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Theory of spherical functions

Theory of spherical functions CHAPTER II w 5" Zonal spherical functions (1). 5. x. The algebra ~(G, U). Let G be a unimodular locally compact group and U a compact subgroup of G. We denote by ~=~-r U) the algebra over C formed of all complex-valued continuous functions q~ on G with compact carrier and satisfying the condition (5. i) 9(ugu') =9(g) for all geG, u, u'~U, the product in ~ being defined by the " convolution " 9 ~b (g) = fa ~ (gg~- ~) + (gl) dgl. If U is an open subgroup and if the Haar measure of G is normalized in such way that _ (vdg= i, the characteristic function c o of U is the unit element of the algebra 5r Moreover, in this case, all Z-valued functions in 5r forms a subring, which we denote by ~q~(G, U)z. From the arithmetical point of view, it is important to consider this ring. The following theorem will be proved: THEOREM I. Let G be a connected algebraic group over a p-adic number field k and U an open compact subgroup of G, satisfying the assumptions (I), (II) (see w 3), and let v be the dimension of a maximal k-trivial torus in G. Then the algebra ~~ U) is an a/fine algebra of (algebraic) dimension v over C, i.e. a (commutative) integral domain with unit element, which is finitely generated over C and of transcendence degree v over C. Moreover, if A (see n o 3-4) is generated (as semi-group) by r(~), ..., r (~1 and if c (~) denotes the characterist# function of UT~r(ilU, ~(G, U) is generated over C by c (~) (I~i~<l). Remark. If, besides the above assumptions, an additional condition that (5.2) UT:'NoUrc'U=Ur~' for all rEA is satisfied, we shall see that ~Cp(G, U)z is generated (over Z) by c (~1 (I<~i<<.l). (a) For the fundamental concepts on spherical functions, see [11], [22]. 247 ICHIRO SATAKE 5.2. G, U being as above, a complex-valued continuous function co on G in called a zonal spherical function (abbreviated in the following as z.s.f.) on G relative to U, if the following conditions are satisfied: (i) co(ugu') = o~(g) for all gEG, u, u'eU. (ii) c0(i) =- I. (iii) For every Os~q', co is an eigen-funetion of the integral operator defined by ~, i.e. we have (5.3) with X~eC. We denote by a=f~(6, u) the totality of z.s.f, on G relative to U. For coef~, we denote X, in (5.3) by ~(q0), i.e. we put (5.4) Co (~) = f o ~? (g) o~ (g- 1)dg. Then it is clear that Co is a homomorphism (of algebras over C) from ~ onto C. Conversely, if U is an open subgroup of G, it is easy to see that any non-trivial homomorphism o~:~ v ~C comes in this way from a uniquely determined z.s.f, co ([22], pp. 366-367). Thus, under the assumptions of Theorem I, f~ may be viewed as a model of the affine algebra ~v. More precisely, the correspondence gives a bijection of f~ onto an affine variety associated with o~ ~ C[c (t~, ..., c(zl]. 5.3. Construction of z.s.f. From now on, until the end of w 6, we assume that G is an algebraic group over k satisfying the assumption (I) with respect to the subgroups U, H, N and that the Haar measures are normalized as stated in N ~ 4. I. In order to construct z.s.f, depending on complex parameters, we make use of the representations of the " principal series " of G. Namely, let e be a quasi-character of H (i.e. a continuous homomorphism of H into C* with respect to the p-adic topology) and call 3r '~ the Hilbert space formed of all complex-valued measurable functions f on G satisfying the following conditions for all geG, hell, heN, (5.5) f (ghn) = o~( h) f (g) (5.6) llf]l ~-- folf(.)