICH STUHLER Next we need some control over how U~, '~ changes if z varies along a geodesic line in X. We fix two different points x and x' in A. The geodesic [xx'] joining x and x' is [xx'] ={(1--r) x+rx':0<~ r<~ 1}. Proposition I. 3.1. -- Assume x to be a special vertex; for any point z ~ [xx'] we have Uce~ ~ 1)'r Tlce) Proof. -- We may assume z to be different from x and x'. Let F, resp. F', denote the facet in A which contains z, resp. x'. Define 9 " := { 9 : < }. There certainly exists a decomposition 9 = ~+ to O- into positive and negative roots such that ~" ~_ q~+. As a consequence of 2.7 it suffices to check that if ~ e O~a\~, Ucx, {- a(z) + el + "Ugot, (- 2a(z) + el + Ue, f,~r + e" U2~, 231~1 + e --- { if e c~F. U~. #,~ + ,. Uz~ ' 21;,~ +, Assume first that e c O~a\~F. Since x is special we have t := -- 0c(x) e F,. If ~(x) = ~(x') then -- e(y) = t for anyy e{ x } to F to F' and hencef;(~) = t+. If ~(x) > e(z) > ~(x') then -- ~(y) > t for anyy 9 F and hencef~;(~) >~ fF(~) >/t+. Now assume that e e W, i.e. that ~(x) < e(z) < ~(x'). If there exists an t' 9 F, such that -- ~(x') < l' ~< -- e(z), then Otherwise there are two successive values g' < t" in P~ such that e',< -- ~(x') < -- ~(z) < t"; hence g' <f~(~) ~< l" and t' <<.f~,(~) ~ t". Using 2.10 we see that in this case U,. r,~.~ + ." U2-. 2r,~ +, : U~,r,+ e. U2~,2r,+ e [] Because of 1.7 the assumption that x and x' are contained in the basic apartment A is unnecessary. Also the statement remains true even if x is not a special vertex. Since it is not needed we do not go into this. But see the proof of III.4.14. Consider the half-line s:--{ (1 --r) x+rx':r>~ 0} REPRESENTATION THEORY AND SHEAVES 119 in A and put Us := subgroup generated by all U~ for ~ e 9 such that c~(x') > o~(x). As will be explained in IV.2 this group is the unipotent radical of some parabolic subgroup of G. Proposition I. 3.9,. __ Assume x to be a special vertex; for any point z ~ ~ we have Proof. -- This is a straighforward (actually simpler) variant of the previous proof. The only additional fact to use is that the product map induces a bijecfion II U~ --% U, whatever ordering of the factors on the left hand side we choose ([Bor] 21.9). [] H. THE HOMOLOGICAL THEORY II. 1. Cellular chains Through its partition into facets the Bruhat-Tits building X acquires the structure of a d-dimensional locally finite polysimplicial complex ([BT] 1.2.1.12 and II. 5.1.32) where d :---= dim A is the semisimple K-rank of G. For 0 ~< q ~< d put Xg := set of all q-dimensional facets of X. In particular we may view X as a d-dimensional CW-complex the q-cells of which are the facets in Xq. The G-action on X is cellular. Let X~:= O FEXq denote the q-skeleton of X, also put X -1 := 0. With the composed maps ~: H~+,(X q+l, Xq; Z) a_~ H~(X~, Z) -+ Hq(X ~, Xq-'; Z) as boundary maps the augmented complex Hd(X d,xd-';z) 0a_g ... a0> Ho(X0, Z) = @ Z r.>Z ~ffXo computes the (singular) homology of X ([Dol] V. 1.3). It is G-equivariant and it is exact since G is contractible ([BT] 1.2.5.16). In order to motivate later constructions we want to give a more combinatorial description of that complex. By [Dol] V.4.4 and V.6.2 we have the direct sum decomposition H~(X q, Xq-'; Z) = ~D Hq(X', X~\F; Z). FEX8 120 PETER SCHNEIDER AND ULRICH STUHLER Consider, for any F ~ XQ+x and F' ~ X~, the composed map Oq > ~, : Hq+,(X '+~, Xq+~\F; Z) c.~ Hq+~(Xq+l, Xq; Z) H,(X q, X ~- 1; Z) H~(X ~, X"\F'; Z). One has: -- H~(X q, Xq\F; Z) ~ Z for F e X~ (but for q> 0 no canonical such isomorphism exists). -- ~, is an isomorphism if F' _c ~ and is the zero map otherwise. (Using [Dol] V. 6.11 this follows from the fact that in our case the characteristic map ~F of F can be chosen to be injective and hence to be a homeomorphism onto F.) Define now an oriented q-facet to be, in case q > 0, a pair (F, c) where F e X~ and c is a generator of Hq(X ~, X~kF; Z); then (F, -- c) is another oriented q-facet. In case q = 0 an oriented 0-facet simply is a 0-facet F which we sometimes also think of as the pair (F, 1) where 1 is the canonical generator of H0(X ~ X~ Z) = Z. Let X(~) denote the set of all oriented q-facets. Observe that for any (F, c) e X(~+x~ with q >/ 1 and any F' e Xq such that F' ~ F we have (F', The group of oriented cellular q-chains of X by definition is C~ Z) := group of all maps co : X(~) ~ Z such that: -- co has finite support, and, if q >t 1, -- co((F, -- c)) = -- ~((F, c)) for any (F, c) E X(,,. Clearly co-x z) H,(X ", X'-I; Z) o k (q), (F, c) ~ X(q) with ~ = -- 1, resp. 0, in case q > 0, resp. = 0, is an isomorphism which is G-equivariant if G acts on the left hand side by (go) ((F, c)) := o~((g -1 F, g-1 c)). The boundary map 0~ becomes 0q : ~'(X(q+l), Z) ~ C~ Z) o~((F, c))). (F, 9 ~ X(q+l) F'-=~ REPRESENTATION THEORY AND SHEAVES 121 The augmentation map becomes or : Co (X{o}, Z) -~ Z co ~ Y, co(F). F ~ X(o) H.2. Representations as coefficient systems A smooth (or algebraic) representation V of G is a complex vector space V together with a linear action of G such that the stabilizer of each vector is open in G. Let AIg(G) denote the category of those smooth representations. On the other hand a coefficient system (of complex vector spaces) V on the Bruhat-Tits building X consists of -- complex vector spaces V F for each facet F ___ X, and -- linear maps r~, : V F ~ V~, for each pair of facets F' ~ F such that r~ = id and ~,, = ~;, o r[, whenever F" _c ~' and F' _~ F. In an obvious way the coefficient systems form a category which we denote by Coeff(X). We fix now an integer e 1> 0. For any representation V in Alg(G) we then have the coefficient system V := (V v~'}) of subspaces of fixed vectors V~:=V U~'): -{veV:gv=v for all g~U~ '~}; because of U~I, _= for F' _c F the transition maps r~, are the obvious inclusions. Since the U~ '~ are profinite groups the functor y, : Alg(G) -+ Coeff(X) V ~ (VU'{')) F is exact. For any 0 ~< q ~< d the space of oriented (cellular) q-chains ofy,(V) by definition is C~ y,(V)) := C-vector space of all maps o~ : X~} -+ V such that: -- {o has finite support, -- {~((F, c)) ~ V v{/), and, if q >/ 1, -- {o((F, -- c)) = -- o~((F, c)) for any (F, c) e X,~}. The group G acts smoothly on these spaces via (g{~) ((F, c)) := g({o((g -x F, g-~ c))). A straightforward computation shows that the boundary map a : c;'(x,~+., v.(v)) -~ c~ v.(v)) ,~ ~ ((F', c') ~ Z ~((F, c))) {F, c} ff X~q§ F'_cF Or~(C) = C' ~,,TT{'} 122 PETER SCHNEIDER AND ULRICH STUHLER fulfills O o 0 = 0. In this way we obtain the augmented chain complex 0 or v.(v)) L ... c~ (x,0,, v,(v)) A v where the augmentation map is given by : r(x,0 ,, y,(v)) -+ v E X(o) The homology of this chain complex could be called the (cellular) homology of the coefficient system y,(V) on the space X. We will not use this terminology since in the next section it will be shown that these complexes under a rather weak assumption are exact. That assumption has to do with the surjecfivity of the augmentation map. For any open subgroup U _~ G we have the full subcategory AlgV(G) := category of those smooth G-representations V which (as G-representation) are generated by their U-fixed vectors V tr of Alg(G). If the representation V lies in AlgV~')(G) for some vertex x in X then the augmentation map r clearly is surjective. The subsequent two statements are immediate generalizations of the Propositions 1 and 2 in [SS] w 3. Recall that a representation V in AIg(G) is called admissible if the subspace V U, for any open subgroup U ___ G, is finite-dimensional. Proposition II.2.1.- If V in Alg(G) is admissible then the complex C~ y,(V)) consists of finitely generated G-representations. Proof. -- By the admissibility assumption the subspace in C~'(X~ql, y,(V)) of q-chains supported on { (F, + c) }, for a given facet F ~ X~, is finite-dimensional. If F runs over a set of representatives for the finitely many G-orbits in Xq then the corres- ponding subspaces together generate C-~r(x~I, y,(V)) as a G-representation. [] For any continuous character Z : C ~ C � of the connected center C of G we define the full subcategory Alg� :---- category of those smooth G-representations V such that gv = z(g).v for all g E C and v E V of AIg(G). Since C acts trivially on X the complex C~ y,(V)) lies in Algx(G ) if V does. Proposition 1"I.2.2. -- For any representation V in Algx(G ) the complex C~ y,(V)) consists of projective objects in Algx(G ). Proof. -- This relies on the fact that the group P~/CU~ '~ is finite for any facet F _c X. As a consequence of 1.2.9 the group G~ ~ is finite. On the other hand P~/CG~ is finite according to [BT] II.4.6 28. [] REPRESENTATION THEORY AND SHEAVES 123 II.3. Homological resolutions In order to formulate the main result of this chapter let e i> 0 be an integer and let x be a special vertex in A. Theorem lI.a.l. -- For any representation V in AlgV~"~(G) the augmented complex cT(x,.,, v is an exact resolution of V in Alg(G). Proof. -- In the case G = GLa+I(K ) this result was established in [SS]. The proof in the general case in its most parts is a direct generalization of the arguments in [SS]. Insofar we will only indicate the main steps. Step 1. -- Let Cc(T ) denote the space of complex-valued functions with finite (e) support on the G-set T:= G/U, . This is a smooth representation of G which acts by left translations. It lies in AlgV~)(G) and one has the surjective G-homomorphism Co(T) | V V + N v w-~ Y~ +(g) .g(v). g E G[U (*) Bernstein's theorem (I. 3) now ensures the existence of an exact resolution in Alg~")(G) of the form ... -* @ Co(T) -~ (9 Co(T) -~ V -~ 0 11 Io with appropriate index sets Io, Ii, . .. Since the functor 7e is exact a standard double complex argument reduces therefore our assertion in case V to the" universal " case C~(T). Step 2. -- A slightly more sophisticated double complex argument for Co(T) ([SS] w 1) further reduces our assertion to a geometric property of the Bruhat-Tits building X. For any facet F we put T~:= U~'I\T. It follows from I. 2.11.ii) that T~ = T~0II ... lI T,~ T T if{ xo, ..., x, } are the vertices in F. The natural projection T -+ T F is finite and induces an isomorphism C.(Tr) -~ u,(x)u(*)" which we will view as an identification. More generally we have the simplicial set T. ~ : ... , ~T�215 >T� >T > TF Tp ) TF 124 PETER SCHNEIDER AND ULRICA-I STUHLER all face maps of which are finite. There are obvious commutative diagrams /\ T ~ T r T. r' ~ T r, and T. F , T F for F' ___ F. By passing to functions we obtain the double complex (A) 0 0 0 0 ~, @ Co(TF) o o ----- ) ... , @ C,(Tr)--- > C,(T) , 0 FUX d F~Xo 0 , @ C,(T) , ... , @ Co(T) ,. C,(T) ~. 0 l o 0 , @ Co(T � T) > ... > @ C,(T ~ T) , C,(T) :, 0 F~X d TF F~Xo 0 > @ C,(T � T � T) , ... 9 0 C,(T � T � T) , C,(T) , 0 F E Xd TF TF F ~ Xo Te TF l o All its columns are exact. This follows from the fact that each T. ~ is the disjoint union T, r = [J TIt' t ~Tp of the simplicial finite sets T t): ... 9 > :' T, X T t � T t :, T t X T t ~ T t where T t denotes the fiber of the map T ~ T r in t. Simplicial sets of the form TIo tl are well-known to be contractible. The top row in (A) is the complex whose exactness REPRESENTATION THEORY AND SHEAVES 125 we want to establish. Here we have fixed for simplicity an orientation of the building X. Next we have to study, for a fixed m/> 0, the row (A,~) 0~ O Co(T~)~... ~ @ C,(T~)~Co(T)-70 F~X d FEXO from (A). If we view each T~, resp. T, as a subset of T m+l := T � ... x T(m + 1 fac- tors) in the obvious way, resp. diagonally, then the differentials in the above complex are induced by the inclusions T_cT~,'~T~ for F'~F. In order to rewrite this row in a more suitable form we introduce certain subcomplexes of the Bruhat-Tits building. For a fixed (to, ..., t,,) e T "+1 we put X (t0 ..... Ira) :-= union of all facets F __ X such that to, . .., t,~ are not mapped to the same element in U~*)\T. It is easy to see that fl T~=T, ~ex. i.e. that X "~ ..... t,~ is empty if and only if t o ..... t,~. If{ to, ..., t,~ } has cardinality at least 2 than X "~ ..... t,~) is a nonempty closed CW-subspace of X (I.2.11 .i) and [Dol] V.2.7). We now have @ C~(T~)= (~ Cr ..... ~m)) F ~ i. { tO, * 9 -, ~m ) E T m+l and the decomposition on the right hand side is compatible with the differentials. As a result of this discussion we obtain that (A,~) = O (augmented complex of relative chains of the pair (X, X "~ ..... t,)). (tO,..., t m) ~T m+l Since X is contractible this means that we are reduced to show the contractibility of the subspaces X"0 ..... t,) for any { to, ..., t,, } of cardinality at least 2. Step 3. -- The special vertex [l(e) Xo:=goX where t o--go_~ is contained in X "0 ..... t,,) ([SS] w 2 Remark 1). We show that with any pointy e X "0 ..... t,,) the whole geodesic [Xoy ] lies in X "~ ..... t,,). This of course implies the wanted contrac- tibility. Fix a point z e [Xoy ] and let F, resp. F', denote the facet in X which contains z, resp. y. Clearly F' _ X "~ ..... tml SO that there exists I ~<j~< m such that gt~ # t o for allgeU[f). 126 PETER SCHNEIDER AND ULRICH STUHLER If F is not contained in X "~ ..... t,,~ then we find gl ~ U[g ~ with gl tj : t 0. According to 1.3.1 we have U~: U~.~ TTce~ -- Z 0 " ~I~' so that gl E hU~ ) for some h e II (e) i --~,o ~ Put g : = h- ~ gl e U~). Then gtj : h- ~ to : to which is a contradiction. [] Corollary II.3.2. -- For any representation V in AlgV(1)(G) n Algx(G ) the augmented complex C;r(X, ,, v.(v)) -+ v is a projective exact resolution of V in Algx(G ). Proof. -- Combine Theorem 1 and 2.2. [] Corollary H.3.3. -- Let V, resp. V', be an admissible representation in AlgU(1)(G) n Algx(G), resp. Algx(G); then the vector spaces Ext],,x,o,(V , V') are finite- dimensional and vanish for 9 ;> d. Proof. -- Combine Corollary 2 and 2.1 (compare [SS] w 3 Cor. 3). [] Note that because of I. 2.9 any finitely generated smooth G-representation lies in Alg~ if e is chosen large enough. HI. DUALITY THEORY II1. |. Cellular cochaius An element of the space C'~c'(XI,I,y,(V)) also can be viewed as an oriented cellular cochain with finite support on X. This suggests that there is a cohomological differential, too. Indeed, for any pair of facets F, F' ~ X such that F' c ~, we have the projection map p~' : VV~ ;) _+ VO, ") Since the partition of X into facets is locally finite the coboundary map d: c;'(x,.,, v.(v)) -+ c;'(x,.+~,, v.(v)) p~'(~o((r', o~,(c))))) ,. ~ ((F,c) ~ Y~ F' ~F REPRESENTATION THEOI~Y AND SHEAVES 127 is well defined; in case q ---- 0 the summands on the right hand side have to be inter- preted as ~,(c).o)(F'). A standard computation (IDol] VI. 7.11) shows that d d o, e o, x v,(v)) co (X,o,, v,(v)) ... is a complex in Alg(G)--the cochain complex (with finite support) of y~(V). We will see in Chapter IV that this complex computes the cohomology with compact support of a certain sheaf on X. Here we are interested in the relation between the chain and the cochain complex. Again let X : C --~ C � be a continuous character. In Alg� there is the " universal " representation ~ := space of all locally constant functions + : G -+ C such that: -- t~(g-~h) = z(g).+(h) for all g EC, h eG, there is a compact subset Z _c G such that + vanishes outside Z. C where G acts by left translations. This is the x-Hecke algebra of G; its algebra structure will be recalled later on. Note that G also acts smoothly on JV x by right translations. Both actions commute with each other. In the second action the connected center C acts through the character X-i. Fix now a representation V in Algx(G). Its smooth dual V lies in Algz_~(G ). For any 0 ~< q ~< d we have the pairing CO,~ X , v,(V)) x v,(v)) -+ defined by LF~.