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Reductive algebraic groups over p-adic fields

Reductive algebraic groups over p-adic fields CH'APTER I REDUCTIVE ALGEBRAIC GROUPS OVER p-ADIC FIELDS (1) w I. k-Borel subgroups. x .x. k-Borel pairs. Let k be a p-adic number field. An algebraic group G over k is called a torus, if it is connected, commutative and consisting only of semi- simple elements. For a torus G over k, there exists a finite extension K/k such that G K is K-isomorphic (i.e. birationally isomorphic over K) to (K*)~; such a field K is called a " splitting field " of G. In case this isomorphism is obtained in k, i.e. in case G is k-isomorphic to (k*) ~, G is called a k-trivial torus. An algebraic group G over k is called unipotent, if it only contains unipotent elements; a unipotent algebraic group G over k is always connected and nilpotent. Let G be an algebraic group over k. A maximal k-trivial torus (resp. maximal k-unipotent subgroup) in G is a k-trivial torus (resp. k-closed unipotent subgroup) in G which is maximal with respect to this property. A pair (A, N) of a maximal k-trivial torus A and a maximal k-unipotent subgroup N in G such that A normalizes N is called a k-Borel pair. For such a pair (A, N), AN becomes a k-closed subgroup of G, called a k-Borel subgroup of G, which is a semi-direct product over k of A and N. A typical example of a k-Borel pair is G = GL(n, k), A=D(n, k) (the group of all diagonal matrices in GL(n, k)), N=TU(n, k) (the group of all upper unipotent matrices, i.e. matrices x=(~i) with ~u=I,~q=o for i>j, in GL(n, k)). It is known (Borel) that, for any pair (A, N) of a k-trivial torus A and a k-closed unipotent subgroup N in G such that A normalizes N, there exists always a k-Borel pair (A', N') with A'3A, N'~N and that all k-Borel pairs in G are conjugate to each other with respect to the inner automorphisms of G [x2]. It follows that, for any Borel pair (A, N) in G, we can transform G by an inner automorphism of the " ambient group " GL(n, k) in such a way that A coincides with the connected component of the neutral element of GraD(n, k) and N coincides with GriTS(n, k). (1) For the fundamental concepts on algebraic groups, see [1]; especially on algebraic toruses, see [18]. Cf. also [7], [12], [20]. 234 THEORY OF SPHERICAL FUNCTIONS tl 9 .2. The following proposition gives a characterization of k-Borel subgroups in terms of the p-adic topology. PROPOSITION I.I. Let G be an algebraic group over a p-adic number field k. Then, for a k-closed subgroup H of G, the homogeneous space G/H is compact (in the p-adic topology), if and only if H contains a k-Borel subgroup of G. Proof. The proof is obviously reduced to the case where G is connected. Hence, assuming G to be connected, let H be a k-Borel subgroup of G. Then since G/H is identified (birationally) with the set of k-rational points in a projective variety defined over k [x2], it is compact; whence follows the " if " part of the Proposition. Conversely, let H' be a k-closed subgroup of G such that G/H' is compact. One can take a k-Borel subgroup H of G containing a k-Borel subgroup of H'. Then, from what we have proved above, H'/(HnH') is compact. Therefore HH' is closed in the p-adic topology, so that H/(HnH')~HH'/H' is compact by the assumption. But, since H is a k-Borel subgroup, it has a composition series (as algebraic group over k) : H = H0~ HI~... ~ H r ={ I } such that the factor groups Hi_l/Hi are isomorphic either to k* or to k; and it is then a trivial matter to prove inductively that H~(HnH')=H(I~<i<r). Thus we get H=HnH', i.e. H'~H, q.e.d. COROLLARY. An algebraic g~oup G over k is compact (in the p-adic topology) if and only if k-Borel subgroups of G reduce to the neutral element. In view of a decomposition theorem of Chevalley ([7], P. I44) it follows that a compact algebraic group over k is necessarily reductive, i.e. isogeneous to the direct product of a semi-simple algebraic group and a torus. Remark. Proposition I. I and its Corollary are also valid for algebraic groups over R or C. (The same proof l) 9 .3. Maximal compact subgroups. PROPOSITXON I .2. If an algebraic group G over k has a maximal compact subgroup, G is reductive. Pro@ Suppose that G is not reductive. Then, by virtue of the decomposition theorem of Chevalley, G has a k-closed unipotent normal subgroup N1 + { I }. The center Nz of N1, being a unipotent commutative group, is k-isomorphic to a vector space over k, i.e. we have a k-isomorphism f : N~ ~k "~ (m> o). Then the inner automorphism I 0 defined by g~G induces an automorphism of N~, which, by f, corresponds to a linear transformation, denoted by pg, of k m, Let U be a maximal compact subgroup of G and let Ll=omck m. Then, Ul={ueUIPuLt=L~} is an open subgroup of U and hence is of finite index in U,. Put U = y u~U1, L = 0 Pu' LI" Then L is an " Q-lattice " in k m (see N ~ 8. i) invariant under p. (u~U). It follows that U(r U.f-l(rc-iL) (i= i, 2, ...) 235 ICHIRO SATAKE I2 are all compact subgroups of G containing U, so that, by the maximality of U, one gets f-I(=-~L) cU (i=I, 2, ...). This implies that N2cU, which is a contra- diction, q.e.d. Now, for a reductive algebraic group G over R, it is well-known that G has always a maximal compact subgroup, which is R-closed, and that all maximal compact subgroups of G are conjugate to each other with respect to the inner automorphisms of G. Moreover, if (A, N) is an R-Borel pair in G and if A+ denotes the connected component (in the sense of the usual topology) of the neutral element of A, we have (I.I) G=U.A+N, Un (A+N)={i} for all maximal compact subgroup U, and (1.2) G=UA+U for a suitable U. Unfortunately, in the p-adic case, we are not yet in possession of such a general result. Since, however, the existence of a maximal compact subgroup with properties similar to (I. i), (I. 2) is indispensable to the theory of spherical functions, we will assume it in w 3; for classical groups, our assumptions will be verified case by case in Chapter III. On the other hand, it should be noted that, in the p-adic case, the maximal compact subgroups of a reductive algebraic group G are, if they exist, not necessarily conjugate to each other with respect to the automorphisms of G and that they are not necessarily corresponding in one-to-one way under a k-isogeny. w 2. Reductive algebraic groups. 2.x. In this section, G denotes a connected reductive algebraic group over a p-adic number field k. Let A be a maximal k-trivial torus in G, Y the character module of A and let dim A = rank Y = ~. If" {~1, .