[ 2du< oo, the inner product in 3f ~ being defined by For go e G, fe o~ ~ we put (5.7) (To~of) (g) =f(go lg). Then we have: 2~8 THEORY OF SPHERICAL FUNCTIONS PROPOSITION 5" I. In the above notations, for every g0eG, feJf ~, we have T~of~Of TM, and T ~go is a bounded operator on ~. The correspondence go ~ T~o is a strongly continuous representation of G (with respect to the p-adic topologr) by bounded operators on 3r ~. Proof. It is clear that T~,,fsatisfies the condition (5.5). If we put U=U/(Un HN), it follows from Lemma 4. I that II T~'0f 11 ~ =- frj [J (go lu) I s da = f lf(u'h'n')I = f~lf(u')1~ [ ~(h')]2 a(h')dff'. Since I~(h')12a(h ') is a continuous function of u(UnHN)eU, we have I~(h')I~a(h')<C with a positive constant C. Therefore we have llW flI <ClIfll , i.e. T ~ is a bounded operator on ~. Moreover, if g0eU, we may take as u'=go~U, g* h'=n'=I, so that T ~ (g0eU) is a unitary operator. Now the correspondence g0~T~o g~ is clearly a representation; to prove that it is strongly continuous, it is enough to show that, for a fixed fe~', go~T~.f is strongly continuous at go= i. Since f can be approximated by a continuous function in 3/Z ~ as closely as we wish (with respect to the norm in a%Z~), we may assume that f is continuous. Then f is uniformly continuous on U and our assertion is obvious, q.e.d. From the above proof, we obtain. COROLLARY I. The representation T ~' is unitary, if and only if (5.8) The representation T ~ is called " of the first kind " (relative to U), if there exists an element d/4~o in W~ such that T~qb=d/ for all uaU. COROLLARY (2. The representation T ~ is of the first kind, if and only if (5.9) a(H") = i. Proof. If there exists + as stated above, one has +(uhn)=oc(h)~b(i), so that +(I)+O, ~(H")=I. Conversely if 0r one can define +~ satisfying the above condition by putting (5- xo) ~,(uhn) = ~r q.e.d. In case cr satisfies condition (5-9), one sees readily that (5. I I) ~(g) = <+~, T;+~> = f +=(g-lu)du is a z.s.f, on G relative to U ([2z], p. 37o). In particular, if e is a restriction on H of a quasi-character of G, denoted also by ~, which is trivial on U and N, we have (5.I 2) o~'(g) = ~-t(g). 5.4. Parametrization ofz.s.f. Now we introduce complex parameters in {r as follows. If ~ is a quasi-character of H satisfying (5.9), e is uniquely determined 249 ICHIRO S~ATAKE by ~(~m) (meM) (see N ~ 2.3), and the correspondence m~logq~(r#) is a homomorphism from M into C. Therefore one can find s~X(H)| such that (5- = q- % m.s denoting the natural pairing of meMoir(H) and seX(H)| such s is uniquely determined modulo 2rd 1VI, where log q (5.14) /~/I={seX(H)| for all meM}. Thus we obtain a canonical isomorphism Hom(H/H", C*)~ X(H)| (5. I5) t \tog q ] If {Zl, ..-, X~} is a system of generators of X(H), our correspondence can also be defined by the relation ~(h) =I] [ z~(h)[;i <=~s = Esd( ~ with s~C. When (5.13) holds, we write ~++s. When S2~++s, we put co~---=%. Now the operation of w aW =W H on X(H) can be canonically extended to a C-linear transformation of X(H)| leaving 1VI invariant. Then we have PROVOSmON 5" 2. We have (5.16) o~_s(g) = %(g-1), (5.17) o~=% for all w~W n. Proof. AsinLemma4.i, put gotu=u'h'n '. Then gou'=u(h'n')-t=uh'-an '' with n"eN. Hence from the definition and Lemma 4. I, we have %(g~ = f~5 +~'(g~ = f ~J ~(h')-a ~(h')-a du ---- f 6 +~_, ~-~(goau) d ff = o)_~(go), 1 1 because, if 