~(g) := Y, v~((F, c)) [(g-1 c0) ((F, c))]. (F, c) ~ X(q) One easily verifies that ~.ho=hLF~.~ and ~Fh~.~----LF~.~(.h) for any h~G. This means that the above pairing induces a homomorphism ~F: C~ y,(~)) ~ Homa(C~'(X,, ,, y,(V)), ~x) ~ (~o ~ ~/'~,~) which moreover is G-equivariant if the action on the right hand side is the one induced by the right translation action on ~fx" Next one checks that ~F0~. ,, ----- ~. d~ and ~Fa~ ' ~ = ~F~, 0~. In other words ~F is a homomorphism of complexes from the chain, resp. cochain, complex of y,(~) into the Horna(., ~'x)-dual of the cochain, resp. chain, complex 128 PETER SCHNEIDER AND ULRICH STUHLER of y,(V). We claim that )'F is injective. For any (F, c) e X(q~ and any v e V, define an oriented q-chain CO(F,c), ~ of y,(V) by if (F', c') -=-- (F, c), (o(~. ~,.,((F', c')) := prF(v ) if ql> 1 and (F',c') : (F,--c), otherwise; here pr F denotes the projection map pr F : V -+ V U~') v~--~ gv if v E V u for some open subgroup : u] U z We then have V,,o(,,o.0(I ) = 2*.B((F, c)) [prF(v)] ----- 2*.~q((F, c)) Iv] with ,----1, resp. 0, in case q >t 1, resp. = 0. The second identity comes from the observation that any linear form in -~u,(') factorizes through pq,. This clearly implies the injectivity of ~F. Lemma III. 1.1. -- Let V be an admissible representation in Algx(G ) ; then the G-equivariant linear map : C~c~(X(.), 7,('~)) ---%- Homa(C~'(X,.,, 7,(V)), ")fix) is an isomorphism; under this identification we have Hon~(O, 9rt~ = d and Homo(d , ,~~ = O. Proof. -- Only the surjectivity of W remains to be established. Let Wo be an element of the right hand side. Define a map ~ : X(,) -+ V by c)) [v] := .,, o) (1). For a fixed (F, c) the function Wo(CO(F. c,,~) in 9~' z only depends on pr~(v). Therefore the admissibility assumption guarantees the existence of a compact subset Z _~ G such that all functions Wo(O(F ' ~),,) for v ~ V vanish outside E.C. Because of ~(g-'(F, c)) Iv] = ~o(CO,F.c,,~) (g) it follows that ~(g-~(F, c)) ---- 0 for g-' r E.C. REPRESENTATION THEORY AND SHEAVES 129 Since G has only finitely many orbits in XIq) we obtain that the map ~ has finite support. It is now straightforward to see that ~ e C~ y,(~)) is a q-chain such that ~t'(~l) = 2'. ~Fo with the same ~ as above. [] This duality between chain and cochain complexes is perfectly suited to analyze the Ext-groups g'*(V) := Ext~,lgx(V, s~x) in the category Algx(G ). Through the right translation action of G upon s~ffx the space 8*(V) in a natural way is a G-representation which in general might not be smooth. As before we fix a special vertex x in A. Lemma 111.1.2. --For any representation V in AlgU(1)(G) n Algx(G ) we have 6"(V) = h'(HomG(C~ y,(V)), "r H~ "~x))" Proof. -- II.3.2. [] Proposition I11.1.3. ~ For any admissible representation V in AIgU~')(G) n Algx(G ) we have g'(v) = v,(V)), d). Proof. -- Lemmata 1 and 2. [] Remark 1II.1.4. i) The category of finitely generated smooth G-representations is stable with respect to the formation of G-equivariant subquotients ; ii) a smooth G-representation is finitely generated and admissible ~ and only if it is of finite length; iii) let V be an admissible representation in AlgV-('>(G) n Algx(G); then ~ is an admissible representation in AlgV(l>(G) n Algr_l(G ) and we have ~r _= V. Proof. -- i) and ii) [Ber] 3.12. iii) [Cas] 2.1.10 and 2.2.3 together with Bernstein's theorem (I.3). [] In particular it follows from these considerations together with II. 2.1 that the spaces ~"(.) form functors finitely generated and finitely generated o ~" : admissible representations --~ representations in Algx(G ) in Algr_l(G ) If * > d then d'* = 0. For later use we need the following technical consequence of the above results. 17 130 PETER SCHNEIDER AND ULRICH STUHLER Lemma m. 1.5. -- Let V be a representation of finite length in Algx(G ) and assume that there is an integer 0 <~ d(V) <~ d such that g*(V) = 0 for 9 # d(V) ; we then have: i) For e big enough the complex Or (~)) d d or (~)) ) kerd d'v) Co (X~o), T. ) ... ) Co (Xld(v)_~), T, ~I~'(V) is an exact projective resolution of o~v'(v) in AIg~-I(G); ii) #'(gn'v'(V)) = { 0 V otherwise./f * = d(V), HI. 2. Parabolic induction The computation of the spaces oa'(V) in an essential way makes use of the theory of parabolic induction. We fix a decomposition 9 = (D + (I)- of the set of roots (I) into positive and negative roots. Let A ~_ (1) + be the corresponding subset of simple roots. The subsets O G A parametrize the conjugacy classes of parabolic subgroups of G in the following way. First we have the torus S O := connected component of n ker ~Ee of dimension dim S -- $| and the Levi subgroup M e := centralizer of S O in G. Second there is the unipotent subgroup Uo := subgroup of G generated by all root subgroups U, for ~ cO+\< 0 > linear combination of roots in 0 }. The product where (O):={~EO:~isa Po := Me Uo is a parabolic subgroup of G; its unipotent radical is U o. Let ~e : Po -~ Me h [ det(Ad(h) ; Lie Uo) ]-' denote the modulus character of Pc- Because of C ~ M e the category Algx(Me) of all smooth Me-representations on which C acts through the character X is defined. We have the " normalized " induction functor Algx(Me) -+ Algx(G ) E ~ Ind(E) REPRESENTATION THEORY AND SHEAVES 131 where Ind(E) := space of all locally constant functions ~ : G -+ E such that (ghu) = ~2 (h) . h- 1(~0 (g) ) for allgeG, h~M o, and u eU o with G acting by left translations. The reason for introducing the character 8o is the formula Ind(E) ~ ---- Ind(F,) ([Gas] 3.1.2) for the smooth dual E. An irreducible representation V in Algx(G ) is called of type | if there is an irreducible supercuspidal representation E in Algz(Mo) such that V is isomorphic to a subquotient of Ind(E). We put AlgfxZ, o(G) :---- category of all smooth G-representations of finite length all of whose irreducible subquotients are of type | Any irreducible representation in Algx(G ) has a type, i.e. lies in some category Alg~x ~, o(G) ([Gas] 5.1.2). Also Ind(E) lies in Alg~x ~, o(G) if E is irreducible supercuspidal in Algx(Mo) ([Gas] 6.3.7). Technically very important is the fact that Alg~x,o(G ) = Algfxio,(G) if | and 0' are associated ([Gas] 6.3.11). We recall that two subsets | and 0' of A are called associated if S o and S o, are conjugate in G; in this case M o and M o, are conjugate in G, too, and $0 = ~| We actually need a more precise version of that fact. Fix a g E G such that 1 and let E be an irreducible supercuspidal representation in Algx(Mo). M o = gM o, g- Via the map M O, -+ M O h~#g -1 E can be considered as an irreducible supercuspidal representation in Algx(Mo, ) which we denote by gE. Obviously the isomorphism class of ~ does not depend on the choice ofg. Proposition HI. 2.1. -- The representations Ind(E) and Ind(~ have (up to isomorphism) the same irreducible subquotients; moreover given an irreducible subquotient V of Ind(E) there is a subset O' =_ A associated to | such that V is a homomorphic image of Ind(aE). Proof. -- [Gas] 6.3.7 and 6.3.11. [] A key result of this paper which will be established in the Chapter IV (IV.4.18) is the following. Theorem lll.2.2. -- Let E be an irreducible supercuspidal representation in Algx(Mo); there is a subset O' c_ A associated to | such that f Ind(']~) /f 9 = d -- ~O, 8*(Ind(E)) 0 otherwise. 132 PETER SCHNEIDER AND ULRICH STUHLER m.3. The involution Theorem 1II.3.1. -- For any representation V in Alg~xi, o(G) we have: i) o~ = O for 9 4= d -- $| ii) o~-*~ lies in AI~_I,o(G). Proof. -- We prove the vanishing of d"(V) for 0 ~< * < d -- ,| and all V in AlgYx~ o(G) by induction with respect to ,. Fix an integer q such that 0 ,< q < d- ~0 and assume that g'(V) = 0 for all 9 < q and all V. We have to show that g*(V) = 0 for all V. By induction with. respect to a Jordan-H/Sider series we obviously may assume that V is irreducible. Then 2.1 says that there is an exact sequence of G-representations 0 -+V' -+ Ind(E) -+V -+0 where E is an irreducible supercuspidal representation in Algx(Mo, ) for some subset | _~ A associated to | We obtain an exact sequence 6'*-1(V ') -+ oaq(V) -> g'(Ind(E)). Because of Al~xl.e,(G)= Alg~x,o(G ) (and fO'= $| the left term vanishes by the induction hypothesis and the right term by 2.2. For proving ii) we again may assume that V is irreducible. Similarly as above we then obtain an injection gn-~o(V) ~ ~-*~ = Ind(~ where the right hand equality comes from 2.2. This shows that o*~-~*~ hes in Alg~x-, o,(G) = Alg~z~_t,o(G). The remaining vanishing assertion in i) also follows by induction. We already know that ~**(V)= 0 for 9 > d. Since, quite generally, Ind(E)~ = Ind(~,) holds 2.1 can be dualized to the statement that in case V is irreducible we find a mono morphism of G-representations V ~ Ind(E') where again E' is an irreducible super- cuspidal representation in Algx(Me, ) for some subset | _~ A associated to | Therefore an induction argument similar to the above one but downwards from 9 = d + 1 to , -- d + 1 -- ~| is possible. [] This result together with 1.5 ii) implies that @ : Algfx~. o(G) -+ AI~_Le(G ) V ~ o~-~~ is an exact (contravariant) functor such that ~ o d' = id. CorollaTy m.3.2. -- If v is an irreducible representation in Algx(G ) then g(V) is irreducible, too. REPRESENTATION THEORY AND SHEAVES Corollary m.3.3. -- For any representation V in Alg~,e(G) we have: i) V has an exact projective resolution in Algx(G ) of length d- 1~| ii) Ext~agxCol(V , V') -= 0 for 9 > d -- $| and any representation V' in Algx(G ). Proof.- Theorem 1 and 1.5 i). [] Remark m.3.4. -- For any representation V in Al~,zx(G ) we have d"(V) = HomG(V , ~x) = V. Proof. -- We may assume V to be irreducible. Then HomG(V , o~fx) is irreducible, too, by Corollary 2. On the other hand the matrix coefficients of V provide an embedding ~r ([Cas] 5.2.1). [] Fix an invariant measure dg on G/C. Then o~z becomes an associative E-algebra (without unit) via the convolution product q~ (+* q~) (h) :=f +(g) r d~, for +, dtl le Also this algebra o~V� acts from the left on each representation V' in Algx(G ) through d?.v:-=( +(g).gvd~ for ddegffx, v V'. dG /C The antiautomorphism g ~-* g-1 of G induces an algebra antiisomorphism e~a Hence any representation in Algx_l(G ) can be viewed as a right 9f'x-module. In this way the tensor product V" ~x V' and its left derived functors Tor~X(V", V') are defined for any V" e Algx_l(G) and V' e Algx(G). Duality theorem. -- Let V be a representation in Alg~x,o(G); we then have a natural isomorphism of functors Extk~x,o,(V , . ) = Tor~__xt~ e_.(@(v), . ) on Algx(G ). Proof. --- For any V' in Algx(G ) consider the smooth representation ~$"x No V' where G acts only on the first factor. The map a~'x| >> V' +| ~-*+*v is a G-equivariant epimorphism. According to (a slight generalization of) a result in [Bla] (compare also [Ca2] a.4) the representation o~V x and hence o~f' x @c V' is a 134 PETER SCHNEIDER AND ULRICH STUHLER projective object in Algx(G ). We see that the objects in Algx(G ) have a functorial projective resolution by representations of the form ~,~o� ~ V'. It follows from Theorem 1 that Ext~,.z,e,(V, "~x ~ V') = Ext~x,a,(V, ~x) ~ V' = 0 for 9 4= d -- ~@; here the first equality is an immediate consequence of the fact that V has a projective resolution by finitely generated G-representations (II.2.1 and II.3.2). These two properties imply by a standard homological algebra argument ([Har] 1.7.4) that Ext~agz(al(V , .) is the left derived functor of EXffA~:l~l(V, .). In order to establish our assertion it therefore remains to exhibit a natural isomorphism V,x ,(v, v') = "" ~(v) | v' = Ex ea ,l ,(v, a~x) ~x | v'. ,'~gx Using the projective resolution in II. 3.2 (for an e large enough) to compute the Ext's on both sides we see that a natural homomorphism Ext*~a~� a~t~ | V' ~ Ext~a,x,G,(V, V') ,,vg x is induced by the homomorphism of complexes HomQ(C~ I, y,(V)), ~ffx) | V' -+ HomG(C~ ,, y,(V)) V') 9 ,~r 9 , '~" | v' ~ (~ ~ 'v(~), v'). In order to establish that the induced homomorphism in degree d- $| in fact is an isomorphism it suffices, again by applying the above constructed resolution of V', to consider the case V' = ~� But then the map in question simply is ~(v) ~ ~ -. ~(v) w| which, although the ring 9ff x has no unit element, is bijective ([BW] XII.0.3 i)). [] In a more elegant but less precise way these results can be formulated on the level of derived categories. Let D~I(Alg x(G)) denote the bounded derived category of complexes in Algx(G ) whose cohomology objects all are of finite length. Then the functor Dx: D~(Algx(G)) -+ D~(algx_~(G)) V" ~-~ R Homm.� ~x) is well-defined and is an anti-equivalence such that 13� o D x = id. The Duality theorem becomes the statement that R Hom~a~�162 , V") = Dz(V') | V'" for V', V'" in Dh(AIgz(G)). Jf'X These facts constitute a kind of " Gorenstein property" ([Har] V.9.1) for the non- commutative ring ovtax. REPRESENTATION THEORY AND SHEAVES 111.4. Euler-Poincar6 functions In this section we assume that the connected center C of G is anisotropic and hence compact. Then all our previous results hold true without fixing a specific central character Z in advance. Dropping X from a notation has the obvious meaning; e.g. ~ is the Hecke algebra of all locally constant functions with compact support on G. We fix a representation V in Alg(G) of finite length. By II. 3.3 we have, for any other admissible representation V' in Alg(G), the Euler-Poincar6 characteristic EP(V, V') := Y~ (-- l)q.dim Ext~a,(G,(V , V'). ~>~0 It will be a consequence of our theory that this Euler-Poincar6 characteristic is a character value of V'. The character of V' is the linear form tr v, :~ -~C ~-, trace(+ 9 . ; V') which exists since the operator + 9 . on V' has finite rank. In order to see this consequence we fix, for any 0 ~< q ~ d, a set ~ of representatives for the G-orbits in X~. By our assumption on G the stabilizer P~ of a facet F is a compact open subgroup in G. Let Cr : Pt F --> { I } be the unique character such that g((r, c)) ---- (F, r for (F, c) e Xta , and any g 9 P~. We also fix a special vertex x in A and an integer e i> 0 such that V lies in AlgV~')(G). --Fj,~F acts on the finite-dimensional Since U~ *~ is normal in P~ the finite group pttTT~*~ space Vtr~ "). The character of this latter representation will be denoted by "r vF,e : Pt~ --> C. We extend the functions ~-~ and ~v by zero to functions on G. With these notations F, e we define := (- q~O u G ..~"q The volume vol(Ptr) is formed with respect to a fixed Haar measure dg on G. In order to be consistent with our earlier conventions we always let the invariant measure dg on G/C be the quotient measure of dg by the unique Haar measure of total volume one on C. Then the operator ~, . on V', for q~ 9 W, can be written 9 v=fGt~(g).gvdg for v 9 The function f~'p obviously lies in o~. It depends on our choices which, for simplicity, we do not indicate in the notation. We call fvp an Euler-Poincar~ function for the repre- sentation V. If V = C is the trivial representation then fg r is the Euler-Poincar6 function of Kottwitz ([Kot]). Our subsequent results generalize corresponding results in [Kot] w 2. 136 PETER SCHNEIDER AND ULRICH STUHLER Proposition I1"!.4.1. -- For any admissible representation V" in Alg(G) we have: trv,(fVp) ~-- EP(V, V'). Proof. -- According to II. 3.2 we have Ext~,g,o)(V , V') ---- h*(HomQ(C~ yc(V)), V')). But each C~ 7,(V)) decomposes as a G-representation into c~ v.(v)) = 9 C~r(F, v,(v)) F ~ ,,arq where C~ y~(V)) := subspace of all those chains with support in the union of the G-orbits of the oriented facets with underlying facet F. We therefore obtain (-- l)q.dim Homa(e~ y,(V)), V'). EP(V,V') = Z Z q=O F ~ ~q~-q Consider now a single facet F ~ .