--, :%} is a system of generators of Y, the correspondence (2. x) A~a ~ (~ql(a), ..., ~%(a)) e(k*) ~ gives a k-isomorphism A~(k*) ". Let g, a be the Lie algebras of G, A, respectively. It is easy to see that, for any k-homomorphism p from A into another algebraic group over k, the closure of the image o(A) is again a k-trivial torus. Applying this to p = adjoint representation of G, we have (2.2) g=g0+ Z g~, yGr where icY is the " restricted root system " relative to A [2o] and (2.3) gv={xegIAd(a)x=y(a)x for all aeA}, go={X~glAd(a)x=o for all aea}. 236 THEORY OF SPHERICAL FUNCTIONS I3 It is clear that g0 is the Lie algebra of Z(A) (= the centralizer of A). On the other hand, for any linear order in Y, put (~.4) n = Z g-c y>0 and call N the corresponding connected subgroup of G. Then we have [2o]: PROPOSITION 2. I The notations being as above, Z(A) is a connected reductive algebraic group over k consisting of only semi-simple elements, N is a maximal k-unipotent subgroup of G, normalized by Z(A), and Z(A).N is a semi-direct product of Z(A) and N over k. 2.2. For any connected reductive algebraic group G over k, we denote by X(G) the module of all k-rational characters (i.e. k-morphisms of G into k*). Furthermore we put (~.5) GX={geGtz(g)=I for all zeX(G)}. Then G 1 is a connected k-closed normal subgroup of G and there exists a k-trivial torus A' contained in the center of G such that G = cl(G1.A'), GlnA'= finite; in other words, the natural homomorphism gives a k-isogeny: G 1 � To see this, let S, T the semi-simple and the torus parts of G, respectively. Since S cG x, it is clear that the closure of the canonical image of G x in cI(G/S) is equal to el(G/S) x. On the other hand, for a torus, our assertion is known ([2o], Prop. I), i.e. irA" denotes the (unique) maximal k-trivial torus in T, we have a k-isogeny T 1 � It follows also that cl(G/S) 1, and consequently G x is connected. Under the k-isogeny T-+el(G/S), induced by the canonical homomorphism, T 1 corresponds to cl(G/S) 1, whence one gets G l=cl(T a.S). Thus one has G~T�215215215 ', where G~G' means that G is isogeneous to G'. This proves our assertion. It follows that the homomorphism X(G) -~X(A') defined by the restriction is injective and has a finite cokernel. In particular, if G consists only of semi-simple elements (which implies necessarily that G is reductive), it follows from Proposition 2.I that G=Z(A) (i.e. A=A' in the above notation), and hence by Corollary to Proposition i. I that G 1 is compact. Thus we obtain the following PROPOSITION ~. 2 Let H be a connected algebraic group over k consisting only of semi-simple elements and A the (unique) maximal k-trivial toms in H. Then H 1 is compact and H is k-isogeneous to the direct product ofH 1 and A. Moreover, X(H) may be identified with a submodule of X(A) =Y with finite index. Let {Z1, -..,~} be a system of independent generators of X(H). Then the mapping (2.6) 9 : U~h > (z~(h), ..., z~(h)) ~(k*) ~ 237 ICHIRO SATAKE x4 defines an injective homomorphism of H/H 1 into (k*)L In case this mapping is surjective, we say that H satisfies the condition (N). 2.3. We denote by u the unit group in k, i.e. u=o*. Let H be a connected algebraic group consisting only of semi-simple elements and put (2.7) HU={h~H[z(h) au for all z~X(H)}. Then we have the following PROPOSITION 2. 3. The notations bei~ as above, H" is the unique maximal compact subgroup of H; it is a normal subgroup of H, containing H 1. Moreover, there exists a subgroup D of H, isomorphic to Z ~ (~=rank X(H)), such that (2.8) H=D.H u, DnHU={:}. Proof. Let (I) : H-+(k*) ~ be as defined by (2.6). Then, since [Y : X(H)]<~ (Proposition 2.2), the restriction of (I) on A is a k-isogeny. Therefore, as is well-known, (I)(A) is an open subgroup (in the sense of the p-adic topology) of (k*)', of finite index, and so is also (I)(H). Now it is clear that H a contains H t, which is compact. Since HU/Ha~(I)(H ~) =(1)(H)ntff, we see that H ~ is compact. Since H/H a is commutative, H" is a normal subgroup of H. On the other hand, H/H"~(I)(H)/(I)(HU), being isomorphic to Z ~, does not contain any compact subgroup. Therefore H u is a maximal compact subgroup and, since it is normal, it is the unique maximal compact subgroup. The existence of the subgroup D is obvious, q.e.d. COROLLARY. A"= A raH u is the unique maximal compact subgroup of A. We put X(H) = Hom(X(H), Z). For every hall, the correspondence l h : X(H) ~Z defined by (2.9) lh(x) = ord,z(h) for Z a X(H) is an element of Ni(H), and the correspondence h ~l~ is a homomorphism from H into :X(H), whose kernel is equal to H ~. Thus, denoting by M the image of this homomorphism, one has (2. IO) H/H"~M. When the decomposition (2.8) is fixed once for all, the homomorphism h~l~, induces an isomorphism of D onto M. Hence, when la=-m with daD, roaM, one writes d=T~ Ba ; by definition, one has Iz( m) Ip=q - for all xeX(H), rneM, (2. and X(H). < > denoting the pairing of X(H) 238 THEORY OF SPHERICAL FUNCTIONS I5 If, in particular, H satisfies the condition (N), one has M=X(H). This is surely the case for A. Thus we obtain the following commutative diagram A/A~ in~, H/HU (2. ~3) 9 inj.> X(H) which allows us to consider that ~'r w 3- Fundamental assumptions. 3. x. Assumption (I). Let G be an algebraic group over k. We shall now make two fundamental assumptions (I), (II) on G. In the first place, we assume (I) There exist, in G, an open compact subgroup U, a connected (reductive) k-closed subgroup H consisting only of semi-simple elements and a k-unipotent subgroup N normalized by H such that the following condition.r are satisfied: (3.1) G=U.HN=U.H.U, (3.2) U~ H u. Let A be the unique maximal k-trivial torus in H; then we have (3.3) AcHcZ(A). Since AN is a k-closed subgroup of G such that G/AN is compact (by (3. i) and Propo- sition 2.2), it follows from Proposition i. I that (A, N) is a k-Borel pair in G. Let D be a subgroup of H as described in Proposition 2.3. Then by (3. i), (3.2) one has (3. I)' G=U.DN-----U.D.U. From this one concludes at once that U is a maximal compact subgroup of G and so, by Proposition 1.2, that G is reductive. Now HN is clearly a semi-direct product of H, N over k. Moreover one has (3.4) HNnU =H". (Nn U). In fact, for ueHNnU, write u=hn with hell, neN. Then the correspondence u-+h being a continuous homomorphism (with respect to the p-adic topology), one concludes that its image is compact and so contained in H", by Proposition 2.3. LEMMA 3" I. Under the assumption (I), suppose further that G is connected, that H satisfies the condition (N) and that HnG 1 is connected, G 1 being defined by (2.5). Then we have (3.5) G = H. G 1, U ----- H". U 1, where U 1 ---- U n G 1. Proof. Since H contains the unique maximal k-trivial torus A' in the center of G and since Hr~G t is connected, it follows from what we stated in n o 2.2 239 ICHIRO SATAKE that H,~(HnG 1) � and G=cl(H.GX). Hence the restriction homomorphism X(G) ~X(H) is injective and, if one identifies X(G) with its image (i.e. the annihilator in X(H) of HnG 1, whichis connected), X(H)/X(G) has no torsion [i8]. Therefore one may take a system of (independent) generators {Z1, ..., Z~} of X(H) in such a way that {;tl, -.