32~+-~s, one has 3~(3-1~-1)+-~--s. This proves (5.16)- The proof of (5.17) (depending on Lemma 4.3) will be given in N ~ 6. I, q.e.d. In the following, we denote by W. 1~1o~ ] the group of" affine transformations 2":i lql. of X(H)| generated by W and by the group of translations defined by log q ( 2,:i ) Since W leaves 19i invariant, W. \l-0-~g q l~I is actually a semi-direct product of these two groups. From (5.x7) it follows that two parameters s, s'eX(H)| which are ( 2hi l~I) give one and the same z.s.f. Actually we equivalent with respect to W. \1o-0-~ will prove the following. 250 THEORY OF SPHERICAL FUNCTIONS THEOREM 2. The assumptions being as in Theorem i, all z.s.f, o~ on G relative to U can be written in the form o~=% with sEX(H)| and we have %=%, for s, s'eX(H)| [ 27:i ) if and only if s, s' are equivalent with respect to W. ~ ~ . Thus f~ = f~(G, U) is analytically isomorphic to the quotient space X(H)| \l-~g ( 27:i q l~I). w 6. Proofs of the main results. For ~oE.Lf, we define 6.t. Fourier transform. We keep the notations in w 5. its Fourier transform ~ by (6.I) ~(s) = ~,(~o) =f~(g)%(g-~)dg. Then we have, if 3~+-',s, (by (5.1 I)) ~(s) = f~f~(g)+=(gu)dg du (by (g, u) -+(gu -t, u)) =: f ~ ~(g)+~(g)dg (by (4-4)) = Un (by (5.1o)). Hence, putting V(h) = ~(h) fNv(hn)dn, (6.2) we have which depends actually only on the class of h (rood. HU), (6.a) ~(s) = m~Z ~,(~m) ~(~m) - ~- 2: ~,~m,q- ~.,j . Since q~ has a compact carrier, one has ~(r~ m) =~o for only finitely many meM. Incidentally we notice that (5.17) is equivalent to the fact that the Fourier coefficients '~(7: m) (mEM) are invariant under the operation of the Weyl group W, which was already established by Lemma 4.3. This remark completes the proof of Proposition 5.2. Now by virtue of (5.17), we can further transform (6.3) in the form (6.4) ~ (s) = 2] '~(~') y, _-wr., re A "to :W/Wr q ' the second summation being taken over a complete set of representatives of W/Wr, W r denoting the subgroup of W consisting of all the w EW leaving r invariant. Thus, if we denote by {m (t), . .., m 0)} a system of generators of M, ~ is a Fourier polynomial (allowing negative powers) in q m(i/'s (I ~<i<v) invariant under W. We denote the totality of such Fourier polynomials by C[qMa] w. 251 28 ICHIRO SATAKE 6.2. Up to now, we have not used the connectedness of G and the assumption (II). Now we shall show that, under these assumptions, the correspondence ~-+~ gives actually an isomorphism of .~q~ U) onto C[qM'] w. For reM, we denote by c r the characteristic function of the double coset Un'U. Then, by assumption (II1) , {c r (reA)} forms a basis of ~(G, U) (over C). On the other hand, if we put (6.5) F,(s)= ~--~"' ,o :W IWr q {F r (teA)} forms a basis of C[q~"] w. By (6.4) , we have (6.6) ~,= Z Z(~")F,,. r'eA Here we have, for every r, meM, (6.7) =~Ta(7~m) fNCr(T~mn) dn = 3~(nm). meas. of ((=-mUn'U) n N). Hence it follows from (II) that we have ~',(~e)=o for r, r'eA, r<r', and it is clear from the definition that cr(n)+ o for all teA. These mean that the infinite matrix ( c,(n )) with the indices r, r'eA arranged in the linear order < is of the lower triangular form with non-zero diagonal elements. r' We shall now show that the matrix (c,(=)) has actually an inverse matrix. For that purpose, let T (resp. S) be the torus (resp. semi-simple) part of G, let A' (resp. A") be a maximal k-trivial torus in T (resp. S) such that A=cl(A'.A"), and put YQ= X(A)| Y~=X(A')|174 