~'q and fix an oriented facet (F, c). Using the oriented q-chains oIF, c),v introduced in III. 1 we have the isomorphism HornG(C~ y,(V)), V') =-~ ,, ,v(/' V,U~"),, Home~v , ,v ~ (~ ~ 'v(~0,F.,,,o)) where the exponent e F on the right hand side stands for the er-eigenspace of P~. It is a standard fact from the representation theory of finite groups ([CR] (32.8)) that dim Home(V U(~'), V'U~')) " ---- vol(Ptr)-1.jp~ ~-~v 9 ~, ,. ~'~. dg. Therefore our assertion finally comes down to the following general observation: For any function d/~ ~ which is supported on P*~ and is constant on the cosets modulo U~ *) one has " F, e dg trv,( ) = fp, +._v ([Car] p. 120). [] The real number (-- 1) ~. vol (PtF) - 1. dim V t-(') f~(~) = z y. REPRESENTATION THEORY AND SHEAVES 137 is independent of the choice of the sets a~-. Moreover the invariant measure dV g := f p(1).dg on G does not depend on the choice of the Haar measure dg. We call it an Euler-Poincard measure for V. The corresponding volume function is denoted by vol v. In case of the trivial representation V-----C the measure dCg is the canonical measure of G in take sense ofSerre ([Ser] 3.3). We recall that d c g is nonzero with sign (-- 1) a ([Ser] Prop. 28). Serre's " Euler-Poincar6 property " of dCg has a counterpart for any V. Proposition hi. 4.2. -- For any cocompact and torsionfree discrete subgroup P in G we have volv(P\G ) = Y,, (-- 1)*.dim He(P , V) ---- Y, (-- 1)*.dim Ext~tr~(V , C). e~o ,>/o Proof. -- Since V is admissible the spaces V re,') are finite-dimensional. Moreover 1-' being torsion free and cocompact acts freely on X, with finitely many orbits. Hence G-~'(Xcr y,(V)) is a finitely generated free C[P]-module. We therefore can use the resolution in II.3.2 in order to compute the homology groups H.(P, V) and we see that those groups are finite-dimensional and vanish in degrees > d. The second identity is also clear from that. We obtain (compare [Ser] p. 140) Y~ (-- 1)*.dim n~(P, V) = 5] (-- 1)*. Y,, dim V v(,') ~...~> 0 a~ > 0 P ff I"~Xq v(,~) = 1~ (-- 1)*. ]~ dim V . ~r\G/P~ ,>.~ o F E.~r -----vol(l'\G). ]~ (--1)*. E vol(P~)-l.dimV U(') ,~o 1~ Esrq = vol (r\G). [] If K has characteristic 0 then a discrete subgroup P as in Proposition 2 always exists ([BH] Thin. A and Remark 2.3). Hence in this case the measure dVg is uniquely determined by the representation V (and does not depend on the choice of U.). I.i We also introduce the rational number fVp(1) dEp(V) := ; it fulfills d v g = dEp(V ) . d c g. The denominator of d~v(V ) is bounded independently of V. If K has characteristic 0 then as a consequence of Proposition 2 the number dEp(V ) only depends on V; in the rare case that V is finite-dimensional we have dEv(V ) = dim V ([Ser] p. 85). We f0p(1------~ 138 PETER SCHNEIDER AND ULRICH STUHLER therefore call dsp(V ) the formal dimension of V. In a moment it will be seen that this is compatible with the notion of the formal dimension (or degree) of a square-integrable representation. Remark 1II.4.3. -- If V lies in Alge~(G) then we have: i) EP(o~(V), V') = (-- 1)a-~~ V')for any admissible representation V'; ii) ]~ (-- 1)q.dim Ha(P , o~ = (-- 1) a-~~ ~ (-- 1)*.dim Ha(P , V) for any cocom- q~>o ~>~o pact and torsionfree discrete subgroup I' in G; iii) d~,(@(~)) = (-- 1)a-~~ ) /f K has characteristic O. Proof. -- In the proofs of the above two Propositions we used the chain complex (C~ -(,(V)), 0) whose only nonvanishing homology is V in degree 0. On the other hand we know from 1.5 i) and 3.1 that the only nonvanishing homology of the cochain complex :r'~ ~, ~ c.I,Y,(V)),d) is @(V) in degree d ~| [] Associated with any irreducible square-integrable representation V is a unique Haar measure d v g on G which makes the Schur orthogonality relations hold (compare [Car] p. 122 and p. 131); dvg is called the formal degree of V. Proposition 11-1.4.4. -- If either V is irreducible supercuspidal or V is irreducible square- integrable and K has characteristic 0 then we have d v g = d v g. Let us draw immediately the following consequence which reproves results of Harish-Chandra, Howe, and Vigneras. Corollary 1II.4.5. -- If either V is supercuspidal or V is square-integrable and K has characteristic 0 then the rational number dEp(V ) only depends on V and has sign (-- 1)a; its denominator is bounded independently of V. Proof. -- The only additional observation which we have to make is that dEp (.) is additive in short exact sequences. [] The proof of Proposition 4 requires the abstract Plancherel formula. Let G denote the unitary dual of G, i.e. ([Car] p. 133), the set (of isomorphism classes) of preunitary irreducible representations in AIg(G). For any ~ e~Y the Fourier transform ~ is the function on (~ defined by ~(V') := try,(+ ). Since the group G is of type I([Car] p. 133) the abstract Plancherel formula ([Dix] 18.8) is available; it says: -- d/(1) = Ia~(~) dff for any + ~ where d~ denotes the Plancherel measure (cor- ~Lt responding to dg) on (~; -- a V" in G is square-integrable if and only if vola~({ V' }) > 0 in which case we have g = vol (( V' }).dg. REPRESENTATION THEORY AND SHEAVES 139 Let us now first consider the case of an irreducible supercuspidal representation V. Since V is a projective object in Alg(G) ([Gas] 5.4.1) Proposition 1 implies that 1 if V' ~ V, (fVP)^(V')---- 0 otherwise. Inserting the function fv e into the abstract Plancherel formula therefore gives d v g =fEvp(1).dg ---- vol~({ V }).dg = d v g which proves Proposition 4 in the special case under consideration. The argument in the other case is the same once we use the following two additional facts. Firsdy the support of the Plancherel measure is contained in the tempered irreducible represen- tations ([Be2] Example 4.3.1). Secondly we have the following result. Theorem Ill. 4.6. -- Assume that K has characteristic O. If V is irreducible square-integrable and V' is irreducible tempered then 1 ,f v' ___- v, EP(V,V') = 0 otherwise. Proof. -- The vanishing assertion is a consequence of the subsequent Theorem 21 and [Kal] Cor. on p. 29. In the case V' ---- V we have to show, again by Theorem 21, that f 0v(c-1).0v(C) de ----1; here C 'u denotes the set of regular elliptic conjugacy ell dc classes in G and de the natural measure on it ([Kal] w 3 Lemma 1). But this can easily be deduced from [Kal] Thin. F and [Kal] w 5 Prop. 3 [] There are two more consequences of this type of arguments. Corollary HI. 4.7. -- Assume that K has characteristic O. If V is irreducible tempered but not square-integrable then d~.p(V) = 0. Proof. -- More generally Theorem 21 and [Kal] Cor. on p. 29 imply that, for V and V' irreducible tempered, we have EP(V, V') = 0 unless V and V' are relatives in the sense of [Kal] p. 10; the latter means that there is a representation which is parabolically induced from a square-integrable represen- tation of a Levi subgroup and of which V and V' both are constituents. None of these finitely many relatives is square-integrable ([Kal] Lemma 1.4). Using Proposition I we see that the Fourier transform (fVp)^ has support in a set of Plancherel measure 0. Hence the abstract Plancherel formula says that fVp(1) = 0. 140 PETER SCHNEIDER AND ULRICH STUHLER Recall that if V is irreducible square-integrable then a function + E ~ is called a pseudo-coefficient for V if 0 for V' irreducible tempered but V' g V, try'(+) = 1 for V'~-V. Corollary III.4.8. -- Assume that K has characteristic O. If V is irreducible square- integrable and lies in AI~o(G ) thenfVi , and (-- 1)d-#e.fff~ g~ are pseudo-coefficients for V. Proof. -- Proposition 1, Remark 3 i), and Theorem 6. [] It is very likely that the restriction to characteristic 0 is unnecessary in all of the above statements. Actually we strongly believe that Ext~og~o~(V, V') = 0 for any * t> 0 holds true whenever V and V' are two irreducible tempered representations which are not relatives in the sense of [Kal] p. 10. A possible strategy to prove this would be the following. Define an appropriate category Temp(G) of tempered representations and show that the Ext-groups in Alg(G) and in Temp(G) of any two admissible tempered representations (which naturally belong to both categories) coincide. Next we will discuss the orbital integrals of the Euler-Poincar6 functions fvp and their relation to the character trv as a locally constant function on the regular elliptic set. Recall that an element of G is regular elliptic if its connected centralizer in G is a compact torus. For the sake of completeness let us include the following well-known fact. Lemma HI. 4.9. -- If h ~ G is regular elliptic then the map G~G g ~g-lhg is proper. Proof. -- It suffices to show that the preimage of a compact subset of the form Ug0 U where U _c G is a compact open subgroup is compact. This preimage equals [.] GhgU g E oh\o/tr a-zhgC Ugo U where G h denotes the centralizer of h in G. The element h being regular elliptic its centralizer is compact. Hence the double cosets G h gU are compact. On the other hand it is a particular case of Lemma 19 in [HCD] that the set over which the above union is taken is finite. [] We denote by G ell the open subset of all regular elliptic elements in G. The above Lemma says that for each h ~ G ~ and each + ~ ~ff the integral + (h) = +(g-1 hg) dg REPRESENTATION THEORY AND SHEAVES 141 exists. As another consequence of that Lemma the set X h of fixed points of a given element h e G en in the Bruhat-Tits building X is compact (compare also [Rog] Lemma 1). To see this fix a facet F e #'.. The Lemma says that the set {g ~ G :g-' hg e pt }/pt = {g e G: h e Pt F }/Pry ={gEO:X nngF+ O}/P~, is finite. It follows that X n is covered by finitely many facets and hence is compact. We have hF=F for a facet F~X, if and only if F(h):=FnXh+0; moreover, as explained in [Kot], F(h) then is a polysimplex whose dimension fulfills ~(h) = (-- 1) ~-"'m~'~'. In this way X h is a finite polysimplicial complex; we denote its set of q-dimensional facets by (Xh),. Lemma Ill. 4.1O. -- For any h e G *n we have (fvx,)V(h) = ~ N (-- 1)'. v--~-, ,(h). Proof. -- We first quite generally consider a function d/~ ~f' whose restriction to P~, for some facet F ~ X,, is a class function and which is zero outside of pt. Let (X~) h denote the set of fixed points of h in X~ and let (G.F) h be the intersection of (X~) h with the G-orbit G. F of F in X~. We compute d?(g-X hg) dg = Y~ d?(g -1 hg).vol(G~ gP t) fG 9 e c,\G/P~ ~r- l hg ff P~ = Y~ +(g-' hg). vol(P;). [Gh: Gh c~ P,tF] oF E Gh\CG. 7# = vol(P;). X; +(g-' hg). Applying this to each summand of our function fv we obtain = ('r~,,. ~,) (g-' hg) q=O FG,~q cF ~ (O.F)h = N 2~ 2~ (-- 1)".%F(h).-rvF,,(h) q=0 FG..~q gFff(tl.F) h ---- X ~ (-- 1)q.~F(h).v[,(h) q = 0 F f3 (Xq) h = Z ~] (-- 1)a.'r~,,(h). [] An element in G is called noncompact if it is not contained in any compact subgroup. 142 PETER SCHNEIDER AND ULRICH STUHLER Remark m.4.11 (The Selberg principle for Euler-Poincar6 functions). -- For any noncompact semisimple element h ~ G we have fo,\o f p(g-1 hg) = 0 where dg' is any Haar measure on the centralizer Gh of h in G. Proof. -- Note that since semisimple orbits are closed their orbital integrals exist in any characteristic. The Euler-Poincar6 function fvp vanishes on noncompact elements. [] Lemma III.4.12. -- The function (fvp)v on G ell is locally constant. Proof. -- This is a consequence of Lemma 10 once we know that for any given h e G *u there is an open subgroup U __c_ G such that X ~'--X n for any h'EhU nG on. First note that X h is never empty ([Tit] 2.3.1). We choose a point y e X h. Since X n is compact we find a constant r > 0 such that x { z ex:d(y, z) < r}. Consider now the open subgroup U:={geG:gz-=z for any zeX with d(y,z)~ r}. Clearly, for any h' E hU, we have Xh~ X h' and Xh'\X h _c { z e X : d(y, z) > r }. Since with any z e X h' the whole geodesic [yz] is contained in X h' the latter inclusion forces Xh'\X h to be empty. [] We define an equivalence relation on G ~u by h,~h' ifXh=X h'. In the preceding proof we have seen that the corresponding equivalence classes are open. Hence any function ~ E o~ ~ with support in G en can in a unique way be written as += 5: h ~ Gellf ~ where d& e ,~' has support in the equivalence class of h. Also the function q-- 0 F(h) E {Xhlq REPRESENTATION THEORY AND SHEAVES 143 in o-~ only depends on the equivalence class of h; here, for any compact open subgroup U _c G, cv denotes the idempotent ~tr(g) := { v~ if g ~ U, 0 otherwise inaff. Lemma 111.4.13. -- For any ~? ~,~ with support in G en we have +,, try( fot~(g) (fvv)V(g-~) dg = Y~ h E Gell/~ Proof. -- It is clear ([Car] p. 120) that 9 ~..(h) = tr~(h(~.~)) holds true for any facet F in X such that F c~ X h + 0. Using this together with Lemma 10 we obtain (fvp) v(h-1) = trv(h(eh)) -= trace (v ~--~ foch(h-~ g) gv dg) = trace (v ~ fo~dg) hgv dg) = trace (v ~ h(eh * v)). Therefore the left hand side in our assertion becomes fo qb(g).trace(v ~-~g(%, v))dg = trace (v ~-'fa ~b(g).g(%, v)dg). If t~ has support in the equivalence class of some h E G en the last expression obviously is equal to ) dg)=trace(v~--,+, (ah,v))= trv(+,~h). [] Lemma HI. 4.14. -- For any h e G eu and any F0(h ) e (Xh)o we have ~;~ , ~ = ~;~. Proof. -- Let F0(h ) = {y } be the given vertex of X h. We introduce a relation between facets F and F' in X as follows: We write F -~ F' if -- V(h), ~. V'(h) + ~, -- F' ~ F (equivalently F'(h) _ F(h)), F' 4: F, and -- there are points z e F(h) and z'e F'(h) such that z e [yz']; trace(v~fod?(g).g(an*v 144 PETER SCHNEIDER AND ULRICH STUHLER moreover in this situation F, resp. F', is called y-large, resp. y-small. These notions have the following elementary geometric properties: I. Any facet F 4:F0 with F(h) 4= 0 is either y-large or y-small. 9. A y-large facet is not y-small and vice versa. 3. For any y-small facet F' there is a unique y-large facet F such that F-~ F'. 4. If F is y-large we have (- 1)"lm~ + X (- 1).~m~,~h~ = 0. In order to see these properties consider an arbitrary point y' E X ^ different from y. The whole geodesic [yy'] then belongs to X h. In addition one has: 5. There are only finitely many facets F0, ..., F,~ in X such that F, n [yy'] 4: O. Indeed, X is locally finite. 6. Each intersection F~ ~ [yy'] either consists of one point or is an open convex subset of [yy']. Choose an apartment A'_ X which contains [yy'] and therefore each F~. Let ( F( ) denote the affine subspace of A' generated by F d note that F( is open in ( F, ). If the intersection F( n [yy'] consists of more than one point then [.yy'] ~ ( F~ ). The enumeration F0, ..., F,. obviously can be made in such a way that d(y, z,) < d(y, z,+l) whenever z, e F, n [yy'] and z,_,, E F,+ 1 n [yy']. Then the intersections consisting of one point precisely are the F2, t~ [yy'] for 0~< i~< -~ and we have F 0c_ F1- F2-~ F3=- F4~- --- It is clear that F,._ 1 -~ F,. if F,. n [yy'] =: {y' }. The converse holds in the following stronger form. 7. If F ~ F,. then F = F,._ 1 and hence F,. n [yy'] = {y' }. Let z E F(h) and z'e F,.(h) be points such that z e [yz']. Choose A' as above so that F= _~ A' and hence [yz'] u F _c A'. Applying the preceding discussion to z' we obtain that (F) contains [yz'] u F., and hence [yy'] and F.,_ 1. We also obtain that F,. n [yz'] -- { z' } so that [yy'] cannot belong to (F,.); this means that F,. n [yy'] = {y' } or in other words that F,.-a - F,.. As a consequence ( F,._~ ) contains [yy'] u F,. and hence [yz'] and F. Therefore F and F=_~ must be facets of the same dimension both having F,. in their boundary. Assuming F and F,._ I to be different there would exist an affine root for A' which is 0 on F,. but has different signs on F and F,._ 1. On the other hand because of F n [yz'] 4: O, reap. F=_ 1 n [yy'] 4: 0, this affine root has the same sign on y and on F, reap. F,._ 1, which is a contradiction. REPRESENTATION THEORY AND SHEAVES 145 This implies 3. together with the following characterization. 