-, Z~,} ('/=rank X(G)) forms a system of generators of X(G). Since H satisfies the condition (N), one concludes from this immediately that for any gaG there exists hall such that z(g)--z(h) for all z~X(G), i.e. h-~g~G 1, which proves the first equality (3.5). If, in the above, gEU, one has z(g)Eu for all zeX(G), so that one may choose the above h so as to belong to H". This proves the second equality (3-5), q.e.d. Remark. In case H=-Z(A), the third assumption in Lemma 3.I is surely satisfied. In fact, call A" the connected component of the neutral element of AnG 1. Then clearly A=cl(A'.A") and one sees that Z(A)oG 1 is equal to the centralizer of A" in G 1. Hence Z(A)nG 1 is connected ([x], Prop. I8.4). It follows that, under the assumptions of Lemma 3. I, one may replace U by U 1 in (3. I), i.e. (3. I )" G = U 1 . HN = U 1 . H. U 1. Furthermore, since N cG 1, one gets also (3.6) G 1 = U 1 . (Hn G1)N = g 1 . (Ha GX). U 1. This shows that G 1, U 1, HoG 1, N also satisfy assumption (I). 3-2. We give here several procedures which allow us to construct groups satisfying assumption (I), starting from other such groups. PROPOSITION 3.I. If G~,Ui, Hi, N~(i=I,2) satisfy (I), so do also GI� Ut < U2, H1 � H2, Nt � N2. Trivial. PROPOSITION 3.2. Let G, U, H, N satisfy (I) and assume further that G, H satisfy the conditions stated in Lemma 3. i. Let X=X(G) and XI a submodule of X such that X/X 1 has no torsion. Put G*={geGlz(g )=I for all z~X~}. Then, G*, U*=Uc~G*, H* =HnG*, N also satisfy assumption (I). Proof. One identifies X with the character module of the k-trivial torus cl(G/GI). Then, (3* is the (complete) inverse image, under the canonical homo- morphism G~cl(G/G1), of the annihilator of Xl in cl(G/G1), which is a subtorus by the assumption [x8]. Therefore G* is connected. As we have done in the proof of Lemma 3-I, one may consider that XcX(H); then, by our assumptions, X(H)/X1 also has no torsion. Hence, by the same reason as above, H* is connected. Now, since G* contains U 1 and N, it follows from (3. I)" that G* = U1.H*N = U1.H*.U 1 and afortiori (3.I) for G*, U*, H*, N. One has also H*"=HUaG*cU *, i.e. (3.2), q.e.d. 240 THEORY OF SPHERICAL FUNCTIONS I7 PROPOSITION 3"3" Let G, U, H, N satisfy (I). Let Z be a k-trivial torus contained in the center of G. Put (3.7) H0={heHlhmeZH u for some meZ} and suppose that H 0 normalizes U. Then G = G/Z, [J= (HoU)/Z, H = H/Z, N---- NZ/Z also satisfy (I). Proof. First we note that, since Z is a k-trivial torus, the canonical homo- morphism G--+cl(G/Z) is surjective, i.e. el(G/Z) =G/Z. Since Z is contained in any maximal k-trivial torus, we have ZcH. Hence, from the definition, it follows that H 0 is an open subgroup of H such that H0/(ZH" ) is the torsion part of H](ZH"). Ho[Z is therefore the unique maximal compact subgroup of H = H/Z. Furthermore, by the assumption, one sees that (HoU)/Z= (H0/Z) . (UZ/Z) is an open compact subgroup of G = G/Z. Our Proposition is now obvious, q.e.d. 3.3. Weyl groups. Let the notations be as in N ~ 3.I. For seN(H) (=the normalizer of H), the inner automorphism I~ defined by s induces an automorphism of H, and hence that of X(H), which we call ws, by the formula (3.8) (w~z)(shs -1) =)~(h) for all hell, ;ceX(H). Since A is the unique maximal k-trivial torus contained in the center of H, I~ leaves A invariant. Therefore, w~ can be extended to a (uniquely determined) automorphism of Y= X(A), which we denote again by w,, by (3.9) (ws~)(sas -1) =~(a) for all aeA, ~eY. The group formed of all w~ (seN(H)) is called the (restricted) Weyl group of G relative to H and is denoted by W~. The kernel of the homomorphism s~w~ being given by N(H) n Z(A), one has (3. io) Wn:~N(H)/(N(H ) ca Z(A)). As stated above, W a may be regarded as a subgroup of the Weyl group W A of G relative to A. In case G is connected, W A is the Weyl group of the restricted root system i: in the usual sense [20]. W a also operates on :~(H)=Hom(X(H), Z) in a natural manner, i.e. by (3. II) <w% wz> =<~, Z> for xeX(H), ~eX(H). Then (in the notation of nO 2.3), for seN(H), hell, one gets from (2.9), (3.8), (3.11) the relation Ws lh = l~h 8_,. (3.'2) Thus W H leaves Mr invariant. Suppose that there is given a subgroup W of W H such 3.4. Assumption (II). that every weW can be written in the form w=w~ with ueN(H)nU. As we have seen in n o 3.3, W operates on M. Taking a linear order in M, put (3.13) A={meM[wm~<m for all weW}. 241 ICHIRO SATAKE Then it is clear that A is a " fundamental domain " of W in M, i.e. every meM is equivalent, under W, to one and only one element in A. From (3. I)', (3.12) and from our assumption on W, it follows that (3" 14) G = O UrdU. r@A Now we state our second assumption: (II) The notations being as defined in n ~ 2.3, 3.3, there exists a subgroup W of W R such that every weW can be written in the form w=-w, with u~N(H)nU and a linear order in M satisfying the following property: If nmNnUnrU+o with meM, reA, we have m~<r, where A is a fundamental domain Of W in M defined by (3. I3)- This implies the following weaker condition: (II1) The notations being as above, the double cosets Urc'U (rEA) are mutually distinct. (In other words, (3.14) is a disjoint union.) In fact, let u~ru=urfu with r, r'~A. Since we have ="cUrt'U, it follows from (II) that r'~<r. Similarly we have r~<r' and so r=r'. Under the assumption (II1), every geG can be expressed in the form g = u~du', u, u' eU with a uniquely determined reA; therefore one puts r=r(g). Then the function r:G-+A is characterized by the following properties (3.15) r(ugu') =r(g) for all geG, u, u'eU, r(~') =r for all reA; if moreover iII) is satisfied, we have (3.16) r(~mn)~>m for all meM, neN. The existence of the function r satisfying (3.15), (3. i6) (resp. (3. I5)) is equivalent to (II) (resp. (II1)). We list below some direct consequences of the assumption (II). i ~ If ~zmNnU~eo, we have m=o. In fact, it follows from (II) that m.<<o. Since we have also n-"NoU=(nmNnU)-l+o, we have m~>o; hence m=o. 2 ~ For h~H, one has with ueN(H) nU. h =- urcr(h)u-1 (rood. H ~) (3.17) if and only if one has UhU = Uh'U It follows that, for h, h'eH, with ueN(H) nU. h' = uhu- 1 (mod. H u) (Note that this is a consequence of only (II1).) 3 0 r(h) for heH is invariant under the inner automorphisms of G, i.e. /f h, h'eH and h'=ghg -x with geG, we" have r(h')=r(h). In fact, it is clear that, in replacing h by uhu -~ with u~U, if necessary, we may assume, without any loss of generality, 242 THEORY OF SPHERICAL FUNCTIONS I9 that lheA , i.e. lh=r(h ) (and similarly that lh,=r(h')). Now, let g-=uhln with ueU, haeH, neN. Then one has h' =ghg -~ -=uhlnhn-lh-;-tu -t = u(hlhh; -1) n'u -1 with n'eN. Hence, by (3. I5), (3. I6), one gets r(h') = r((hlhh~)n ') ~ lh,hhr, = l h = r(h). Similarly, one gets r(h)>_.r(h'); hence r(h') =r(h), as desired. 