and YQ=X(A)| then YQ is identified with the direct sum of Y~ and Y~', and YQ with the dual space of YQ over Q. Moreover, call M" the intersection of M cYQ with the annihilator of Y~. From what we have stated in n ~ 2.2, M" can be also defined as the submodule of M formed of all meM such that ]x(~m) lp=I for all zeX(G). Then we have LEMMA 5.I. If rcmNnU~'U:~o with m, reM, we have m=r(mod. M"). Proof. Let X be any k-rational character of G. Then, since N has no non-trivial k-rational character, we have z(N) = I. On the other hand, we have clearly IX(u)Ip = I for all ueU. Therefore, if genmNnU='U, we have ]x(g)[, = IX(~) I,--IX(n') I~ and so [Z(n m-') [p= I. As this holds for all ~(eX(G), we have m--reM", q.e.d. LEMMA 5.2. For every reA, the set of r'eA such that r=r'(mod. M") and r'<r is finite in number. Proof. We extend the linear order in M to that in #/Q in a natural manner. Then there exists a Q-linear form L on YQ, not identically zero, such that, for xeYQ, L(x)>o implies x>o (and hence that x>o implies L(x)~>o). For each weW, w+ 1, we can define an order in YQ by x>o (i--w)x>o, ("t'~ar(TI~m) THEORY OF SPHERICAL FUNCTIONS 29 which induces a linear order on the factor space of YQ modulo the subspace formed of all x eYQ such that wx =x. Hence, similarly as above, there exists a non-trivial linear form L w on ~'Q having the property that Lw(x)>o implies x>wx (and hence that x>wx implies L~0(x)>~o and that x=wx implies Lw(x) =o). Therefore, if we put AQ={X~XZQI x>>.wx for all w~W}, A~={xeY, ILw(x)>~o for all weW), AQ is contained in A~ and the interior of A~ is contained in AQ. Since W = W A is the Weyl group of the restricted root system of G relative to A, we can conclude from this that we have AQ = A~ and that AQ is a (closed) " Weyl chamber " of W. Thus AQ is actually a closed cone defined by v"(= rank Y") linear inequalities Li(x) = Lwi(x ) >I o (I < i<, ~"). Now consider first the case where X(G) = I, i.e. YQ=Y~'. The Li's being linearly independent, we can write as L = EX~L~ with X ieQ. Here we assert that all Xi are >o. Clearly, it is enough to show that, if L~(x)>~o for all i (i.e. x~AQ) and x4=o, then L(x)>o. From the assumption, it follows that x>~wx and so L(x)>lL(wx) for all weW. But, since Y~={o}, we have ~ wx=o. Hence, if L(x)~<o, we would have L(wx)=o wEW forall wsW. But this is impossible, because forany x4:o, the set {wxlweW } contains always v" linear independent vectors. It follows that, for any reA, the set {xeAQlx<r } is bounded, and therefore that {r'EAIr'<r} is finite. In the general case, we have, from what we have proved above, L--Y~NLi (mod. Y~) with ?,~>o. Hence, if x--r is in the annihilator of Y~ and x~<r, we have L(r--x)=XXiLi(r--x)>~o. Therefore the set formed of all xeA~ satisfying these conditions is bounded, and so the intersection of it with A is finite, q.e.d. From Lemmas 5-i, 2, we see that the matrix (~',(~*'))(r, r'eA) can be decomposed FP into the direct sum of the (countably many) matrices (c,(r~))(r, r'eA, r_ r'(mod. M")), each of which is of the lower triangular form with respect to a set of indices isomorphic to {i, 2, ...}. Hence (c,(~)) has an inverse matrix of the same form. Thus we conclude that the mapping q0-+~ is an isomorphism (of vector spaces over 13) from .WiG , U) onto C[qg'~] w. Since it is also a homomorphism of algebras over 13, we have proved the following theorem: THEOREM 3" The assumptions being as in Theorem I, the Fourier transformation r gives an isomorphism (of algebras over 13) from -~a(G, U) onto 13[qM.