8. A facet F' 4: F 0 in X such that F'(h) 4:0 isy-small if and only if F' c~ [yz'] == { z' } for some (or any) z' r F'(h). It also implies the direct implication in the following analogous characterization of y-large facets. 9. A facet F # F 0 in X such that F(h) # 0 is y-large if and only if F n [yz] is open in [yz] for some (or any) z ~ F(h). For the reverse implication let A'c X be an apartment which contains ~z] and let L ___ A' denote the affine line generated by [yz]. It follows from I. 1.5 that rn L ~ X h. The intersection (F\F)c~ L consists of exactly two points and those belong to X h. Taking as y' that one of bigger distance to y we obtain, with the previous notations, that F --~ F,. Clearly 6., 8., and 9. imply 1. and 2. It remains to discuss the property 4. Fix a y-large facet F. In particular F ~ F 0. _-'F'_ "F" 10. Let F" and F' + F be facets such that F" c F, c ~, and F'(h) ~: 0; if F-+ then F -L F'. We choose points z e F(h) and z" e F"(h) such that z e [yz"]. We also choose an apartment A'c_ X containing y and F and therefore also F' and F". Fix a point ~" e F'(h) and consider the euclidean triangle in A' with verticesy, z", and ~'. It follows that (z~') ::= [z~']\{ z, 7' } is nonempty and is contained in F(h). Fix a point ~e (z~"). The two affine lines in A' through y and ~ and through z" and 7' intersect in a point z' ~ F'(h). Then~ e [yz'] and hence F -~ F'. The last statement means that Y:=union of all F'(h) where F'___F, F'~e F, and not F~F' is a subcomplex of F n X h. Since the latter is contractible we have Z (-- 1)d'~ r'(h) = 1. F'--q~ ]~"(h) ~ o The property 4. therefore is equivalent to Z (-- 1)aJmv'(a) = 1. , F'ff~) _r Y The latter certainly holds if we show Y to be contractible. We may assume thaty and F are contained in the standard apartment A. Let < F > denote the affine subspace of A generated by F. Since F is y-large we necessarily have y e < F >. Let H~, ..., H, _r A be affine root hyperplanes such that the Fx, ..., F, defined by F, : H~ n 19 146 PETER SCHNEIDER AND ULRICH STUHLER precisely are the codimension 1 facets ofF. For cacti H, fix a defining affine root cq(. ) + t, in such a way that F ={xc(F):~,(x) +t, 2>0 for any 1~< i~< s}. Since F 4= F 0 the set I :={ 1 ~< i< s : g,(y) +t,.< 0} is nonempty. Using [Bou] V. 3.9.8 ii) one sees that the ~, for i e I restricted to the linear subspace parallel to ( F ) are linearly independent. Hence there is a facet F' _c such that fl- ,~zF, 9 Since h fixesy and F it permutes the F, with i c I and fixes F'. This means that F'(h) 4 ~ 0. Once we show that Y = 0 n X h iEI it is then clear that Y can be contracted to any point in F'(h). Consider first a point xcF~nX h for some ieI. Then oq(x) + g, -= 0 and ~,(y) + l, ~< 0. This implies (~,(.) +gi) l[yx]<<. 0 and hence [yx] nF=0. If F_c ~ is the facet containing x then it follows that not F ~ F. We conclude that x e ~'(h) _~ Y. Now let x be a point in Y. Then [yx] n F = 0. Moreover we have (~,(-) +t,)] [yx]\{x}>0 for any ir The case x =y is clear since, in that case x 9 F~ for all i 9 I. Assume therefore that x 4: y and that x is not contained in the right hand side of our claimed equality. Then =,(x) +t,> 0 for any i e I. Hence we would find a x'e [_yx], x'+ x, such that ,ti(x' ) + g, > 0 for any 1 .< i ~< s. The latter means that x' 9 F which is a contradiction. This finishes the proof of the properties 1.-4. It follows from 1.-3. that eh = ~u~, "~. + Y' ((-- 1) "'=''~' ~-~,,) + Z (-- 1) ~='''~' e~:~). F y--lart~ F~F' Because of 4. it is therefore sufficient to show that e,c,~. * eo~,~ = so~;) * ~;~, whenever F -~ F'. Fixing a pair F ~ F' we have to check that U~,~ U~ U ~.~ 11',' F0' ~ ~- Fo'~F' REPRESENTATION THEORY AND SHEAVES 147 holds true. The inclusion U~', ) _c U~, ~) is clear from I. 2.11. Hence it remains to establish the other inclusion U,., c U~.~ U~' This is a variant of I.3.1. We may and will assume that the facets Fo, F and F' lie in the basic apartment A. Let z ~ F(h) and z'~ F'(h) be points such that z ~ [yz']. It is trivial to see that, for , ~ @~a, we have f~(~) >~f~,(~) and hence U~*' c~ U~ _= U(~? except possibly in the case fF(~) =fF,(~), :c [F not constant, but a iF' constant. In that case we have -- e(y) < -- .(z) < -- ~(z') -----fi(e). If fFo(~) ~< -- e(z') then f;(e) /> fx~o(~) and hence U(v e, n U~ _c Otherwise there are two consecutive values g < t' in P~ such that t < -- ~(y) < -- ~(z') < t'. Using 1.2.10 we obtain TT{e) A U~ = Ug t,.U2~,~t,+ -- - U~ ~ n U~ [] ~Fo , r 9 To get further we need the subsequent result which doubtlessly holds true in general but which we can establish, at present, only under some additional assumption. Let [ogf, og'] be the additive subgroup of .gf generated by all commutators ~b, q) -- r for % tp ~ o~f and put oVf "b : ---- o~f/[o~f, ogf]. Proposition 111.4.15. -- If G is split or if K has characteristic 0 then the class of fVv in ~b and hence the function (fv)v is uniquely determined by the representation V (and the Haar measure dg) and does not depend on the choice of U(~ ~. Proof. -- Proposition 1, [Kal] Thm. O, and [Ka2] Thm. B. [] Theorem III.4.16. -- If G is split or if K has characteristic 0 then we have try(+) = fa +(g) (fvv) V (g-~) dg for any qJ ~ ~ with support in G% Proof. -- By Proposition 15 we may choose the number e as large as we want. Let hi,..., h,~ e G ~a be representatives of those equivalence classes which meet the ~Fo'TT'e) 148 PETER SCHNEIDER AND ULRICH STUHLER support of + and, for each 1 ~< i ~< m, fix a F~(h~) e (X~)0. We now choose e large enough so that +hi is U~]-right invariant for any 1 ~< i ~< m. This means that +~i * *v~]) = +hi and hence, by Lemma 14, that +h~* ah~ = +h~. The statement follows then from Lemma 13. [] Of course we know from Harish-Chandra that the distribution trv is given by a locally constant function on the regular semisimple subset G ~ in G; let 0 v denote the restriction of that function to G en. The last Theorem can then be rephrased by saying that under the assumption made there we have 0v(h) = (f~)V(h-X). This might be viewed as a kind of explicit formula for the character values on the regular elliptic set; compare also the Hopf-Lefschetz type formula in IV. 1.5. Corollary HI. 4.17. -- If G is split or K has characteristic 0 then we have: i) Ov(h -1) = Ov(h ) for h e G~n; ii) 0,,~)= (-- 1)a-~~ /f V lies in Alg~o(G). Proof. -- i) We obviously have fVp(g-1) =fV(g) for any g e G. ii) It follows from Proposition 1 and Remark 3 i) that trv,~j~. P _ (_ l)d-~o.fvp) = 0 for any admissible V'. Hence that function is contained in [~,~] by [Kal] Thm. 0 and [Ka2] Thm. B, respectively. [] J~mma HI.4.18. i) EP(V, V') = EP(V', V)for V and V' of finite length in Alg(G); ii) let | C A be a proper subset and let E be a representation of finite length in Alg(Mo) ; then EP(V, Ind(E)) = 0. Proof. -- i) The symmetry is obvious from the expression = 1 e 1 $-x v v' EP(V, V') ~ ~ (--) .vo (PF) . ag which was given in the proof of Proposition 1. Of course e >/ 0 here should be chosen u(e) C. in such a way that both V and V' lie in Alg 9 (_). ii) The subsequent argument is due to Kazhdan. The set of unramified characters of M o is a complex algebraic torus of dimension d -- $O (compare [Car] 3.2). The function REPRESENTATION THEORY AND SHEAVES 149 on this torus is regular according to [BDK] w 1.2. Using Proposition 1 we see that the function ~-~ EP(V, Ind(E | ~)) is regular and integral valued; therefore it is constant. On the other hand it is shown in [Ca2] A. 12 that Ext~,o~(V , Ind(E | ~)) = Ext~.~(~o,(V~o , E | ~) holds true. But for any character ~0 such that the central torus So in M o acts on the Jacquet module V~o and on E | ~0 by different characters we have Ext~,,,o)(V~o , E | ~0) = 0 ([BW] IX.1.9). [] Lemma 1II.4.19. --- If K has characteristk 0 then we have dg h\o ~g, = 0 for any h e fo fVp(g-1 hg) G~g\G 'n. Proof. -- Proposition I, Lemma 18, and [Kal] Thm. A. [] Following [Kal] we put A(G) :: { qj eoCf :fo +(g_Xhg ) dg = 0 for any h e G~'\G "n } h \G ~g' and A(G) := A(G)/[.~,.,~V]. Let R(G) := Grothendieck group of representations of finite length in AIg(G) (w.r.t. exact sequences) tensorized by C. The induction functor Ind(.) induces a homomorphism R(Mo) -+ R(G) for any subset 0 ~ A. We put R~(G) := E image of R(Mo) and R(G) := R(G)/R~(G). OCA If K has characteristic 0 then it follows from Proposition 15 and Lemma 19 that R(G) -+ ~,(G) class of V ~ class offVl, is a well-defined homomorphism; as a consequence of Proposition 1, Lemma 18, and [Kal] Thin. 0 this map is trivial on RI(G ). 150 PETER SCHNEIDER AND ULRICH STUHLER Proposition 111.4.20. -- If K has characteristic 0 then the map R(G) ~-~ A(G) class of V v--> class of fVp is an isomorphism. Pro@ -- It follows from Theorem 16 that, up to the substitution h ~h -1, the map in question is the inverse of the isomorphism in [Kal] Thm. E. [] Our approach allows to establish a kind of orthogonality formula for characters which was conjectured by Kazhdan and which generalizes [Kal] Cor. on p. 29. Let C ~n denote the set of all regular elliptic conjugacy classes in G; then +, for any + e ~, as well as 0 v can be viewed as functions on C en. According to [Kal] w 3 Lemma 1 there is a unique measure dc on C eu such that d? dg = climb(c) dc for any t~ e ~ with support in G "n. Theorem 111.4.21 (Orthogonality). -- If K has characteristic 0 then, for any two representations V and V' of finite length in Alg(G), we have fO Or(C--l).Ov,(C )dc = EP(V, Vt). ell Proof. -- According to [Kal] Thm. F we have the identity try'(+) = ,n 0v'(c)" +(c) dc for any function + e A(G). The Lemma 19 allows to apply this identity to the function d/ =f~p. Using Theorem 16 we obtain trv(fNVp) = fGel I 0V(C-- 1). 0V ,(C) rig. It remains to apply Proposition 1 to the left hand side. [] By Lemma 18 the Euler-Poincar~ characteristic induces a symmetric bilinear form EP(., .) : R(G) x R(G) -+ G. Because of Theorem 21 this form coincides with the form [., . ] considered on p. 5 in [Kal] provided K has characteristic 0; hence it is nondegenerate in this case. As is pointed out in [Clo] w 5 a better undcrstanding of this form is tied up with the study of the L-packets. Finally we want to relate our concepts to the notion of the rank of V introduced by Vigneras ([Vig]). She extends the formalism of the Hattori-Stallings trace to the REPRESENTATION THEORY AND SHEAVES 151 context of smooth representations. The technical difficulty which arises is that the Hecke algebra .OF has no unit element in general. For us it is most natural to work with the subalgebras o,~(e) := ~ui,~ , ~, ~,) for e t> 0 in which ~v~*~ is the unit element (recall that x is a fixed special vertex); by I. 2.9 we have 9~' = U ~,~(e). e~>0 The point is that UI, *1 fulfills the assumptions of [Ber] 3.9 as we noted already in I. 3 so that the functor AIgU~')(G) _Z_> category of unital left ~,Vg(e)-modules V' ~ V'(e) := (V') of*) is an equivalence of categories which in addition ([Ber] 3.3) respects the property of being finitely generated. First let V' be a finitely generated projective representation in AlgV~')(G); then V'(e) is a finitely generated projective ~ff(e)-module and we have the obvious isomorphism Homae,,l(V'(e),.,Cd'(e)) Q V'(e).~ Endae,,,(V'(e)). .~(e) If E v~ | v, is the element in the left hand side which corresponds to the identity endo- morphism in the right hand side then the rank of V' is defined as r v, := class of Z v~ (v~) in ~,b. Now consider an arbitrary finitely generated representation V' in Alg(G). Bernstein has shown ([Vig] Prop. 37; alternatively we can use II.3.2) that V' has a resolution 0 ~V~ -+ ... -+V~ -+V' -+0 by finitely generated projective representations V~ in Alg(G). Choosing e large enough so that the whole resolution lies in AlgV~')(G) the rank of V' then is defined to be r v, := 5", (-- 1)i.rv ,. t~0 It is shown in [Vig] Prop. 39 that if G is split or K has characteristic 0 then the class r v, E w,b only depends on V' and is characterized by the property that trw(rv, ) = EP(V', V") holds true for any irreducible representation V" in Alg(G). Combining this with our Proposition 1 we see that the Euler-Poincar6 functions fvp of our representation of finite length V are representatives of its rank r v. 152 PETER SCHNEIDER AND ULRICH STUHLER Proposition Ill. 4. '2"2. -- If G is split or if K has characteristic 0 then we have r v = class offVv in ,~b. Actually a more precise result holds true. Each individual summand offVv is the rank of a corresponding direct summand of our projective resolution in II. 3.2: As already used in the proof of Proposition 1 there is the decomposition C~r(X(ql ~'e(V))= O c~ , o, , v,(v)). PG.~q Fix a facet F e ~'~ and put V' := C~ ~ , y~(V)). Since V' is projective its rank is characterized by the property that trv,,(rv, ) = dim HomG(V' , V") for any irreducible representation V". But in the proof of Proposition 1 it was shown that the latter dimension is equal to trv,,(vol(P~)-I xv ~r). 9 F, r Hence we obtain that rank of C~ yr = class of vol(ptv)-l.xvv~.~-r in ~t '~. Propositions 2 and 22 together give a different proof, for a representation of finite length, of the dimension formula in the Main Theorem 36 in [Vig]; the positivity statement in loc. cit. is, by the above discussion, trivial for the projective representations appearing in our resolution II.3.2. More importantly we obtain a relation between the rank r v and the trace 0 v which is entirely similar to the case of a finite group ([Hat]). Note that the function + on G en only depends on the class of + in ~ff~b. Theorem HI. 4.23. -- If G is split or if K has characteristic 0 then we have Ov(h ) = (rv)V(h -x) for heG ell. Proof. -- Theorem 16 and Proposition 22. [] IV. REPRESENTATIONS AS SHEAVES ON THE BOREL-SERRE COMPACTIFICATION IV. 1. Representations as sheaves on the Bruhat-Tits building Let V be a smooth representation of G. For any open subgroup U_ G we have the space V v := maximal quotient of V on which the U-action is trivial of U-coinvariants of V. We write v mod U for the image in V v of a vector v ~ V. Fix an integer e >/ 0. In order to simplify the notation we sometimes will suppress indicating REPRESENTATION THEORY AND SHEAVES 153 the dependence on e in the notions to be introduced. Let F be a facet of X. Since l;~*~ ~F is profinite the projection map prr : V ~> V U~(') in III. 1 induces an isomorphism ~ VU?) Vr{,) =, Whenever F' is another facet such that F' _ F then we have the commutative square where p~, is the other projection map from III. 1 and pr is the obvious quotient map (coming from the fact that U~,}~ U~*}). The representation V gives rise to a sheaf V on the Bruhat-Tits building X in ev the following way: For any open subset fl ~ X put V(~) := C-vector space of all maps s: ~ -+ 0 Vu~, ~ such that z~fl -- s(z) c Vu~,) for any z c t, -- there is an open covering fl = [J fl~ and vectors v~ ~ V with ~I s(z) = v~modU~- *) for any z~"/i and icI. The stalks of the sheaf V are the expected ones as we will see in a moment. The star of a facet F' in X is the subset of X defined by St(F') := union of all facets F ~ X such that F' __q F. These stars form a locally finite open covering of X. Lemma IV.I.1. i) (V), = Vv(/) for any z ~ X; ii) the restriction of V to any facet F of X is the constant sheaf with value Vv6,). Proof. -- There is the obvious map (v), vo{,, germ of s ~ s(z). 