4 ~ We have W=WH=W A. In fact, if W.Wa, there would be r.r' in An~" such that r'=wr with weWa, because Y is of finite index in M and An~" is a fundamental domain of W in Y. Then we would have ="--- srds -1 (mod. H ~) with seN(A), which contradicts 3 ~ . w 4. Haar measures. 4.x. In this section, G denotes an algebraic group over k satisfying the assumption (I) with respect to U, H, N. The groups G, U, H, N are then all " unimodular ", i.e. their left-invariant Haar measures are also right-invariant. We denote by dg, du, dh, dn the volume-elements of the (both-sides-invariant) Haar measures of G, U, H, N, respectively, normalized as follows: (4. 1 fjg=fJu=f .dh=f o an=,. Then the left- and right-invariant Haar measures of HN are given by (4.2) d,(hn) =dh.dn, dr(hn ) =~(h)dh.dn, being a positive quasi-character of H (i.e. a continuous homomorphism of H into the multiplicative group of positive real numbers with respect to the p-adic topology) defined by (4-3) d(hnh-~) = 3(h)dn. For any integrable function f on G, one has or symbolically (4.4) dg=du, dr(hn) = du. 3(h)dh dn. We need in Chapter II, w 5 the following transformation formula of the relatively invariant measure on U/(UnHN). LEMMA 4.I. Let g0eG. For ueU, write golU=u'h'n ' with u'eU, h'eH, n'eN. Then the cosets u'(Uc~HN), h'H u are uniquely determined by go and u(UnHN). Denoting by d K the volume-element of a relatively invariant measure on U/(U c~ HN), we have (4.5) dg=~(h')dK'. (Note that, since ~(H")= i, 3(h') depends only on the coset h'HU.) 243 20 ICHIRO SATAKE Proof. By (3.4), the first statement is obvious. To prove the second, let h, n be " generic " elements in H, N, respectively, and put g=uhn. Then go~g=u'h'n'hn=u'(h'h)(h-ln'hn). Hence, by (4.4) and by the invariance of the Haar measures, one has d(gol g) = du'. ~( h'h)d(h'h)d(h-ln'hn) = du'. 3(h')3(h)dh dn whence follows (4.5), q.e.d. 4.2. An integral formula. Let 2gY g = go q- be the decomposition of g given in N ~ 2. I. Since HcZ(A), all the subspaees g3" (ye~:) are invariant under Ad h (hell), Ad denoting the adjoint representation of G. One denotes by Rv(h ) the restriction of Ad h to g3". Then det(R3,(h)) is a k-rational character of H, whose restriction to A is equal to d3,. Y (in the additive notation), d3, denoting the dimension of g3,- Thus, identifying X(H) with a submodule of Y=X(A), one gets d3,.yeX(H ) and (4.6) det (R3, (h)) = (d3,. u (h) for h e S. Moreover, taking a so-called Weyl basis of g~= g| (k--algebraic closure of k), one sees immediately that g-3, may be identified with the dual of gv with respect to the inner product induced by the Killing form. Since this inner product is invariant under Ad h (hell), one has (4.7) R_3" is equivalent to tR~-l. Now, for hell, denote by Ad,(h) the restriction of Ad h on n=5093", and put (4.8) A(h) = I det(Ad.(h) -- x.)I. = 1-I l det(R3,(h)-- I3,)tp, 3">0 i,, I3, denoting the identity transformations on n, g3", respectively. Then we have LEMMA 4.2. For heI-I with A(h)+o, the mapping 9 ~ : Nsn ---, n' =hnh-ln -1 is an injective rational mapping from N into itself, the image ~h(N) contains a Zariski open set in N and one has (4.9) an' = A(h),Zn. Proof. Every neN can be written uniquely in the form n=exp x=exp(Sx3" ) with x=3,>0 ~ x3"en' x3"~g3" and one has, for hell, hnh -~ = exp(Ad(h)x) = exp( ~2 R3,(h)x3, ). 3'>0 244 THEORY OF SPHERICAL FUNCTIONS 2I Therefore one has hnh-~=n it" and only if Rv(h)xv=x v for all y>o; in particular, if A(h) 4:o, the latter condition implies that x----o, i.e. n=i. Thus, for h~H with A(h) 4: o, ~h is an injective rational mapping from N into itself. Applying the same consideration to N ~, k denoting the algebraic closure of k, one sees that Wh(N ~) contains a Zariski open set in N ~ ([x6], p. 88, Prop. 4)- Now, let neN ~. Since one has ~h(n ~) = tFh(n)~ for all automorphisms ~ of k over k, it follows from the injectivity of ~h that, if qPh(n)eN, one has n"=n for all a, i.e. heN. In other words, one has tFh(N ) -----~h(N~)nN, which proves that ~(N) contains a Zariski open set in N. Now to prove the last assertion, we regard x6N as a left-invariant vector-field on N (in the algebraic sense) and denote by x n the tangent vector at n6N determined by x. Then one has (4. Io) dtt~(Xn) = (Ad(n). (Ad(h)--I)X)h~h-,.-, , dtFh denoting the " differential " of the rational mapping ~h. In fact, by definitions, one has for any rational funcfionf on N, defined over k and regular at no, and therefore (dVh.x..)(f) ---- x,,(fotFh) which proves (4. IO). Now if we denote by o~ an invariant differential form of the highest degree on N and by td~ h the linear mapping on the space of differential forms on N extending the dual of d~n, it follows tYom (4. IO) and from the fact that det(Ad,(n0) ) = I that 'd~ h . %,~_,,_, = det (Ad,,(h) -- I,)%. Since we have symbolically dn=[%[p, up to a constant multiple ([26], 2.2), we obtain (4-9), q.e.d. By the similar argument as above, we get also ~(h) = [det(Ad,(h))[p = II [det(Rv(h))Iv, "f>0 or by (4.6) (in the multiplicative notation). (4. II) y>O 245 22 ICHIRO SATAKE Put further 1/2 (4.12) D(h) -= l-I [det(Rv(h ) -- Iv) p Then from the definitions and from (4.7) one gets easily the following relations (4.13) D(h -1) = D(h), (4.14) D(h) =~ 2(h)A(h). It should be also noted that one has A(hlhh7 -1) ----A(h) for all h, hleH. (4.15) Let f be a (complex-valued),function on G with a compact carrier, satisfying LEMMA 4" 3" the relation for all geG, u, u'eU. f (ugu') =f(g) Then, for hell with A(h) @ o, we have (4.16) D(h) dg denoting the volume-element of a (suitably normalized) relatively invariant measure on G/A (1). Proof. Since one has dg = du. dn. dh for g = unh, one has symbolically dg=du.dn.dh, dh denoting the volume-element of a relatively invariant measure on H/A; here we normalize dh in such a way that fH dh = I. Then the left-hand side /A of (4.16) is equal to = D(h) fI~a (f, f(nhihh-Ztn-') dn)dh, (by the assumption) =D(h)A(h-')-tfm A (fNf(hthh-:tn')dn')dh, (by (4.9), (4.i5)) 9 Since hthh~ -1 -h (mod. Hu), one gets from the assumption f(hlhh3tn ') =f(hn') ; therefore, by (4.13), (4.14), this last expression is equal to = ~(h) f~f(hn) dn, q.e.d. Since D(h) is invariant under the inner automorphisms defined by elements in N(H), this Lemma implies that, if one puts /(h) = 2(h) f f(hn) dn, t~J f(n'), viewed as a function of meM, is invariant under the operation of the Weyl group W~. (x) This is an analogue of an integral formula of Harish-Chandra ([18], p. 261). :j(hn)dn, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Reductive algebraic groups over p-adic fields

Publications mathématiques de l'IHÉS , Volume 18 (1) – Aug 7, 2007

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Copyright © 1963 by Publications mathématiques de l’I.H.É.S