20 154 PETER SCHNEIDER AND ULRICH STUHLER It is an isomorphism since if z lies in the facet F then St(F) is an open neighbourhood of z with the property that U~/~_c _,,II ~"~ for any z' e St(F) (by 1.2.11 i). The same argument shows more generally that, for any nonempty subset 2; open in F, we have lira> V(f2) = locally constant Vv~,)-valued functions on Y~. [] -~ St(F) open ~F-~ Lemma IV. 1.2. -- Let F be any facet in X; then Vu~,) f* ~0~ H*(St(F), V [ St(F)) -= H*(F, V I F) = ( /f, > 0. Proof. -- This is (a polysimplicial version of) [KS] 8.1.4. [] It follows from Lemma 1 that the functor Alg(G) -+ sheaves on X V~V is exact. Our aim in this Chapter is to compute the cohomology with compact support H~(X, V) of the sheaf V.~ The interest in this comes from the fact that this cohomology can be calculated from the cochain complex of y,(V) considered in III. 1. * h,/Corfx Proposition IV.1.3. -- We have the equality H,(X, V) = ,.~ ,~ ~.,, y,(V)), a). t~J Proof. -- The filtration X = ~0~ f~l~_ ... ~ Oa of X by the open subsets ~2" := X\X "-1 induces the filtration ro(x, .) _= ro( l, .) =_ ... r4n', .). Because of [God] II.4.10.1 the spectral sequence of this filtration reads El"" := (~ H:+'~(F, V [ F) ~ H:+'~(X, V). F~Xn 9 ,~ According to Lemma 1 ii) we have ifFeX, and * =n, H;(F, V I F) ---- ( H~(F,o Z) | Vu~,) otherwise. Inserting this into the spectral sequence we obtain dd- l, O 0 Hd(F, Z) | Vu~,)]. H:(X, V) = h'[F2x ~ S~ Z) | Vv#, ... F E X d The description of the cellular coboundary in [Dol] V. 6 and VI. 7.11 shows that the complex on the right hand side coincides with (C~ y,(V)), d). [] REPRESENTATION THEORY AND SHEAVES Corollary IV.1.4. -- Let V be a representation of finite length in Alg� ~e is chosen big enough then we have oP'(V) = H:(X, ~). Proof. -- Proposition 3 and III. 1.3. [3 The sheaf V at hand, we can reformulate III.4.16 as a trace formula. Proposition IV. 1.5 (Hopf-Lefschetz trace formula). -- We assume that the connected center C of G is anisotropic and that either G is split or K has characteristic O. Let V be a repre- sentation of finite length in Alg(G) and choose e big enough. For any h e G ~1I we have 0v(h ) ----= Y, (-- 1)a.trace(h; H*(X h, V)). q~O Proof. -- First of all note that since X h is compact and V is admissible the coho- mology H*(X h, V) has finite dimension. By III.4.16 (as explained in the paragraph after that Theorem) we have 0v(h ) = (fvp)V(h-'). Moreover III.4.10 says that (fVp) V (h- 1 ) = Y, Y~ (-- 1)*.trace(h; VOW')). Proposition 3 of course is completely formal and applies to the finite polysimplicial complex X h as well. Therefore the right hand side in the last identity is equaI to Y~ (-- 1)Ltrace(h; C~ y,(V))) ~=0 ---- Y, (-- 1)q.trace(h; Hq(X h, V)). [] q=O IV. 2. Extension to the boundary In [BS] Borel and Serre have constructed a compactification X of the Bruhat-Tits building X with the help of which they could determine the cohomology with compact support of a constant sheaf on X. Our strategy for computing the cohomology with compact support of our sheaves V will be similar. In this section we will defne an appro- priate " smooth" extension j, V of V to a sheaf on X. The boundary cohomology of that extension will be discussed in the next section. Finally in the section after the next one it wilt be shown, for V of finite length at least, that j,, o~ V is cohomologically trivial on X. The result about the cohomology with compact support of V will then be obtained from the long exact cohomology sequence. 156 PETER SCHNEIDER AND ULRICH STUHLER We first give a description of the Borel-Serre compactification X which is adapted to our purposes. Let A denote the compactification of the basic apartment A by " the directions of half-lines " ([BS] 5.1). As an explicit model one can take A:={x~A:d(O,x)<~ 1} together with the embedding j:A~A x~ t 1 -- d(0, e -d(~ x) .x if x+ 0, 0 ifx =0. A boundary point x e A| := A\j(A) then corresponds to the half-line [0x) := { rx : r >>. 0 } in A. The N-action on A extends uniquely to a continuous action of N on A. Note that Z acts trivially on the boundary A~. For any boundary point x e A~ we have the parabolic subgroup P, := subgroup generated by Z and all U, for ~ e 9 such that ~(x) >/ 0 in G; its unipotent radical is U, := subgroup generated by all U~ for ~ e@ such that ~(x)> 0; clearly nP, n-l=P,, for heN holds. Moreover [BoT] 5.17 and 5.20 imply that P, nN=N~:={n~N:nx=x} for any xeAoo. These two properties allow to formally imitate the defnition of X by setting X:= G x A/,-~ with the equivalence relation ~ on G x P. defined by (g, x) ~ (h, y) if there is an heN such that nx=y and g-ihneP,. The group G acts on X through left multiplication on the first factor. The map Y~->X x ~ class of (1, x) is injective and N-equivariant. There is an obvious G-equivariant map j:X -~X REPRESENTATION THEORY AaND SHEAVES 157 which is injective. The latter fact follows from the observation that, because P. = U.. N. for x ~ A, we could have used the groups Px instead of U x in the definition of X. On the other hand the boundary X~ := X\X is X~ = G � Ar Hence X~ as a G-set coincides with the Tits building of parabolic subgroups in G (compare [GLT] 6.1). We see that at least as a G-set X is the Borel-Serre compactifi- cation of X. We equip .~ with the quotient topology of the product topology on A given by the natural topology on A and the ::-adic topology on G. Zemm~ IV. 2.1. -- The space X is the Borel-Serre compactification of X. Proof. -- Without loss of generality we may assume that the origin 0 in A is a spccial vertex. Also fix a decomposition (I) = 9 + w (I)- into positive and negative roots; this corresponds to fLxing a fundamental Wcyl chamber D:----{x~A:~(x) t> 0 for any ~O +} in A. Let D denote the closure of D in A. Then the obvious map P0 � b -~ X is surjective. The Borel-Serre topology on X is the quotient topology with respect to this map if the left hand term is equipped with the product topology of the n-adic topology on P0 and the natural topology on D (see [BS] 5.4.1). Therefore the Borel-Serre topology is finer than our topology. But it is also shown in loc. cit. that the former one induces on D its natural topology and that the G-action on X is continuous in the Borel-Serre topology. This implies that the two topologies under consideration actually coincide. [] We have ([BS] 5.4): -- X is compact and contractible; -- X is open in X and the topology induced by X on X is the metric topology of X; -- X~o with the topology induced by X is the :z-adic Tits building of G ([BS] w 1) ; -- the topology induced by X on A is the natural topology of A; -- the G-action on X is continuous. In the following we keep the assumptions and notations introduced in the proof of Lemma 1. One advantage of viewing X as the quotient of P0 � ~j is that since is a fundamental domain for Po in X ([BS] 4.9 iii)) the equivalence relation ~ for (g, x) and (h,y) ~ P0 � ~) simplifies to (g, x) ~ (h, y) if and only ifx=y and g-lheP0np,. For later purposes it is necessary to explicitly construct appropriate neighbourhoods in X of any point in the boundary X,. Since D~, := D\D is a fundamental domain 158 PETER SCHNEIDER AND ULRICH S'I'UHLER for Po in X,o it suffices to consider a point x e D,o which is fixed throughout the following. The set St.(x) :={x' eDoo :P., _c p=} is an open neighbourhood of x in Do. Put r := { ~ 9 r : ~(x) > o } _ r We also fix an open normal subgroup U in P0 and a real number r >/ 0 such that U nU~_D U=,, for any ~eq)(x). Lemma IV.2.~,. -- Let f~ c_ Stw(X) u{y eD: a(y) > r for any o: er } be any subset; then the subset U(P0 n P=) x ~ is ,,~-saturated in Po � i). Proof. -- Consider a point y 9 and elements g e P0 and h 9 U(P o n P=) such that g-lh e P0 n P~. We have to prove that then necessarily g 9 U(Po n P,). Since by assumption g c h(Po n P,) c_ U(Po n P=) (Po n P,) it suffices to show that Po n Pv -~ U(Po n P=) holds true. In case y ~ StD(x) we even have Py _c p=. Therefore we may assume that y 9 with ct(y) > r for any a 9 According to 1.1.2 and 1.1.4 we have Po n P~ = Pio~j = l-I U= Stoyj,~" Nf~l r ~ ~red for an appropriate ordering of the factors on the right hand side. Since N acts on A by affine automorphisms Nc0,~ is contained in Nw=, j where x' e Do is such that y lies on the half-line [0x'). But x'e Stw(X); this follows from ~(x')> 0 for any 9 e q)(x) which amounts to U=_ U=,. We obtain Nt~ c_ NLo~, ~ _c Po n P=, _q Po n P=. For a root 0t eO ~ we distinguish two cases: ~ eO\(--O(x)) or at ~--O(x). In the first case we have ~(x) >/ 0 which means Us __c_ P= and hence U~,,Ico~c~,~ _c U,,.o c_ Po n P=. In the second case we have -- e(y) > r and hence U,~. ftoylr ~ U=,_ =~v~ c U=., __c U. [] REPRESENTATION THEORY AND SHEAVES 159 For any subset f~o c StD(x), we now define the subset C,(~o0) := ~o u { y 9 D : y 9 [0x') for some x' 9 f~oo and a(y) > r for any ~ 9 ~(x) } in D. For any x' 9 f~o there is a unique point y' 9 [0x') such that [0x') n C,(~) = [0x')\[0y']. Lemma IV.B.a. -- Let f~o---Stw(x) be an open neighbourhood of x in D~; then U(P 0 c~ Px).C,(f~) is an open neighbourhood of x in X. Proof. -- It is easy to see that C,(f~oo) is open in D. Hence U(Po n P,) � Cl,(f~oo ) is open and w-saturated in P0 x D. [] These neighbourhoods have the disadvantage not to reflect the cellular structure of X. We therefore define Cl',(~o) := f~ u [J { St(y) n D :y 9 C,(fl~o) n D a vertex such that St(y) c~ D _ Cl,(f~oo) }. /,emma IV.2.4. -- Let ~ _c StD(x ) be an open neighbourhood of x in Do~ and put := U(P 0 n P,).C',(fl~) ; we then have: i) G',(fl~o ) is an open neighbourhood of x in I); ii) f~ is an open neighbourhood of x in X; iii) f~ nX-- [.J St(y). V 9 a vertex Proof. -- The set U(Po n P,) � C',(f~o) is open in P0 � D if we assume i) and is w-saturated by Lemma 2. Hence ii) is a consequence of i). Moreover iii) follows from ii). Indeed, by construction fl n X is a union of facets. But any open subset of X which is a union of facets contains with any facet F the whole star St(F). This in particular shows that the right hand side in iii) lies in f~ n X. To see the reverse inclusion first note that the right hand side is invariant under U(P0 c~ P,). It therefore suffices to consider a point z EC',(~)~)riD. Then by definition there is a vertex y 9 C~(f~) n D c f~ n X such that z 9 St(y). The crucial assertion to establish is i). Since St(y) n D is open in D for any vertex y 9 D it remains to ensure that any point x' 9 f~o~ has an open neighbourhood in which is contained in C',(f~,~). For this it is convenient to use certain standard neigh- bourhoods of x' in Doo. Thinking of A~ as being the unit sphere in A (as we do in our explicit model) we have, for any 0 < ~ < 1, the open neighbourhood f~, := { x" eD= "d(x',x")< ~} of x' in Doo. We may choose r small enough so that ~2, _~ fl~. Then C,(~,) is an open neighbourhood of x' in D which lies in C,(~o). Let now c > 0 be a fixed real constant. 160 PETER SCHNEIDER AND ULRICH STUHLER It is an elementary computation to show that by decreasing c and increasing r appro- priately we obtain a C,,(f~,,) _c C,(f2,) with the property that { z' e D : d(z, z') < c } C,(f2.) for any z e C,,(f~,,) n D. We choose the constant c in such a way that d(z, z')< c whenever z E D and z' e U { St(y) n D: y any vertex of the facet containing z }; this is possible by I. 2.10. It follows easily that then C,,(f~,,) _c C',(f~,). [] Lemma IV. 2.5. -- Let f~ c__ X be an open neigkbourhood of x; then we can choose U and r in suck a way that U(P0 n P,).C',(f~) ___ f~ for some open neighbourhood f~ c_ StD(x ) of x in D o . Proof. -- Consider the quotient map ~ : Po � b ~ X. The subset ~t-x(f~) is open in P0 X D and contains (P0 n P~) x { x }. We therefore find, for any h e Po n P~, an open normal subgroup U(h) ___ P0 and an open neighbourhood f~o(h) of x in D such that u(h) k x no(k) _ By the compactness of (P0 n P~)/C we have U(h~) hx C u ... w U(h,.) h,. C _~ Pon P. for finitely many appropriate elements hi, 9 9 h,, e P0 n P,. Now put f~o := f~o(hl) n ... n f~o(h,,) and U := U(hx) n ... n U(h=). We then obtain U(Po n P,) � f~o c V.-~(~) and hence U(Po n P~).f~o c_ fL It remains to observe the elementary geometric fact that for any open neighbourhood f~o ofx in D we find an open neighbourhood f~g StD(x ) ofx in D~o and a r/> 0 such that C,(f~) _c Do. p, Lemma IV.2.6. -- Any boundary point in X~ has a fundamental system of open neigh- bourhoods D in X such that f~ n X = [.J St(y). w~t~x & vertex Proof. -- Lemmata 4 and 5. [] Lemma IV.2.7. -- Let c _ U. be a compact subset; then there is an open neighbourhood D of x in ~, such that r =_ U~ ~) for any facet F c_ f~ n X. __%_ REPRESENTATION THEORY AND SHEAVES Proof. -- (Recall that e >/ 0 is fixed throughout this Chapter.) Fixing an enume- ration of the roots in ~ c~ ~(x) any element g e c can be written in a unique way as g = l-I g~ where g~ ~ U~. The compactness of r implies that for any such root ~ the set t,:={t(g,):gec such thatg~4= 1} is bounded below; we put r : = min { t : t e G }- Define now f~o:={yeA:~(y)>e+ 1--/, for any ~e~ar3~(x)). Clearly there is an open neighbourhood f~ of x in ,~ such that f~ n A = f~0. It therefore suffices to show that tg Uk "~ for any facet F_~D0. Fix a root =t ~ ~ c~ ~(x), an element g E c, and a facet F __q f~o- We actually check that g~ e U,,~ +, __q U~ 'J c~ U, holds true. The case g, = 1 is trivial. Otherwise we have f~(~,) ~< -- inf a(y) ~< -- (e q- 1 --l,) ~< -- e -- 1 q-t(g,,) w Et'l, and hence f;(~) + e < t(g.). [] The sheaf V has the two obvious extensions j, V _ j. V to sheaves on X. We will work with a third " intermediate" or " smooth " extension j,V---~j.,| ,j.V which is constructed as follows. Let i : X~ -+ X denote the inclusion of the boundary. The stabilizer P, of any boundary point z ~ X~ is a parabolic subgroup of G; let U, denote the unipotent radical of P,. For any z e X~o we may form the Jacquet module Vtr ~ of U,-coinvariants of V; similarly as before we write v mod U, for the image in Vtr * of a vector ~ ~ V. Analogously to V we can define a sheaf _V_V on X| in the following way: For any open ~ _~ X~ put V(t2) := G-vector space of all maps s : l) -~ 6 Vv, such that -- s~t'l --s(z) e Vv, for any z eO, -- there is an open covering f~ = L) f~ and vectors v~ 9 V with (EI s(z) =v~modU, for any zet2, and ieI. 21 162 PETER SCHNEIDER AND ULRICH STUHLER It was mentioned already that X~ is a simplicial complex but equipped with a topology which is coarser than the simplicial one. For any point z ~ X~ we put St(z) :={ z' ~X~ :P,, _~ P,}. Remark IV.2.8. -- Let z ~ X~ be a boundary point; for any open subgroup U ~_ G the subset U.St(z) is an open neighbourhood of z in X~. Proof. -- Because of St(z') _~ St(z) for any z' E St(z) it suffices to prove that there is a subset fl _ St(z) containing z such that U .fl is open in X~. We may assume that z c D~o. Choose an open neighbourhood ~ ___ StD(z ) = St(z)nDo of z in D,. According to Lemma 3 the subset U(P0 n P,).