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Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
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10.1007/BF02684782
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Abstract

CH'APTER I REDUCTIVE ALGEBRAIC GROUPS OVER p-ADIC FIELDS (1) w I. k-Borel subgroups. x .x. k-Borel pairs. Let k be a p-adic number field. An algebraic group G over k is called a torus, if it is connected, commutative and consisting only of semi- simple elements. For a torus G over k, there exists a finite extension K/k such that G K is K-isomorphic (i.e. birationally isomorphic over K) to (K*)~; such a field K is called a " splitting field " of G. In case this isomorphism is obtained in k, i.e. in case G is k-isomorphic to (k*) ~, G is called a k-trivial torus. An algebraic group G over k is called unipotent, if it only contains unipotent elements; a unipotent algebraic group G over k is always connected and nilpotent. Let G be an algebraic group over k. A maximal k-trivial torus (resp. maximal k-unipotent subgroup) in G is a k-trivial torus (resp. k-closed unipotent subgroup) in G which is maximal with respect to this property. A pair (A, N) of a maximal k-trivial torus A and a maximal k-unipotent subgroup N in G such that A normalizes N is called a k-Borel pair. For such a pair (A, N), AN becomes a k-closed subgroup of G, called a k-Borel subgroup of G, which is a semi-direct product over k of A and N. A typical example of a k-Borel pair is G = GL(n, k), A=D(n, k) (the group of all diagonal matrices in GL(n, k)), N=TU(n, k) (the group of all upper unipotent matrices, i.e. matrices x=(~i) with ~u=I,~q=o for i>j, in GL(n, k)). It is known (Borel) that, for any pair (A, N) of a k-trivial torus A and a k-closed unipotent subgroup N in G such that A normalizes N, there exists always a k-Borel pair (A', N') with A'3A, N'~N and that all k-Borel pairs in G are conjugate to each other with respect to the inner automorphisms of G [x2]. It follows that, for any Borel pair (A, N) in G, we can transform G by an inner automorphism of the " ambient group " GL(n, k) in such a way that A coincides with the connected component of the neutral element of GraD(n, k) and N coincides with GriTS(n, k). (1) For the fundamental concepts on algebraic groups, see [1]; especially on algebraic toruses, see [18]. Cf. also [7], [12], [20]. 234 THEORY OF SPHERICAL FUNCTIONS tl 9 .2. The following proposition gives a characterization of k-Borel subgroups in terms of the p-adic topology. PROPOSITION I.I. Let G be an algebraic group over a p-adic number field k. Then, for a k-closed subgroup H of G, the homogeneous space G/H is compact (in the p-adic topology), if and only if H contains a k-Borel subgroup of G. Proof. The proof is obviously reduced to the case where G is connected. Hence, assuming G to be connected, let H be a k-Borel subgroup of G. Then since G/H is identified (birationally) with the set of k-rational points in a projective variety defined over k [x2], it is compact; whence follows the " if " part of the Proposition. Conversely, let H' be a k-closed subgroup of G such that G/H' is compact. One can take a k-Borel subgroup H of G containing a k-Borel subgroup of H'. Then, from what we have proved above, H'/(HnH') is compact. Therefore HH' is closed in the p-adic topology, so that H/(HnH')~HH'/H' is compact by the assumption. But, since H is a k-Borel subgroup, it has a composition series (as algebraic group over k) : H = H0~ HI~... ~ H r ={ I } such that the factor groups Hi_l/Hi are isomorphic either to k* or to k; and it is then a trivial matter to prove inductively that H~(HnH')=H(I~<i<r). Thus we get H=HnH', i.e. H'~H, q.e.d. COROLLARY. An algebraic g~oup G over k is compact (in the p-adic topology) if and only if k-Borel subgroups of G reduce to the neutral element. In view of a decomposition theorem of Chevalley ([7], P. I44) it follows that a compact algebraic group over k is necessarily reductive, i.e. isogeneous to the direct product of a semi-simple algebraic group and a torus. Remark. Proposition I. I and its Corollary are also valid for algebraic groups over R or C. (The same proof l) 9 .3. Maximal compact subgroups. PROPOSITXON I .2. If an algebraic group G over k has a maximal compact subgroup, G is reductive. Pro@ Suppose that G is not reductive. Then, by virtue of the decomposition theorem of Chevalley, G has a k-closed unipotent normal subgroup N1 + { I }. The center Nz of N1, being a unipotent commutative group, is k-isomorphic to a vector space over k, i.e. we have a k-isomorphism f : N~ ~k "~ (m> o). Then the inner automorphism I 0 defined by g~G induces an automorphism of N~, which, by f, corresponds to a linear transformation, denoted by pg, of k m, Let U be a maximal compact subgroup of G and let Ll=omck m. Then, Ul={ueUIPuLt=L~} is an open subgroup of U and hence is of finite index in U,. Put U = y u~U1, L = 0 Pu' LI" Then L is an " Q-lattice " in k m (see N ~ 8. i) invariant under p. (u~U). It follows that U(r U.f-l(rc-iL) (i= i, 2, ...) 235 ICHIRO SATAKE I2 are all compact subgroups of G containing U, so that, by the maximality of U, one gets f-I(=-~L) cU (i=I, 2, ...). This implies that N2cU, which is a contra- diction, q.e.d. Now, for a reductive algebraic group G over R, it is well-known that G has always a maximal compact subgroup, which is R-closed, and that all maximal compact subgroups of G are conjugate to each other with respect to the inner automorphisms of G. Moreover, if (A, N) is an R-Borel pair in G and if A+ denotes the connected component (in the sense of the usual topology) of the neutral element of A, we have (I.I) G=U.A+N, Un (A+N)={i} for all maximal compact subgroup U, and (1.2) G=UA+U for a suitable U. Unfortunately, in the p-adic case, we are not yet in possession of such a general result. Since, however, the existence of a maximal compact subgroup with properties similar to (I. i), (I. 2) is indispensable to the theory of spherical functions, we will assume it in w 3; for classical groups, our assumptions will be verified case by case in Chapter III. On the other hand, it should be noted that, in the p-adic case, the maximal compact subgroups of a reductive algebraic group G are, if they exist, not necessarily conjugate to each other with respect to the automorphisms of G and that they are not necessarily corresponding in one-to-one way under a k-isogeny. w 2. Reductive algebraic groups. 2.x. In this section, G denotes a connected reductive algebraic group over a p-adic number field k. Let A be a maximal k-trivial torus in G, Y the character module of A and let dim A = rank Y = ~. If" {~1, .--, :%} is a system of generators of Y, the correspondence (2. x) A~a ~ (~ql(a), ..., ~%(a)) e(k*) ~ gives a k-isomorphism A~(k*) ". Let g, a be the Lie algebras of G, A, respectively. It is easy to see that, for any k-homomorphism p from A into another algebraic group over k, the closure of the image o(A) is again a k-trivial torus. Applying this to p = adjoint representation of G, we have (2.2) g=g0+ Z g~, yGr where icY is the " restricted root system " relative to A [2o] and (2.3) gv={xegIAd(a)x=y(a)x for all aeA}, go={X~glAd(a)x=o for all aea}. 236 THEORY OF SPHERICAL FUNCTIONS I3 It is clear that g0 is the Lie algebra of Z(A) (= the centralizer of A). On the other hand, for any linear order in Y, put (~.4) n = Z g-c y>0 and call N the corresponding connected subgroup of G. Then we have [2o]: PROPOSITION 2. I The notations being as above, Z(A) is a connected reductive algebraic group over k consisting of only semi-simple elements, N is a maximal k-unipotent subgroup of G, normalized by Z(A), and Z(A).N is a semi-direct product of Z(A) and N over k. 2.2. For any connected reductive algebraic group G over k, we denote by X(G) the module of all k-rational characters (i.e. k-morphisms of G into k*). Furthermore we put (~.5) GX={geGtz(g)=I for all zeX(G)}. Then G 1 is a connected k-closed normal subgroup of G and there exists a k-trivial torus A' contained in the center of G such that G = cl(G1.A'), GlnA'= finite; in other words, the natural homomorphism gives a k-isogeny: G 1 � To see this, let S, T the semi-simple and the torus parts of G, respectively. Since S cG x, it is clear that the closure of the canonical image of G x in cI(G/S) is equal to el(G/S) x. On the other hand, for a torus, our assertion is known ([2o], Prop. I), i.e. irA" denotes the (unique) maximal k-trivial torus in T, we have a k-isogeny T 1 � It follows also that cl(G/S) 1, and consequently G x is connected. Under the k-isogeny T-+el(G/S), induced by the canonical homomorphism, T 1 corresponds to cl(G/S) 1, whence one gets G l=cl(T a.S). Thus one has G~T�215215215 ', where G~G' means that G is isogeneous to G'. This proves our assertion. It follows that the homomorphism X(G) -~X(A') defined by the restriction is injective and has a finite cokernel. In particular, if G consists only of semi-simple elements (which implies necessarily that G is reductive), it follows from Proposition 2.I that G=Z(A) (i.e. A=A' in the above notation), and hence by Corollary to Proposition i. I that G 1 is compact. Thus we obtain the following PROPOSITION ~. 2 Let H be a connected algebraic group over k consisting only of semi-simple elements and A the (unique) maximal k-trivial toms in H. Then H 1 is compact and H is k-isogeneous to the direct product ofH 1 and A. Moreover, X(H) may be identified with a submodule of X(A) =Y with finite index. Let {Z1, -..,~} be a system of independent generators of X(H). Then the mapping (2.6) 9 : U~h > (z~(h), ..., z~(h)) ~(k*) ~ 237 ICHIRO SATAKE x4 defines an injective homomorphism of H/H 1 into (k*)L In case this mapping is surjective, we say that H satisfies the condition (N). 2.3. We denote by u the unit group in k, i.e. u=o*. Let H be a connected algebraic group consisting only of semi-simple elements and put (2.7) HU={h~H[z(h) au for all z~X(H)}. Then we have the following PROPOSITION 2. 3. The notations bei~ as above, H" is the unique maximal compact subgroup of H; it is a normal subgroup of H, containing H 1. Moreover, there exists a subgroup D of H, isomorphic to Z ~ (~=rank X(H)), such that (2.8) H=D.H u, DnHU={:}. Proof. Let (I) : H-+(k*) ~ be as defined by (2.6). Then, since [Y : X(H)]<~ (Proposition 2.2), the restriction of (I) on A is a k-isogeny. Therefore, as is well-known, (I)(A) is an open subgroup (in the sense of the p-adic topology) of (k*)', of finite index, and so is also (I)(H). Now it is clear that H a contains H t, which is compact. Since HU/Ha~(I)(H ~) =(1)(H)ntff, we see that H ~ is compact. Since H/H a is commutative, H" is a normal subgroup of H. On the other hand, H/H"~(I)(H)/(I)(HU), being isomorphic to Z ~, does not contain any compact subgroup. Therefore H u is a maximal compact subgroup and, since it is normal, it is the unique maximal compact subgroup. The existence of the subgroup D is obvious, q.e.d. COROLLARY. A"= A raH u is the unique maximal compact subgroup of A. We put X(H) = Hom(X(H), Z). For every hall, the correspondence l h : X(H) ~Z defined by (2.9) lh(x) = ord,z(h) for Z a X(H) is an element of Ni(H), and the correspondence h ~l~ is a homomorphism from H into :X(H), whose kernel is equal to H ~. Thus, denoting by M the image of this homomorphism, one has (2. IO) H/H"~M. When the decomposition (2.8) is fixed once for all, the homomorphism h~l~, induces an isomorphism of D onto M. Hence, when la=-m with daD, roaM, one writes d=T~ Ba ; by definition, one has Iz( m) Ip=q - for all xeX(H), rneM, (2. and X(H). < > denoting the pairing of X(H) 238 THEORY OF SPHERICAL FUNCTIONS I5 If, in particular, H satisfies the condition (N), one has M=X(H). This is surely the case for A. Thus we obtain the following commutative diagram A/A~ in~, H/HU (2. ~3) 9 inj.> X(H) which allows us to consider that ~'r w 3- Fundamental assumptions. 3. x. Assumption (I). Let G be an algebraic group over k. We shall now make two fundamental assumptions (I), (II) on G. In the first place, we assume (I) There exist, in G, an open compact subgroup U, a connected (reductive) k-closed subgroup H consisting only of semi-simple elements and a k-unipotent subgroup N normalized by H such that the following condition.r are satisfied: (3.1) G=U.HN=U.H.U, (3.2) U~ H u. Let A be the unique maximal k-trivial torus in H; then we have (3.3) AcHcZ(A). Since AN is a k-closed subgroup of G such that G/AN is compact (by (3. i) and Propo- sition 2.2), it follows from Proposition i. I that (A, N) is a k-Borel pair in G. Let D be a subgroup of H as described in Proposition 2.3. Then by (3. i), (3.2) one has (3. I)' G=U.DN-----U.D.U. From this one concludes at once that U is a maximal compact subgroup of G and so, by Proposition 1.2, that G is reductive. Now HN is clearly a semi-direct product of H, N over k. Moreover one has (3.4) HNnU =H". (Nn U). In fact, for ueHNnU, write u=hn with hell, neN. Then the correspondence u-+h being a continuous homomorphism (with respect to the p-adic topology), one concludes that its image is compact and so contained in H", by Proposition 2.3. LEMMA 3" I. Under the assumption (I), suppose further that G is connected, that H satisfies the condition (N) and that HnG 1 is connected, G 1 being defined by (2.5). Then we have (3.5) G = H. G 1, U ----- H". U 1, where U 1 ---- U n G 1. Proof. Since H contains the unique maximal k-trivial torus A' in the center of G and since Hr~G t is connected, it follows from what we stated in n o 2.2 239 ICHIRO SATAKE that H,~(HnG 1) � and G=cl(H.GX). Hence the restriction homomorphism X(G) ~X(H) is injective and, if one identifies X(G) with its image (i.e. the annihilator in X(H) of HnG 1, whichis connected), X(H)/X(G) has no torsion [i8]. Therefore one may take a system of (independent) generators {Z1, ..., Z~} of X(H) in such a way that {;tl, -.