~ is open in X,. Therefore f~ := (P0 n P,).fl~ meets the requirement. [] Lemma IV.2.9. -- We have V, = Vv, for any z E X| Proof. -- There is the obvious map Zz ~ VU z germ of s ~-* s(z) which clearly is surjective. In order to see the injectivity let s be a section of V in a neighbourhood f~ _~ X~ of z such that s(z) = 0. By shrinking the neighbourhood we may assume that s is represented by a single vector v ~ V, i.e. s(z') = v mod U,, for any z' e ~. For z' ~ St(z) we have U, _ U,, and hence v rood U,, = 0. Let U be the stabilizer of v in G. It easily follows that actually v mod U,, = 0 for any z' e U. St(z). We obtain s]U.St(z) n~ = O. [] Since the formation of Jacquet modules is exact ([Car] p. 128) Lemma 9 implies that the functor AIg(G) -+ sheaves on X~ V~V_V_ is exact. By construction the sheaf V, resp. V, is a quotient V r-~V, resp. V-~-V, of the constant sheaf with value V on X, resp. X~. The first arrow induces by adjunction and restriction a not necessarily surjective homomorphism V ~i*j.V. REPRESENTATION THEORY AND SHEAVES 163 /,emma IV.2.10. -- We have the commutative triangle /\ v , i.s.v where the upper term is the constant sheaf with value V on X= and the oblique arrows are the natural sheaf homomorphisms. Proof. -- We have to show that, for any point z c X,~, the natural map v (j. v). contains in its kernel all vectors of the form gv -- v for some g e U, and some v ~ V. By the G-equivariance of this assertion we may assume that z c Doo. Choose an open subgroup U ~ G such that v, gv c V v. By Lemma 7 and the fact that (P0 n P,)/C is compact we find an open neighbourhood ~o of z in )k such that { h- 1 gh : h c P0 n P, } ~ U TM for any vertex Yo ~ ~o n X. -- it 0 Consider now a vertex y ~ U(Po n P,) .~0 n X, say, y-~uhyo with ucU, hcPonP,, andyo cY~onX. We then have U~*~ = uhU~*~ h-X u -1 and g' :=- h-l gh cU ~*) and hence gv -- v = u(gu -I v - u -iv) ---- uhg' h-a u -av-v=OmodU~*'. It is quite clear that ~0 contains a subset of the form C',(~oo) as considered above. Using Lemma 4 we therefore see that z has an open neighbourhood ~ in X such that n X = U St(y) ~G~nx & vertex and gv--v=OmodU --y I*l for any vertexycf~nX. If y' ~ fln X is an arbitrary point, say, y' e St(y) for some vertex y e fln X then UI~I ~, = -- li~,) ~ by 1.2 9 11. We obtain that gv--v = 0modU I*1 for anyy'efl nX. --I/" This means that the image of gv -- v in (2". V) (f~) and a forfiori its image in (j, V), is zero. [] 164 PETER SCHNEIDER AND ULRICH STUHLER We now define j.,~o V to be that sheaf on X which makes the diagram j..oo V ,j.V i,V__ - > i.i*j.V cartesian; the right perpendicular arrow hereby is given by adjunction and the lower horizontal arrow is the direct image of the arrow in Lemma 10. By construction we have 9 , 9 3 d.,| V = V and z "* J,,~o " g =- V. In particular the functor AIg(G) -+ sheaves on X V~j., V co~ is exact; of course this functor depends on the choice of the number e whereas the sheaf V does not. We also obtain the short exact sequence of sheaves O~j,V -+j.,| V ~i.V -+0. Since H*(X,j~ V) ----- H:(X, V) ([KS] 2.5.4 i) and (2.6.6)) the associated long exact cohomology sequence reads ... -+ H'(X,j.,~ V) -+ H'(X=, V) -> H',+'(X, V) 1 X 9 ... -+H '+ ( ,),,~oV)-+ (e) Later on it is technically important that for the representation V := C~(G/U= ), x a special vertex in A, our " smooth" extension has a simpler description. Proposition IV.2 11. For the representation V ~ special vertex in A, 9 -- = Co(G/U~ ), x a We haY8 j.,~ V = image(V --+j. V). Proof. -- By Lemma 10 we quite generally have natural homomorphisms V >>j.,~V---~j.V the first one of which is surjective. We therefore have to show that in case of our parti- cular V the second one is injective, i.e. that for any boundary point z 9 X.~ the natural map between stalks Vv, -+ (j. V). REPRESENTATION THEORY AND SHEAVES 165 is injective. Let us first make this map more explicit. Put l~(z) := system of all open neighbourhoods fl of z in X with the property that n n X = O St(y). & vertex Because of Lemma 6 we have (J. v), = lim V(n n X). The sheaf axiom says that, for any ~ e lI(z), the restriction map v(n n x) II v(st(y)) vertex is injective. According to 1.2 the terms on the right hand side are V(St(y)) = Vv~,,. Putting this together we see that what we have to show is the injectivity of the natural map Vu~-+ lim 1-[ Vv~. fl ~ 11(~) ~t~X ~ett*x -~- G (e) In other words fix a function + ~V C,( /U, ) and a neighbourhood ~ e lI(z) such that + = 0 mod II TM for any vertex y e ~) n X. --y We have to cheek that then necessarily + = 0 mod U,. We write ~b as a sum += Z +4 h e u~\G/ui,~ of functions ~b n ~ Co(G/U! ')) in such a way that ~b h has support in U, hU(~I/U~ ~ and we will show that each summand fulfills +h = 0 mod U,. Consider an individual element h e G. There is an apartment which contains both points x and h-Xz in its closure ([Bro] Thin. VI.8). By I. 1.6 we find an element g e P, such that x, gh -~ z e A. Using the G-equivariance of our problem and replacing z and + by gh-~ z and gh-a ~, respec- tively, we see that it suffices to deal with the case z e A and h e P,. We choose ele- ments u~,..., u,, e U, such that hit ~e) , u~ .__= }. hU~e~ supp(+n) = { u 1,.~, , .., Note that u~hll _~ TM = u~ _~ IT (~ h since P~ normalizes _~1I c*~. According to Lemma 7 we find an open neighbourhood ~' of z in ~, such that ~' ___ ~ n A and ux, .,u m~U (~) for any vertexye~' AA. 9 . ~Lr 166 PETER SCHNEIDER AND ULRICH STUHLER In particular we have = 0 mod IT TM for any vertex y ~ f~' n A. On the other hand it was shown in I. 3.2 that U(e) ~ TT IT (e) for any x' [xz). Choosing x' ~ [xz) close enough to z in such a way that the closure of the facet F which contains x' still lies in f~' and choosing a vertex Y0 ~ F we obtain -- TT(~) 1. 2L 1~ . . .~ 2L m ~ UVO 2. ~ = 0 mod U TM and YO ~ 3. IT ('1 c U(F ~ IF ~) ~ U,.TT ~> The properties 1. and 3. imply that 4, IT (~) hi? ~) (TT (~) hTT TM __,0 ___, n supp(~b) = U, ___,hit 'e) n supp(~b) = ,--,o n U,) .__, ~ supp(~b). It follows from 2. and 4. that IT (e) n U, [] ~b h = 0 mod IT (`) and even ~b h = 0 mod --~0 --VO Appendix. -- Geodesics in X. Implicitly in our thinking about the compactification X is the existence of a unique " half-line " in X between any given point in X and any given boundary point. In the section after the next one we will have to make explicit use of this fact. Also it is the link between [BS] and [Bro] VI. 9. Since we could not find an appropriate reference this will be justified in the following. We fix a point x e X and a boundary point z ~ X~. According to [BT] I. 7.4.18 ii) or [Bro] Thm. VI.8 there is an apartment A' _ X such that x ~ A' and z e (A')~. Let [xz)a. denote the half-line in A' in direction z emanating from x and put [xz]A, := v ( z }. The subsequent result allows to simply write [xz) and to view the latter as the geodesic between x and z in X. Proposition. -- The half-line [xz),' does not depend on the choice of the apartment A'. Proof -- Let A" _~ X be a second apartment such that x ~ A" and z e (A")~. We have to show that [xz)A, = [xz)A,. We first treat in several steps special cases where additional assumptions about A' and A" are made. Step 1. -- Here we assume that x is a special vertex. For notational simplicity we may assume by G-equivariance that A" = A is our standard apartment and that REPRESENTATION THEORY AND SHEAVES 167 z z D~. By I. 1.6 we find an dement h e P, such that A' = hA. Write z = hz o with z0 z A~o. Since x is a special vertex we find an n z N, such that nzo ~ D~o. We then must have hn -a E P, because D o is a fundamental domain for Pz in X~o ([BS] 4.9 iii)). Therefore replacing k by hn-1 we may assume that A'=hA with hEP, nP,. Again since x is a special vertex we obtain from [BS] 4.10 that h fixes the half-line [xz)a pointwise. We now conclude that = [hx, hz)h,, = h[xz), = [xz)A. Step 2. -- Here we assume that the intersection A' n A" contains a special vertex Xo. From the first step we know that [x o z)~, = Ix 0 z)x,,. Hence [xz)A, and [xz)A,, are parallel rays in the sense of [Bro] VI. 9A and therefore have to coincide by the Lemma 1 in loc. cit. Step 3. -- Here we assume that the intersection A' n A" contains a sector D' such that z e (D')~. Choose any x 0 e D'. Then clearly Ix o z)a,--Ix 0 z)a,,. Hence [xz)a, and [xz)a,, again are parallel and therefore equal. In order to establish the general case we will show that there is an apartment A ~ X such that x ~ A, z e A| A' n A contains a special vertex, and A" n A contains a sector D with z ~ ~=. Using steps 2 and 3 we then obtain = [x )x = In order to find A we choose an h e G with A' = hA", a facet of maximal dimension F ~ A" with h-1 x z F, a special vertex x 1 z F ([BT] I. 1.3.7), and a sector D" _c A" such that z z (D")| By [BT] I. 7.4.18 ii) or [Bro] Thm. VI. 8 there exists an apartment A_ X which contains kF and an appropriate suhsector D c D". Then x z hF ~ ~,, z z D~ ___ A~, ~A"nA, and x o:=hxxzA'nA. [] Corollary. -- Any element in Pz n P, fixes [xz] pointwise. IV.3. Cohomology on the boundary In this section we explicitly compute the boundary cohomology H*(Xoo, V) in the case of an induced representation. Throughout the notation introduced in III.2 168 PETER SCHNEIDER AND ULRICH STUHLER will be in order. In particular A _~ r is a fixed choice of simple roots. Corresponding to A we had defined in IV. 2 the subset D~o ~ Xoo, it is a (d -- 1)-dimensional simplex whose simplicial structure is given by the subsets D ~ := { x e Do~ : U, = Uo } for any proper subset | C A. The closure D ~ of D ~ in Do| (equivalently in Xoo) is a (d- 1 -- $| simplex. Since Do| is a fundamental domain for the G-action on X~o ([BS] w 1) we have an obvious projection map "r : X~o -~" D~ = G\X~o ; it is proper and has totally disconnected fibers. The proper base change theorem ([God] 11.4.17.1) therefore implies H'(X=, V) = H'(D| v. V). For any subset | _~ A we introduce the space Ind~,o(Vvo ) := space of all locally constant functions ~:G-* Vvo such that e?(ghu) -= h- l(~?(g) ) for allgeG, heMo, and ueUo on which G acts smoothly by left translations. (This is unnormalized induction!) Lemma IV.3.1. -- For any proper subset | C A the restriction of% V_ to D ~ is the constant sheaf with value Ind~o(Vvo ). Proof. -- The map G/P o x D~ --% ~-~ D~ (gPo, x) ~ gx is a homeomorphism and x corresponds to the second projection map on the left hand side. The inverse image of V on the left hand space can be computed as follows. For any x e D ~ and any g e G we have the obvious map Vvo = Vv~ -~ Voo ~. Also let p:=pr � x D ~ o � D ~ 9 REPRESENTATION THEORY AND SHEAVES 169 An argument as in the proof of 2.9 shows that the inverse image in question can be identified with the sheaf on G/P o � D ~ whose space of sections in an open subset f~ is the C-vector space of all locally constant maps r : p- 1 ~ ~ Vu ~ such that ~(gh, x) =h -lq~(g,x) for any (g,x) eiz -1~ and hePo. On the level of sections this identification is given by s(gx) --- g~(g, x). It is quite clear that the direct image of this latter sheaf under the projection map to D ~ is the constant sheaf with value Indr~ Our assertion follows now by an application of the proper base change theorem. [] Specializing to the case of an induced representation we once and for all fix a subset 19o-~ A and an irreducible supercuspidal representation E of Moo and we put V := Ind(E). For any @_ A we need the subgroup W o:= (s=:x~O) of W; moreover let [W/Wo0], resp. [Wo\W/Wo0], denote the subset in W of representatives of minimal length for the cosets in W/Wo0 , resp. the double cosets in Wo\W/Woo. The Weyl group W acts on the set of roots ~. According to [Ca.s] I. 3.4 two subsets 19 and | in A are associated if and only if | = w| for some w e W. Lemma IV. 8.2. -- For any proper subset | C A the following assertions are equivalent: i) Vuo + O; ii) woo ~ | for some w ~ [Wo\W/Woo]; iii) O contains a subset which is associated to @o. Proof. -- The equivalence of i) and ii) follows from [Cas] 6.3.5. The third assertion is a trivial consequence of the second one. To see the reverse implication assume that WOo~ t9 for some wEW. We then have wE[W,,oo\W/Wo0 ] by [Cas] 1.1.3 and hence Vv, o, 4: 0. But Uo -~ U~oo so that Vtr o 4: 0, too. [] Corollary IV.3.3. -- The support of the sheaf v, V is equal to the (d- I -- $| dimensional simplicial subcoraplex D| := U{D~ wO0-c 0C A for some w ~W} of O~. Let ~< on W denote the Bruhat order. We now fix an enumeration /W/Woo/ = { 1 = Wo, wl, W 2 .'. } 22 170 PETER SCHNEIDER AND ULRICH STUHLER in such a way that m~< n ifw~< w.. For any proper subset | C A, this allows us to define a decreasing filtration F]V:={~elnd(E):c?iPow,~Poo=0 for any m<n} of V by Po-invariant subspaces. It induces corresponding filtrations (F~ V)v o of Voo (forming the Jacquet module is exact!) and F" IndP~ ) :---- Ind~ V)vo) of Ind~ ). The latter clearly is G-equivariant. Most importantly we obtain a G-equivariant filtration F" ~. V of the sheaf ~, V defined by F" - V :--~ subsheaf of all sections s such that s(z) ~ F" Ind~ for any z~D ~ and any | Our further computation is based on the associated G-equivariant spectral sequence E~" := n "+"(D~o, gr~ ~, V) :~ H ~+ "(D~, ~, V) = H "+ "(X| V). First of all we have (gr~.V), =Ind~ for any z~D ~ and | A. (The functor Ind~ is exact by [Car] I. 1.8.) By construction gr~oV=0 ifw. r ]. For any w, e [W/W| let us fix a lifting g, e N. If w, e [WokW / Woo] then the compu- tation in [Cas] 6.3.1 and 6.3.4 shows that (gr~o V)v o = 0 if and only if w, G 0 $ | and 9 . g-Z1 [normalized mductaon of . E] = 8o v~ @ [ from g, Poo g~- 1 c~ M| to Mo] if w, | _c | Since, by [Cas] 1.3.3, we have the Levi decomposition g, Poo g-- 1 c~ M o = M~, Oo. (g, Uoo g,- ~ c~ Mo) , the last formula simplifies to [normalized parabolic induction] if w, O| -~ O. (gr~) V)u O = ~1[2 ~ I_ Of ~nXE from M,,,o0 to M o ] REPRESENTATION THEORY AND SHEAV'ES 171 Using the transitivity of parabolic induction ([BZ] 1.9 (c)) we therefore obtain that if w, e [Wo\W/Wo 0] and w. O0 --= O, (gr~, v, Y)'- = { Ind(g~'E)o otherwise for any zeD ~ and OCA. Put D~(n):= u{D ~174 OCA and w, [wo\w/Woo] } r }; = w{D~:| and W. 0o_~ | w. the equality is a consequence of [Cas] 1.1.3. This set is empty if w, | is not properly contained in A; otherwise it is an open subset in the closure of D~ .~176 which contains D~-o0. Altogether this establishes the following fact. Lemma IV.3.4.- The sheaf gr~v.V is the constant sheaf with value Ind(~E) on D| extended by zero to all of D| For the corresponding cohomology groups this has the consequence that H'(Doo, gr~, % V) = H;(D| Z) | Ind(~lE) H'-a(D~"~176 Z) | Ind(r"lE) if * 1> 2, if* ----- 1, = coker(Z -+ H~176176 Z)) | Ind(~ if * =0 ker(Z -+ H~176174 (n), Z)) | Ind(O~'E) provided D| + 0. Lemma IV. 3.5. -- Assume that w, Oo is a proper subset of A and that n 4= O, ~tW/Woo; then D~,~176 is contractible. Proof. -- The set D~ .~176 is the geometric realization of the abstract simplex given by the poset (w.r.t. inclusion) of all nonempty subsets of A\w. O o (note that by assumption A\w. O0 4= 0). The set Do~.~176 is the geometric realization of the subcomplex given by the subposet of all those subsets which do not contain A\w, ~+. Our assumption that n 4= 0, resp. 4= SW/Wo,, implies that A\w. ~+ 4= 0, resp. + A\w. | The first implication is a consequence of [Bor] 21.3. In order to see the second implication assume n + SW/Wo, and put w := w.. Let wa, resp. Woo , be the unique maximal (w.r.t. the Bruhat order) element in W, resp. Woo; then w + wa woo so that WWoo + w,x. Hence there is an e0 e A such that s~, WWoo 7> WWeo and a fortiori s~, w > w 172 PETER SCHNEIDER AND ULRICH STUH.LER which means ~o e w~ +. On the other hand denoting by t(. ) the length function on W w.r.t. A we have, for any 0~ e | wwo0) = t(ws wo.) = t(w) + t(s wo~ = e(w) + t(w..) - I = t(wwoo ) -- 1. It follows that ~0 r woo. What we have to convince ourselves of therefore is the following. Let C be a non- empty finite set, let 0 C C O C C be a nonempty proper subset, and denote by II the poser (w.r.t. inclusion) of all nonempty subsets of C which do not contain (3 o. Then the geometric realization ]II [ of the abstract simplicial complex given by II is contractible. But this is clear: Fix an element c e (3\C 0. The subset { c } corresponds to a vertex in [ H [. Since for any C' e II also (3' u { c } e H, it follows that [ H I can be contracted onto that vertex. [] Lerama IV.3.6. -- i) D| = D~ o/f Oo C A. ii) D~,(n) = D~, Oo /f Oo C A and n = SW/Woo. Proof. -- i) Obvious. ii) This follows from the fact that w. | = w. 9 + n A if w. is the unique maximal element in [W/Wo0 ] ([(:]as] 1.1.4). Theorem IV.3.7. -- Assume that V = Ind(E) for some irreducible supercuspidal repre- sentation E 0f Moo; let g-a eN be a lifting of the unique maximal element in [W/Weo]; we then have V@Ind('E) /f. =0, ~| 1, V /f, =0, $O0<d--1, H*(X~, V) -~ Ind(gE) /f * = d -- 1 -- $0 0 > 0, 0 otherwise. Proof. -- In case Oo = A we have V = 0. In the following we therefore assume that Oo C A. According to the previous results the only nonzero El-terms in our spectral sequence then are V ifn=m=O, E~"'~ ~ Ind(~E) ifn=~W/Woo, n+m=d-- 1 --$0o. Moreover since V is a quotient of the constant sheaf with value V on X| we have a natural augmentation map V--~ H~ V) which splits the edge homomorphism H~ V) = H~ % V) -+ H~ gr ~ v, V) = E ~176 = V. Hence the spectral sequence degenerates and the assertion follows. [] REPRESENTATION THEORY AND SHEAVES 173 IV.4. Cohomology with compact support Our main aim in this section is to establish the following result. In the proof we will follow the strategy developed for II. 3. I; but the tools used there have to be analyzed in more depth. Theorem IV.4.1. -- Let x be a special vertex in A and let e >>. 0 be an integer;for any representation V in AlgU"~')(G) we have: i) the natural map V _2~ H0(X,j,,oo V) is an isomorphism; ii) H,(X,j,,= V) = 0 for * > O. In the proof we will make use twice of homological resolutions. This is made possible by the following observation. Lemma IV.4.2. i) cd(X) = d; ii) cd(X| /fd~> 1; iii) cd(X) = d. Proof. -- (Here cd(.) refers to the cohomology with compact support.) iii) is a consequence of i) and ii) by the additivity of the cohomological dimension. Concerning i) it is a standard fact that a d-dimensional locally finite polysimplicial complex has cohomo- logical dimension d. Finally ii) follows from the existence of a proper map from X~ onto a (d -- 1)-simplex whose fibers are compact and totally disconnected ([BS] 3. I). [] The functor V ~j,,~ V is exact and commutes with arbitrary direct sums as can be seen most easily from the description of the stalks. Moreover X being compact the cohomology functor H*(X, .) commutes with arbitrary direct sums. In the proof of II. 3.1 we had seen that any V as in Theorem 1 has an exact homological resolution in AlgV~')(G) by representations which are direct sums of the " universal " representation C~(T) with T := G/U~ "1. Using Lemma 2 and the facts given in the above paragraph we conclude by standard arguments of homological algebra that in order to prove Theorem 1 it suffices to treat the case V = Co(T). For the rest of the proof V always denotes the representation C~(T). We begin by constructing a very convenient simplicial resolution of the sheaf V on X. Fix an integer m >/ 0 and consider the (m + 1)-fold product T~:=T~+I:=T x ... x T. As in the proof of II. 3.1 we put, for any facet F in X, TF:= V~')\T and T~:= T X ... � T (m -t- 1 factors). TF TF 174 PETER SCHNEIDER AND ULRICH STUHLER The latter is a subset in T m+l. For facets F'_c F we have T~' _c T~. Extending functions by zero therefore induces inclusions Co(T ) _= Co( +1) in such a way that Co(T~') ~ Co(T~) if F'__q I ~. We recall that C~(. ) stands for the space of complex valued functions with finite support. If the point z e X is contained in the facet F we write T, := T r and T~, := T~. A sheaf $',. on X can now be defined in the following way: For any open subset f~ _ X put g'..(f~) := C-vector space of all maps s : f2 ~ LJ C~(T~,) such that sell -- s(z) ~ Co(T~) for any z ~ f2, --there is an open covering f~ = U f~i and functions +~ eC~(T "+1) with s(z) -= qb i for any z ef~ and i eI. /.emma IV.4.3. i) (W=), = C~(T~) for any z e X; ii) the restriction of g',~ to any facet F of X is the constant sheaf with value Cr iii) for any facet F in X we have C~(T~) 0, H'(St(F), St(F)) = H*(F, g'. I F) = { 0 /f, > 0. Proof. -- Entirely analogous to I. I and 1.2. [] In order to distinguish various constant sheaves in the following it is convenient to follow the convention that M/y , for any abelian group M and any topological space Y, denotes the constant sheaf with value M on Y. Obviously ). T.:... ~ >TxT� >TxT ~T as wcll as T F : >TxTxT ~TxT ~T 9 TF TF > TF REPRESENTATION THEORY AND SHEAVES |7b for any facet F are simplicial sets in a natural way. The push-forward of functions with finite support with respect to these face maps commutes with extension by zero. In this way we obtain simplicial sheaves ). ( ) C+.T../x : ... ) ) C~(T2),x ~ Co(T1)/x ~ Cr and W. : " ~'2 , --'gO on X together with an inclusion .g" _c Cr which in degree 0 is an equality W 0 = Co(T)/x. The obvious surjection Co(T)/x ~V defines an augmentation ~" -7 V. Applying j, we obtain the augmented simplicial sheaf j.r.-+j.v on X, Lemma IV.4.4, -- For any abelian group M we have j,(M/x ) = M/~. Proof. -- We will establish a slightly stronger fact. Fix a boundary point z E X| We will show that z has a fundamental system of open neighbourhoods ~ in X such that both ~ and ~ n X are path-connected. The tool to construct such neighbourhoods is the notion of the angle between two intersecting geodesics in X ([Bro] VI. 7 Ex. 1). We first need some notation. Let x ~ X be any point. For any other pointy E X different from x we put [xy) := ( [xy]\{y} ify ~ X, half-line emanating from x in direction y if y ~ X~o ([Bro] VI.9 A). In either case [xy] := [xy) ~3 {y } is a path from x toy in X. We also put (xy] := [xy]\{ x } and (xy) :--- [xy]\{ x,y }. There is a unique facet F(x;y) in X such that x ~ F(x;y) and (xy) n FIx;y ) 0; 176 PETER SCHNEIDER AND ULRICH STUHLER clearly the latter intersection is of the form (xy) n F(x;y) = (xy,) for some y~ e (xy]. -- t Given now two points y,y' e X different from x the two geodesics [xy,] and [xy,] lie in a common apartment (which is euclidean) so that the angle 0 ~< "r(x;y,y') ~ 3, 14... between them is defined (and, in fact, is independent of the chosen apartment). For any real number 0 < ~- < 1 we consider the subset n(x; z; .-) := {y ~ x\{ x }: v(x;y, z) < ~ } of X which contains z. We will successively prove: 1. ~(x; z; ,) and a(x; z; ~) n X are path-connected. 2. The function x\{ x } --, R+ y ~ 7(x;y, z) is continuous. 3. There is a constant 0 < ,(x; z) < 1 such that F(x; z) ~ F(x;y) for any y ca(x; z; ~(x; z)). 4. For any z' e f~(x; z; ,) n Xo~ and any d< min(a -- 7(x; z', z), a(x; z')) we have ~(x; z " , ~') _ c a(x; z; ~) 5. f~(x; z; ~) is open in X. 6. For x' e [xz) and 0 < d ~< ~ we have a(x'; z; ~') _q ~(x; z; q. 7. Let a(.) : R+ -+ (0, 1) be a decreasing function; then fl n(~'; ~; ~(a(x, x'))) = { ~ }. 8. The f~(x; z; ~) for varying x and ~ form a fundamental system of neighbourhoods of z in X. The assertions 1., 5., and 8. contain what we wanted to establish. Ad 1. -- Lety andy' be points in fl(x; z; ,). Then obviously [y,,y] u [y~y'] ~_ f~(x; z; ~). By looking at an apartment which contains F(x;y) and F(x; z) and hence the convex hull of{ x,y,, z, }one sees that [y, z,] _c f~(x; z; ~). We similarly have [y'_ z,] __c_ ~)(x; z; ~). Ad 2. -- The sets F\{ x } with F running through all the facets of X form a locally finite closed covering of X\{ x }. It therefore suffices to check the continuity of the function REPRESENTATION THEORY AND SHEAVES restricted to each such set. But the latter is clear again by looking at an apartment which contains F(x; z) and F. Ad 3. -- This follows from [BT] I. 2.5.11 and the elementary geometry of an apartment containing F(x; z) and F(x;y). Ad 4. -- Lety be a point in ~(x; z'; ~'). Looking at an apartment which contains F(x; z) and F(x;y) and hence, by the assumption on ~', also F(x; z') we find v(x;y, z) .< + v(x; z', z) < Ad 8. -- As a consequence of 2, we know already that fl(x; z; r n X lies in the interior of~(x; z; ~). It remains to consider a boundary point. Because of 4. it is actually sufficient to show that ~(x; z; ~) is a neighbourhood of z where in addition we may assume that z ~ Doo. Suppose this would not be true. Then we find a sequence (z,),E M of points in X\(fl(x; z; ~) u { x }) which converges to z. This means in particular that y(x; z,, z) >/ r for all n e N. On the other hand we will show below that lirn y(x;z.,z) =0 whenever limz,= z. This gives a contradiction and proves our claim. By the construction of X we have z, = tt,,y,, with h, ~ P0 and y, e D. Since P0/C and D are compact we may assume by passing to a subsequence that the h, C converge to hC for some h ~ P0 and that the y, converge to some y E D. It follows that hy = z and hence even y ---- z so that h ~ P0 n P,. Passing again to a subsequence we further assume that all tth-~ 1 fix F(x; z) pointwise. Using that the G-action on X respects angles ([BT] 1.7.4.11) we conclude that v(x; z., z) = v(hh; -1 x; by., hh. z) = v(x; by., z). This argument shows that it suffices (replace D by hD) to consider a sequence z, contained in D. Now using [BT] 1.7.4.18 ii) or [Bro] VI.8 Thm. we find a subsector D' := x' + D ___ D such that x and D' are contained in a common apartment A'. In particular z e D' because of (D'), = D| moreover (x' + z.).E! is a sequence in D' which converges to z (note that x' + z. = z. ff z, ~ D~). It is clear from the definition of the topology of A' that lim y(x; x' + z., z) = 0. m ,--~ O0 From the cosine inequality in [Bro] VI. 7 Ex. 2 it is also clear that lim.. T(x; z., x' + z.) = O. We finally pass for a last time to subsequences of (z,) and (x' + z,) in such a way that the facet F := F(x; z,), resp. F' := F(x; x' + z,), is independent of n ~N. Then 23 17B PETER SCHNEIDER AND ULRICH STUHLER necessarily F(x; z) _~ F'. In this situation we see by working in an apartment which contains F and F' that "~(x; z,, z) ~< ~,(x; z,, x' + z,) + ~'(x; x' + z,, z). Combining the last three formulas we obtain lirn y(x; z,, z) = 0. Ad 6. -- Lety be any point in f~(x'; z; d). Assume that y r [xz) because otherwise y lies in f~(x; z; r for trivial reasons. Then the function x' ~ V(x';y, z) on [xz) is defined. We show that it is increasing as a function of d(x, x'). Choose an apartment A' which contains F(x'; z) u {y }. Looking at the convex hull of { x', z,,,y } in A' it is clear that T(x";y, z) >>. "~(x';y, z) for any x" ~ [x' z,,]. Ad 7. -- Let y be any point in the intersection on the left hand side. Then y r [xz) and T(x';y, z) <~ ,(d(x, x')) for any x' ~ [xz). The left hand side is increasing with d(x, x') by the previous argument whereas the right hand side is decreasing by assumption. It follows that T(x';y, z) = 0 for any x'~ [xz). Would y and z be different then we would have [xz] n [xy] = [xx'] for some x' e [xz). On the other hand one easily deduces from y(x ,y, z) 0 that (x'y.,) n (x' z..) + r Hence we would obtain a contradiction. Ad 8. -- Choose a sequence of points xl, x2,,., in [xz) such that d(x,x,) is increasing and tends to oo and choose a decreasing sequence of real numbers 0 < E 1, ~2, 9 9 9 < 1 which converges to 0. ]'hen, by 5. and 6., the subsets f~(x,; z; ~) form a decreasing sequence of open neighbourhoods of z such that the intersection of their closures is, by 6. and 7., equal to { z }. It is a general fact about compact Hausdorff spaces that such a sequence has to be a fundamental system of neighbourhoocks. [] This has two consequences: First of all we obtain an inclusion j. ~. _~ Co(T.)~ of simplicial sheaves on X. Secondly because of j. W 0 ---- Vt2 it follows from 2.11 that j,, ~ V is the image of the augmentation map. Therefore we actually have an augmented simplicial sheaf j.g'.-~j., V with surjective augmentation map. REPRESENTATION THEORY AND SHEAVES 179 Proposition IV. 4.5. -- The associated complex of sheaves ....~j,~'~j.~-~.L, V~O is exact. Proof. -- This is shown stalkwise. First let z be a point in X. By 1.1 i) and Lemma 3 i) the sequence of stalks in z is the complex of functions with finite support associated with the augmented simplicial set T; -+ T,. It is exact since the fibers of that augmen- tat-ion are contractible simplicial sets (compare [SS] p. 22). The same reasoning works for a boundary point z e X~ where we use the unipotent subgroup U, in order to analogously define T,:=U,\T and T~,:=T x ... � T(mq- l factors) T~ T z once we show that the augmented simplicial vector spaces (3", ~,), -, Co(T)~ h and Co(T*. ) --~ Co(T,) coincide. Both simplicial vector spaces are contained in Cc(T. ) so that the comparison can be done termwise. We need the subsequent two facts. Lemma IV.4.6. -- Let z e X~ be a boundary point;fir any open ndghbourhood Q of z in X we have fl T~,~ ~_ T~. wE~t~X Proof. -- Assume that (to, . .., t,,) is a tuple which is contained in the left hand side but not in the right hand side. Then there is 1 ~< j ~< m such that gt o + tj for anygeU,. Let go e G be a coset representative of to =-- go--, IP '~ 9 Choose a point y e Q n [g0(x) z). By assumption we have ht o=t~ for some h~U c') In 1.3.2 it was shown that UCel c U,.U~o~ * or equivalently U(e) iT(e) ij(e) holds true. (Observe that by [Bro] Thin. VI. 8 there is an apartment which contains the special point go x and the boundary point z in its closure.) But this implies UC,~ t c U,.t 0 which is a contradiction. [] V " 0-- 180 PETER SCHNEIDER AND ULRICH STUHLER /.emma IV.4,.7. -- Let z e X~ be a boundary point; for any tuple (to, . .., t,,) E T~ there is an open neighbourhood f~ of z in X such that (to, ..., t,,) e TV,, for any y e f~ n X. Proof. -- By G-equivariance we may assume that z ~ D~o where D is the funda- mental Weyl chamber introduced in 2.1. Let U' G G be an open subgroup such that gt~-=t t for all geU' and 0,<j,< m. By assumption there are elements u t e U, such that t t=u~t o for l~<j,< m. The subset c:= U {hu~h-l:hePonP,} X~</~<m of U, is compact. Hence we find, by 2.7, an open neighbourhood f2 o of z in A such that c __c_ U~f ) for any facet F ___ f/o n X. According to 2.5 there is now an open neighbourhood f~ of z in X of the form = U(P0 n P,).f21 where -- U __G U' is an open subgroup, -- ~1 is an open neighbourhood of z in D such that f~x n X is a union of facets, and -- ~1 _c ~o. In particular we have r U c') for anyy'ef2 lnx. Consider an arbitrary point y=ghy" with geU, heP0nP,, andy'ef~x nX in f~ n X. By construction we obtain tj = gt t = gu t t o =- gu t g-1 to ~T'r(e) h--1 g-a T'T(e) 9 = gh(h -1 u~ h) h-lg -1 t o egnur t o = for any 1 ~< j ~< m which means that (to, ..., t,.) ~ T~,.. [] Returning to the proof of Proposition 5 let first ~ be a function in (j. $',.), ~ Co(T,. ). This means that there is an appropriate open neighbourhood ~ of z in X such that ~_ Co(T~.,) for anyy ~ f~ n X. In other words the support of + is contained in ~ T*., and hence in T~. by Lemma 6. ~Etanx REPRESENTATION THEORY AND SHEAVES 181 Conversely let d? be a function in Co(T~). Then it follows from Lemma 7 that + can be viewed as a section in g',~(f2 ra X) ----j. g',,(~) for some neighbourhood of z in X. Finally in order to compare the two augmentation maps we have to check that the push-forward of functions with respect to the projection map T ~ T, induces an isomorphism Co(T)v, ~ C~(T,). The surjectivity is trivial. For the injcctivity it suffices to consider a function ~ e Co(T) whose image in Co(T,) vanishes and which is supported on a single U,-orbit in T. Wc then find a compact opcn subgroup U in U, such that ~ even is supported on a single U-orbit in T. Hence the image of~ in C~(U\T) vanishes which implies that ~/= 0 mod U. This finishes the proof of Proposition 5. [] The computation of the cohomology of a sheaf j. g;, will be based on the obser- vation that this sheaf has a natural direct sum decomposition. For each tuplc (to, ..., t,~) e T,, we introduce the subset X.0 ..... t.) := { z e X : (to, ..., t,~) r T~ } of X and we put X,o ..... t~.) : -= X(to ..... t..) n X and X"~ ..... ~.,) := ~,..o ..... '") ~ X~. Clearly ~,o ..... ")=0 if t o ..... t,. Assuming therefore that {to, ...,t,,} has cardinality at least 2 let us first collect a number of properties of the subspace X"* ..... ~,,) which will be of use later on. IV.4.8. X "~ ..... t,.) is a nonempty closed geodesically contractible CW-subspace of X. (This was already noted in the proof of II. 3.1.) IV.4.9. For any open subset ~__c X such that Y2 n X"J ..... t~)+ 0 we have f2 n X "~ ..... t,.