-, Z~,} ('/=rank X(G)) forms a system of generators of X(G). Since H satisfies the condition (N), one concludes from this immediately that for any gaG there exists hall such that z(g)--z(h) for all z~X(G), i.e. h-~g~G 1, which proves the first equality (3.5). If, in the above, gEU, one has z(g)Eu for all zeX(G), so that one may choose the above h so as to belong to H". This proves the second equality (3-5), q.e.d. Remark. In case H=-Z(A), the third assumption in Lemma 3.I is surely satisfied. In fact, call A" the connected component of the neutral element of AnG 1. Then clearly A=cl(A'.A") and one sees that Z(A)oG 1 is equal to the centralizer of A" in G 1. Hence Z(A)nG 1 is connected ([x], Prop. I8.4). It follows that, under the assumptions of Lemma 3. I, one may replace U by U 1 in (3. I), i.e. (3. I )" G = U 1 . HN = U 1 . H. U 1. Furthermore, since N cG 1, one gets also (3.6) G 1 = U 1 . (Hn G1)N = g 1 . (Ha GX). U 1. This shows that G 1, U 1, HoG 1, N also satisfy assumption (I). 3-2. We give here several procedures which allow us to construct groups satisfying assumption (I), starting from other such groups. PROPOSITION 3.I. If G~,Ui, Hi, N~(i=I,2) satisfy (I), so do also GI� Ut < U2, H1 � H2, Nt � N2. Trivial. PROPOSITION 3.2. Let G, U, H, N satisfy (I) and assume further that G, H satisfy the conditions stated in Lemma 3. i. Let X=X(G) and XI a submodule of X such that X/X 1 has no torsion. Put G*={geGlz(g )=I for all z~X~}. Then, G*, U*=Uc~G*, H* =HnG*, N also satisfy assumption (I). Proof. One identifies X with the character module of the k-trivial torus cl(G/GI). Then, (3* is the (complete) inverse image, under the canonical homo- morphism G~cl(G/G1), of the annihilator of Xl in cl(G/G1), which is a subtorus by the assumption [x8]. Therefore G* is connected. As we have done in the proof of Lemma 3-I, one may consider that XcX(H); then, by our assumptions, X(H)/X1 also has no torsion. Hence, by the same reason as above, H* is connected. Now, since G* contains U 1 and N, it follows from (3. I)" that G* = U1.H*N = U1.H*.U 1 and afortiori (3.I) for G*, U*, H*, N. One has also H*"=HUaG*cU *, i.e. (3.2), q.e.d. 240 THEORY OF SPHERICAL FUNCTIONS I7 PROPOSITION 3"3" Let G, U, H, N satisfy (I). Let Z be a k-trivial torus contained in the center of G. Put (3.7) H0={heHlhmeZH u for some meZ} and suppose that H 0 normalizes U. Then G = G/Z, [J= (HoU)/Z, H = H/Z, N---- NZ/Z also satisfy (I). Proof. First we note that, since Z is a k-trivial torus, the canonical homo- morphism G--+cl(G/Z) is surjective, i.e. el(G/Z) =G/Z. Since Z is contained in any maximal k-trivial torus, we have ZcH. Hence, from the definition, it follows that H 0 is an open subgroup of H such that H0/(ZH" ) is the torsion part of H](ZH"). Ho[Z is therefore the unique maximal compact subgroup of H = H/Z. Furthermore, by the assumption, one sees that (HoU)/Z= (H0/Z) . (UZ/Z) is an open compact subgroup of G = G/Z. Our Proposition is now obvious, q.e.d. 3.3. Weyl groups. Let the notations be as in N ~ 3.I. For seN(H) (=the normalizer of H), the inner automorphism I~ defined by s induces an automorphism of H, and hence that of X(H), which we call ws, by the formula (3.8) (w~z)(shs -1) =)~(h) for all hell, ;ceX(H). Since A is the unique maximal k-trivial torus contained in the center of H, I~ leaves A invariant. Therefore, w~ can be extended to a (uniquely determined) automorphism of Y= X(A), which we denote again by w,, by (3.9) (ws~)(sas -1) =~(a) for all aeA, ~eY. The group formed of all w~ (seN(H)) is called the (restricted) Weyl group of G relative to H and is denoted by W~. The kernel of the homomorphism s~w~ being given by N(H) n Z(A), one has (3. io) Wn:~N(H)/(N(H ) ca Z(A)). As stated above, W a may be regarded as a subgroup of the Weyl group W A of G relative to A. In case G is connected, W A is the Weyl group of the restricted root system i: in the usual sense [20]. W a also operates on :~(H)=Hom(X(H), Z) in a natural manner, i.e. by (3. II) <w% wz> =<~, Z> for xeX(H), ~eX(H). Then (in the notation of nO 2.3), for seN(H), hell, one gets from (2.9), (3.8), (3.11) the relation Ws lh = l~h 8_,. (3.'2) Thus W H leaves Mr invariant. Suppose that there is given a subgroup W of W H such 3.4. Assumption (II). that every weW can be written in the form w=w~ with ueN(H)nU. As we have seen in n o 3.3, W operates on M. Taking a linear order in M, put (3.13) A={meM[wm~<m for all weW}. 241 ICHIRO SATAKE Then it is clear that A is a " fundamental domain " of W in M, i.e. every meM is equivalent, under W, to one and only one element in A. From (3. I)', (3.12) and from our assumption on W, it follows that (3" 14) G = O UrdU. r@A Now we state our second assumption: (II) The notations being as defined in n ~ 2.3, 3.3, there exists a subgroup W of W R such that every weW can be written in the form w=-w, with u~N(H)nU and a linear order in M satisfying the following property: If nmNnUnrU+o with meM, reA, we have m~<r, where A is a fundamental domain Of W in M defined by (3. I3)- This implies the following weaker condition: (II1) The notations being as above, the double cosets Urc'U (rEA) are mutually distinct. (In other words, (3.14) is a disjoint union.) In fact, let u~ru=urfu with r, r'~A. Since we have ="cUrt'U, it follows from (II) that r'~<r. Similarly we have r~<r' and so r=r'. Under the assumption (II1), every geG can be expressed in the form g = u~du', u, u' eU with a uniquely determined reA; therefore one puts r=r(g). Then the function r:G-+A is characterized by the following properties (3.15) r(ugu') =r(g) for all geG, u, u'eU, r(~') =r for all reA; if moreover iII) is satisfied, we have (3.16) r(~mn)~>m for all meM, neN. The existence of the function r satisfying (3.15), (3. i6) (resp. (3. I5)) is equivalent to (II) (resp. (II1)). We list below some direct consequences of the assumption (II). i ~ If ~zmNnU~eo, we have m=o. In fact, it follows from (II) that m.<<o. Since we have also n-"NoU=(nmNnU)-l+o, we have m~>o; hence m=o. 2 ~ For h~H, one has with ueN(H) nU. h =- urcr(h)u-1 (rood. H ~) (3.17) if and only if one has UhU = Uh'U It follows that, for h, h'eH, with ueN(H) nU. h' = uhu- 1 (mod. H u) (Note that this is a consequence of only (II1).) 3 0 r(h) for heH is invariant under the inner automorphisms of G, i.e. /f h, h'eH and h'=ghg -x with geG, we" have r(h')=r(h). In fact, it is clear that, in replacing h by uhu -~ with u~U, if necessary, we may assume, without any loss of generality, 242 THEORY OF SPHERICAL FUNCTIONS I9 that lheA , i.e. lh=r(h ) (and similarly that lh,=r(h')). Now, let g-=uhln with ueU, haeH, neN. Then one has h' =ghg -~ -=uhlnhn-lh-;-tu -t = u(hlhh; -1) n'u -1 with n'eN. Hence, by (3. I5), (3. I6), one gets r(h') = r((hlhh~)n ') ~ lh,hhr, = l h = r(h). Similarly, one gets r(h)>_.r(h'); hence r(h') =r(h), as desired. 