~ 4: 0. (This follows from Lemma 6.) IV.4.10. Any boundary point z e X~\X~ ~ ..... t~,~ has an open neighbourhood -Q such that ~ n X "0 ..... t.,) = 0. in X (This follows from Lemma 7.) 11(,). for any z e X"o ..... e,) we have IV.4.11. Choose go e G such that t o ---go--~ , --~ [g~ z) ___ X"~ ..... '-'. (This is proved literally in the same way as Lemma 6.) 182 PETER SCHNEIDER AND ULRICH STUHLER IV. 4.12. We have X(to ..... tm) z X(t0 ..... tm); in particular ~(to ..... tin' is closed in X. (This is a consequence of 10. and 11.) For an arbitrary tuple we now define the sheaf C,o ..... ,=) on X to be the constant sheaf with value C on the open subset X\X It~ ..... tin) extended by zero to all of X. Lemma IV.4.13. -- We have $',, = C(to .... , ira). (to, ..., trn) E T m Proof. -- Straightforward. [] Lemma IV.4.14. -- The sheaf j, C(t ~ ..... t,,) is the constant sheaf with value C on the open subset X\X "~ ..... t,.) extended by zero to all of X. Proof. -- This is a consequence of 12 and Lemma 4. [] By comparing stalks we obtain j. g'= = 9 j. C,,o ..... ,=, (tO, ..., tm)ET m and hence , -- o H (X,3 , #',.) = 0 H (X,j. C.o ..... t.,)) ( tO,..., tm)U. T m = | H*(X, X"~ ..... '"; C). (tO,..., tm)~T m Proposition IV. 4.15. -- We have /f* -----0, n (x,j. ~-,~) = Co(T) 0 /f.> 0. Proof. -- The space X is contractible by [BS] 5.4.5. The same argument together with 8. and 11. shows that X "~ ..... t,,), for {to,..., t,.} of cardinality at least 2, is contractible as well. [] Because of Lemma 2 the resolution in Proposition 5 gives rise to the hypercoho- mology spectral sequence , H,+,,X 9 v~ I, = u,(x,j. ~,) ~ v ,a.,. v). It degenerates by Proposition 15 and exhibits H'(X,j,,~o V) as the homology of the complex ...~176 This proves Theorem I. REPRESENTATION THEORY AND SHEAVES 183 Corollary IV. 4.16. -- Let V be a representation of finite length in Alg(G); /f e is chosen large enough then we have (ker(V /f 9 = 0, H:(X, V) = ~ coker(V ~ V(X.~)) /f 9 = 1, kH'-I(X~o,V) /f* /> 2. We fix a subset @ __c A. As in the proof of 3.5 let wa, rasp. wo, denote the unique maximal (w.r.t. the Bruhat order) element in W, resp. W o. We also fix a lifting g e N of w o w~; note that (w o wA) -x is the unique maximal element in [W/W| Theorem IV.4.17. -- Assume that V = Ind(E)for some irreducible supercuspidal representation E of Mo; /f e is chosen large enough then we have , (Ind(OE) /f, =d-- $| He(X, V)~ - [ 0 otherwise. Proof. -- Corollary 16 and 3.7. [] Corollary IV.4.18. -- Let E be an irreducible supercuspidal representation in Alga(M| we then have [Ind(~ /f* =d--~| o"*(Ind(E)) 0 otherwise. Proof. -- Theorem 17 and 1.4. [] Corollary IV.4.19. -- Let V be a representation in AlgSx~,o(G) ; /re is chosen large enough then we have HI(X, V) ---- 0 for 9 oe d -- ~| Proof. -- This follows from Theorem 17 by an induction argument as in the proof ofIII. 3.1. (Because of 1.4 the present assertion and III. 3.1 i) actually are equivalent.) [] IV. 5. The Zelevinsky involution We fix a central character )~ and let Rz(G; )~) be the Grothendieck group of representations of finite length in Algx(G ) (w.r.t. exact sequences); the class in Rz(G; ?() of a representation V is denoted by [V]. It follows from III. 3.1 that ~: Rz(G; Z) -~ Rz(G; Z) [V] ~ Y,, (-- 1)'. [~(~)] is a well-defined homomorphism such that t([V]) = (-- 1)a-lt~ for any V in Algexl, o0(G). 184 PETER SCHNEIDER AND ULRICH STUHLER Proposition IV. 5.1. -- The homomorphism ~ respects up to sign the classes of irreducible representations. Proof. -- This is III. 3.2. [] Consider a representation V in Alg~x, o0(G). It is a consequence of 3.1 that there is an augmented complex Ind~o (Vvo) =V O~A ~O=d @ Ind~ (V) ~ ... , @ Ind~ (V) 0 _~ A 0 UO 0 ~ A 0 UO ~o=~-a ~o~o which computes the cohomology H*(X~, V). By combining 1.4 and 4.16 we see that the only nonvanishing homology group of that complex is ~(~); it sits in degree d -- 1 -- $00 if the complex is put in degree -- 1 up to d -- 1. Since the formation of Jacquet modules as well as the parabolic induction respect representations of finite length ([Ber] 3.1) each term in the above complex has a well-defined class in Rz(G; Z). Proposition IV.5.2. -- For any representation V of finite length in Alggxl(G) we have t([V]) = E (-- 1) a-~~176 O_=A Proof. -- Obvious from the preceding discussion, rq For any O c A let P_ o be the parabolic subgroup of G which contains M o and is opposite to Po; then P_ o n Po = Mo. Let U_ o denote the unipotent radical of P o. The modulus character of P-o is pr P-o ,~ Mo 8~ R~ (leas] 1.6). Lemma lWV.fi.3. -- Let E be a representation of finite length in Algx(Mo) for some | ~_ A; we then have [Ind~ = [Ind(~o u~ | E)]. Proof. -- If g ~ N lifts wzx then g- i p_ o g = P~ ~e o. We obtain Ind~ ) _~ Indpwaw e o(~ = Ind(8~ we o | ~ = Ind(a(8o v' | E)). Because of g-1 Mo g = M~ A ~o o we may apply III. 2.1 and we see that the latter representation has the same irreducible constituents as Ind(8ol~| E). [] REPRESENTATION THEORY AND SHEAVES Proposition IV.5.4. -- For any representation V in AlgSxl, o0(G) we have [~(~)] = [d(V)~]. Proof. -- Dualizing the discussion preceding Proposition 2 we obtain [e(V)~] = E (-I)*~176176 ] = E (-- 1)~~176174 t~_cA where the second equality even holds termwise by [Ca.s] 4.2.5. On the other hand Proposition 2 holds true, of course, for any choice of simple roots, e.g. --A. Hence we have &,o E (-- 1) ~~176176 [Ind~_o(Vu_o)]. O_cA Apply now Lemma 3. [] Corollary IV.5.5. -- The map ~ is an involution, i.e. to t= id. Proof. -- Combine Proposition 4 and III. I. 5. [] The Proposition 2 shows that (-- 1) a. t coincides with the involution D o studied in [Aul] 5.24. It therefore follows from [Aul] 5.36 that in case G = GLa+I(K ) the Zelevinsky involution i considered in [Zel] 9.16 is equal to -- t. Hence Proposition 1 proves the Duality Conjecture 9.17 in [Zel]. It also follows that the orthogonality property discussed in [Aul] 5.D holds true. Assume K to have characteristic 0 and the center of G to be compact. Then the above results hold without having specified a central character. In III. 4 after Thin. 21 we had seen that the Euler-Poincar~ characteristic EP(., .) induces a nondegenerate symmetric bilinear form on the quotient R(G) = R(G)/R~(G) of the Grothendieck group R(G)-Rz(G)| It follows from Proposition 2 or from III.4.3 i) that EP(t., .) = EP(., .). Hence the involution t respects the subgroup RI(G ) and induces the identity on the quotient i~(G) or, equivalently, (id -- ~) (R(G)) _ Rz(G ). V. THE FUNCTOR FROM EQUIVARIANT COEFFICIENT SYSTEMS TO REPRESENTATIONS In this final Chapter we want to examine more closely the relation between representations and coefficient systems. The category Coeff(X) of coefficient systems (of complex vector spaces) on X was introduced in II.2. We say the group G acts on the coefficient system (VF) v if, for any g E G and any facet F_ X, there is given a linear map gv : V F ~ Vg~ 24 186 PETER SCHNEIDER AND ULRICH STUHLER in such a way that -- ghr o hF = (gh)F for any g, h 9 G and any F, -- le = idv~ for any F, and -- the diagram gp V F > Vg F Vr o,,> Vo r is commutative for any g 9 G and any pair of facets F' _~ F. In particular the stabilizer PtF, for any facet F, acts linearly on V F. Definition. -- An equivariant coefficient system on X is a coefficient system (V~) F on X together with a G-action on it which has the property that, for any facet F, the stabilizer P~ acts on V F through a discrete quotient. Let Coeffo(X ) denote the category of all equivariant coefficient systems on X. This is an abelian category. Fix an object r = (VF) F in Coeffo(X ). By definition, for any 0 ~< q ~< d the space of oriented q-chains of r is ". C~j(X~,), r := C-vector space of all maps to : X(,) --+ U V F such that FEXq to has finite support, -- to((F, c)) 9 VF, and, if q >/ 1, to((F, -- c)) = -- r c)) for any (F, c) 9 X(,,. The group G acts smoothly on these spaces via (gto) ((F, c)) := gc_~r(to((g-~ F, g-X c))). The boundary map o~ ~) -, ~r(X,q, ~) 0 " ~'Jc k (~z+l), r~,(to((F, c)))) to ((F', c') Z (F, e)~ X(q+x ) F'~ O~,(c) : c' is G-equivariant. Hence we obtain the chain complex cor(x, ), & ... & co,(X,o,, REPRESENTATION THEORY AND SHEAVES 187 in AIg(G). Its homology is denoted by H.(X, "//). It is not difficult to see that the above complex as well as its homology actually lies in the full subcategory AlgC(G) :== category of those smooth G-representations V which are generated by V ~ for some open subgroup U ~ G. As a consequence of Bernstein's theorem (I.3) the category AlgC(G) is stable with respect to the formation of G-equivariant subquotients; moreover it is closed under extensions. In the following only the right exact functor Ho(X , .): CoeffQ(X) ~ Alg"(G) will be of importance for us. Let E be the class of morphisms s in Coeffa(X ) such that H0(X , s) is an isomorphism. We then have a unique commutative diagram of functors ~(X,. ) Coeff~, (X) , AlgC(G) where Q is the canonical functor into the category of fractions with respect to Z. Theorem V. 1. -- The functor p : Coeffo(X)[Y~-' ] -~ Alga(G) is an equivalence of categories. Proof. -- The following properties are a consequence of the right exactness of Ho(X , .) ([GZ] 1.3): 1. 2; admits a calculus of left fractions. 2. Coeffo(X) [Z-~] is additive and has finite direct limits. 3. The functors O and p are additive and respect finite direct limits. 4. The functor p detects isomorphisms. The latter two properties imply: 5. The functor ~ is faithful. Namely, let a and b be two morphisms in Coeffo,(X ) [~-~] such that p(a) ---- p(b). Using 3. we have that ~(coker(a -- b)) = coker(p(a -- b)) ---- coker(p(a) -- p(b)) = coker(0) is an isomorphism. By 4. then coker(a -- b) is an isomorphism, too; hence a = b. Fixing a special vertex x in A we have Alga(G) :: O Alg~'~(G). e~>0 188 PETER SCHNEIDER AND ULRICH STUHLER In II. 2 we have constructed, for any e >1 0, an exact functor (e) y, : Alg~x (G) ~ Coeffo(X) ; moreover there is an obvious natural transformation Ye ~T,+I I AlgU~')(G) 9 It follows from II. 3.1 that the latter induces a natural isomorphism in homology H,(X, V,(-)) -~ H,(X, y,+,(.)) on AlgV))(G). After composing with the functor Q the above natural transformation therefore becomes a natural isomorphism = uic) Qo~tr QO~e+l IAlg , (G). Hence we obtain in the direct limit the functor 3' := lira Qo y, : Alg"(G) -+ CoeffQ(X) [Z-a]. Again by II.3.1 we have 6. p o 3' _-_ id~cla ~. It is an immediate consequence of 5. and 6. that P and ~, are quasi-inverse to each other. [3 Because of their practical importance let us state separately the following facts which were established in the course of the previous proof. Lemma V . 2. i) Z admits a calculus of left fractions; ii) thefunctor ~, : AlgC(G) -+ Coeffc(X ) [Z -a] is quasi-inverse to O. INDEX OF NOTATIONS a nonarchimedean locally compact field o the ring of integers in K a fixed prime element in o r the discrete valuation of K normalized by c0(n) = I the residue class field of o the base change to K of some object X over o G a connected reductive group over K the group of K-rational points of G the center of G a maximal K-split torus in G W the Weyl group of G the set of roots of G ~:)red the set of reduced roots of G q)+, ~)- the set of positive resp. negative roots in REPRESENTATION THEORY AND SHEAVES 189 A the set of simple roots in ~+ U~ the root subgroup corresponding to the root a subset of the set A of simple roots <| the subset { c~ E 9 : cc is a integral linear combination of roots from | } the connected component of f] ker(~) So ctEo M| the centralizer of S o in G, i.e. the Levi subgroup corresponding to O U| the unipotent subgroup of G generated by all root subgroups U~, for ~ E ~+\( | ) P| = M0 U0 the parabolic subgroup of type | with respect to the choices S, ~+ the modulus character of the parabolic subgroup P| 8o X the Bruhat-Tits building corresponding to G d the distance function on the metric space X X(q~(Xr the set of oriented (nonoriented) q-dimensional polysimplices of X X(q) the q-skeleton of X A a fixed (basic) apartment of X, a d-dimensional affine space D a fundamental Weyl chamber in the apartment A F a polysimplex of X St(F) the star of the facet F the Borel-Serre compactification of X x~ the boundary of X in Pfl the pointwise stabilizer in G of a subset fl _ X the stabilizer in G of a subset fl c X uT for any integer e >I 0, the e-th filtration subgroup of P~ subgroup of U~ of t-value >/ r Uot, r T = C,/U~ ) for a special vertex x ~ A, the basic homogeneous G-set the subset of all regular elliptic elements of G Gell the set of conjugacy classes of regular elliptic elements of G Cell V a smooth representation of G Alg(O) the category of smooth G-modules AlgU(G) for U a compact open subgroup, the subcategory of Alg(G) of tho~e G-modules which are gene- rated by the subspaee of their U-fixed vectors V ~ for Z a character on the connected center C of G the full subcategory of Alg(G) of those Algx(G) G-modules on which C acts by the character X the full subcategory of Algx(G ) of those G-modules which are of finite length and whose irre- Alga, o (G) ducible subquotients are all of type | R(G) the Grothendieck group of representations of finite length in Alg(G) tensorized by r the (urmormalized) induction to G of a smooth Po-module e Ind~o(e) lad(e) the normalized induction to G of a smooth Po-module e for a character X of the connected center C, the Hecke algebra of locally constant functions on G, ~x compactly supported modulo C and transforming with respect to the action of C by the character Z Ee(V, V') the euler-Poincard characteristic of the smooth G-modules V, V', V finite length, V' admissible d v g the euler-Poincard measure for V volv the volume corresponding to dVg ca - ~o the involution functor on Alg~x ' o(G) Coeff(X) the category of coefficient systems on X y~(V) the coefficient system associated to a smooth G-module V for a fixed integer e >/ 0 ye the corresponding flmctor from Alg(G) to Coeff(X) ~r(x(.),-r,(v)) the oriented chain complex associated to a coefficient system y,(V) v the sheaf on the Bruhat-Tits building associated to the smooth representation V the smooth extension of V to the compactification Z,~ j,,=o v V the sheaf on the boundary Xco of X in "X corresponding to the smooth representation V 190 PETER SCHNEIDER AND ULRICH STUHLER REFERENCES [Aul] A.-M. 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Stuhler Mathematisches Institut der Universit~it G6ttingen Bunsenstr. 3-5 37073 G6tfingen Manuscrit re~u le 28 juin 1995.

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Representation theory and sheaves on the bruhat-tits building

Representation theory and sheaves on the bruhat-tits building

Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

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Representation theory and sheaves on the bruhat-tits building

Representation theory and sheaves on the bruhat-tits building

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