4 ~ We have W=WH=W A. In fact, if W.Wa, there would be r.r' in An~" such that r'=wr with weWa, because Y is of finite index in M and An~" is a fundamental domain of W in Y. Then we would have ="--- srds -1 (mod. H ~) with seN(A), which contradicts 3 ~ . w 4. Haar measures. 4.x. In this section, G denotes an algebraic group over k satisfying the assumption (I) with respect to U, H, N. The groups G, U, H, N are then all " unimodular ", i.e. their left-invariant Haar measures are also right-invariant. We denote by dg, du, dh, dn the volume-elements of the (both-sides-invariant) Haar measures of G, U, H, N, respectively, normalized as follows: (4. 1 fjg=fJu=f .dh=f o an=,. Then the left- and right-invariant Haar measures of HN are given by (4.2) d,(hn) =dh.dn, dr(hn ) =~(h)dh.dn, being a positive quasi-character of H (i.e. a continuous homomorphism of H into the multiplicative group of positive real numbers with respect to the p-adic topology) defined by (4-3) d(hnh-~) = 3(h)dn. For any integrable function f on G, one has or symbolically (4.4) dg=du, dr(hn) = du. 3(h)dh dn. We need in Chapter II, w 5 the following transformation formula of the relatively invariant measure on U/(UnHN). LEMMA 4.I. Let g0eG. For ueU, write golU=u'h'n ' with u'eU, h'eH, n'eN. Then the cosets u'(Uc~HN), h'H u are uniquely determined by go and u(UnHN). Denoting by d K the volume-element of a relatively invariant measure on U/(U c~ HN), we have (4.5) dg=~(h')dK'. (Note that, since ~(H")= i, 3(h') depends only on the coset h'HU.) 243 20 ICHIRO SATAKE Proof. By (3.4), the first statement is obvious. To prove the second, let h, n be " generic " elements in H, N, respectively, and put g=uhn. Then go~g=u'h'n'hn=u'(h'h)(h-ln'hn). Hence, by (4.4) and by the invariance of the Haar measures, one has d(gol g) = du'. ~( h'h)d(h'h)d(h-ln'hn) = du'. 3(h')3(h)dh dn whence follows (4.5), q.e.d. 4.2. An integral formula. Let 2gY g = go q- be the decomposition of g given in N ~ 2. I. Since HcZ(A), all the subspaees g3" (ye~:) are invariant under Ad h (hell), Ad denoting the adjoint representation of G. One denotes by Rv(h ) the restriction of Ad h to g3". Then det(R3,(h)) is a k-rational character of H, whose restriction to A is equal to d3,. Y (in the additive notation), d3, denoting the dimension of g3,- Thus, identifying X(H) with a submodule of Y=X(A), one gets d3,.yeX(H ) and (4.6) det (R3, (h)) = (d3,. u (h) for h e S. Moreover, taking a so-called Weyl basis of g~= g| (k--algebraic closure of k), one sees immediately that g-3, may be identified with the dual of gv with respect to the inner product induced by the Killing form. Since this inner product is invariant under Ad h (hell), one has (4.7) R_3" is equivalent to tR~-l. Now, for hell, denote by Ad,(h) the restriction of Ad h on n=5093", and put (4.8) A(h) = I det(Ad.(h) -- x.)I. = 1-I l det(R3,(h)-- I3,)tp, 3">0 i,, I3, denoting the identity transformations on n, g3", respectively. Then we have LEMMA 4.2. For heI-I with A(h)+o, the mapping 9 ~ : Nsn ---, n' =hnh-ln -1 is an injective rational mapping from N into itself, the image ~h(N) contains a Zariski open set in N and one has (4.9) an' = A(h),Zn. Proof. Every neN can be written uniquely in the form n=exp x=exp(Sx3" ) with x=3,>0 ~ x3"en' x3"~g3" and one has, for hell, hnh -~ = exp(Ad(h)x) = exp( ~2 R3,(h)x3, ). 3'>0 244 THEORY OF SPHERICAL FUNCTIONS 2I Therefore one has hnh-~=n it" and only if Rv(h)xv=x v for all y>o; in particular, if A(h) 4:o, the latter condition implies that x----o, i.e. n=i. Thus, for h~H with A(h) 4: o, ~h is an injective rational mapping from N into itself. Applying the same consideration to N ~, k denoting the algebraic closure of k, one sees that Wh(N ~) contains a Zariski open set in N ~ ([x6], p. 88, Prop. 4)- Now, let neN ~. Since one has ~h(n ~) = tFh(n)~ for all automorphisms ~ of k over k, it follows from the injectivity of ~h that, if qPh(n)eN, one has n"=n for all a, i.e. heN. In other words, one has tFh(N ) -----~h(N~)nN, which proves that ~(N) contains a Zariski open set in N. Now to prove the last assertion, we regard x6N as a left-invariant vector-field on N (in the algebraic sense) and denote by x n the tangent vector at n6N determined by x. Then one has (4. Io) dtt~(Xn) = (Ad(n). (Ad(h)--I)X)h~h-,.-, , dtFh denoting the " differential " of the rational mapping ~h. In fact, by definitions, one has for any rational funcfionf on N, defined over k and regular at no, and therefore (dVh.x..)(f) ---- x,,(fotFh) which proves (4. IO). Now if we denote by o~ an invariant differential form of the highest degree on N and by td~ h the linear mapping on the space of differential forms on N extending the dual of d~n, it follows tYom (4. IO) and from the fact that det(Ad,(n0) ) = I that 'd~ h . %,~_,,_, = det (Ad,,(h) -- I,)%. Since we have symbolically dn=[%[p, up to a constant multiple ([26], 2.2), we obtain (4-9), q.e.d. By the similar argument as above, we get also ~(h) = [det(Ad,(h))[p = II [det(Rv(h))Iv, "f>0 or by (4.6) (in the multiplicative notation). (4. II) y>O 245 22 ICHIRO SATAKE Put further 1/2 (4.12) D(h) -= l-I [det(Rv(h ) -- Iv) p Then from the definitions and from (4.7) one gets easily the following relations (4.13) D(h -1) = D(h), (4.14) D(h) =~ 2(h)A(h). It should be also noted that one has A(hlhh7 -1) ----A(h) for all h, hleH. (4.15) Let f be a (complex-valued),function on G with a compact carrier, satisfying LEMMA 4" 3" the relation for all geG, u, u'eU. f (ugu') =f(g) Then, for hell with A(h) @ o, we have (4.16) D(h) dg denoting the volume-element of a (suitably normalized) relatively invariant measure on G/A (1). Proof. Since one has dg = du. dn. dh for g = unh, one has symbolically dg=du.dn.dh, dh denoting the volume-element of a relatively invariant measure on H/A; here we normalize dh in such a way that fH dh = I. Then the left-hand side /A of (4.16) is equal to = D(h) fI~a (f, f(nhihh-Ztn-') dn)dh, (by the assumption) =D(h)A(h-')-tfm A (fNf(hthh-:tn')dn')dh, (by (4.9), (4.i5)) 9 Since hthh~ -1 -h (mod. Hu), one gets from the assumption f(hlhh3tn ') =f(hn') ; therefore, by (4.13), (4.14), this last expression is equal to = ~(h) f~f(hn) dn, q.e.d. Since D(h) is invariant under the inner automorphisms defined by elements in N(H), this Lemma implies that, if one puts /(h) = 2(h) f f(hn) dn, t~J f(n'), viewed as a function of meM, is invariant under the operation of the Weyl group W~. (x) This is an analogue of an integral formula of Harish-Chandra ([18], p. 261). :j(hn)dn,

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Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 7, 2007

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