~]w, the algebra of all W-invariant Fourier polynomials in q~:m(% with coedficients in C, {m/~t, ..., m ~'/} denoting a system of (independenO generators of M. Remark 1. The connectedness of G was needed essentially only in the proof of Lemma 5-I. (We used it also in the proof of Lemma 5.2, but this is not indis- pensable.) As is seen from that proof, this condition may be weakened, without changing the conclusion, into any one of the following conditions, where G O denotes the connected component of the neutral element of G : COt) The rank of the character module X(G) is equal to that of X(G~ 253 ICHIRO SATAKE 3o (02) In the notation of the assumption (I), we have G~176176 On the other hand, it follows from Lemma 5. I that, if the condition (II) is satisfied for a linear order in M, the same is also true for any linear order in M inducing the same order in M". Remark 2. The formula (6.7) can be further transformed in the following form: (6.8) m = r ) =~-i(v:m)u ym # (U\(U~:mNnUCU)). In fact, one first observes that by (4.3) meas. of (Nnv:-~Uv: m) = ~(nan) -1. On the other hand, it is easy to see that the cosets in (NnT:-mu~m)\(Nn~-mu~ru) are in one-to-one correspondence with those in U\(UC"NnU~rU). From these follows (6.8). Now, if the additional condition (5.2) is satisfied, we have Yrr= I for all reA. Therefore, in that case, the matrix (y,,) (r, r'eA) and its inverse arc integral, so that the Fourier transformation q~-+(~ gives actually an isomorphism (of rings) of the subring Lf(G, U)z= Z crZ of ~q~(G, U) onto the subring  (8-~(zcr)Fr)Z of rCA r6A C[q~ '] w . Remark 3. From the definitions, we have %(s) = # (U\U~'U)~,(~-'). Therefore we have (6.9) %(,-') = # (U\U~'U) -t Z y~,.8-~(~")Fr ,, r'EA rt~r which shows that c%(7: -r) is a W-invariant Fourier polynomial in q*m(~)'t But it seems far more difficult to describe how c%(~ -r) depends on reA. In any case, it is one of fundamental problems in our theory to obtain a handy expression (analogous to the " character formula "?) for (%(v: -r) or for Y,e- 6.3. Proofs of Theorems 1, 2. The first half of Theorem I follows from Theorem 3 immediately. To prove the second half, let {r (1), ...,r (~)} be a set of generators of A (as semi-group) and put c (~) = Or(i). (Note that A, being defined by a finite number of linear inequalities with integral coefficients, has clearly a finite set of generators.) The notations being as in the preceding paragraph, let A. denote an intersection of A with a coset modulo M"; by Lemma 5.2, A. is isomorphic (as ordered set) with the set of natural numbers. We prove, by induction on reA., that F r can be expressed as polynomial in ~(t), ..., ~(*) with coefficients in C. For that purpose, denote, for every r~A, by ~Jt r the vector space over C generated by F r, with r'eA, r'<r, r=r'(mod. M"). Then it is clear that we have (6. Io) Fr.F ,, --- Fr+ r, (mod. 93tr + r, ) for every r, r'eA. It follows that (6.1I) F,. ~r, C931,+,,, ~01,. 99~, Cg~r +,, 9 254 THEORY OF SPHERICAL FUNCTIONS 3~ On the other hand, by n ~ 5.2, we have (6.12) ~r -%rFr (rood. 9J~) with ?~---=y,,3 ~(r~')~=o. Now let teA. and suppose that our assertion is true for all r'eA., r'<r. Since {r (1/, ... r (~)} is a set of generators of A, r can be expressed in the form r = Y, nir (~) ' i=1 with n~Z, ni>>-o. Then from (6.1o), (6.11), (6.12)we have ni I1 ~(~)"i _= 1-I = (IIxr(il)Fr (mod. ~J'~). i i i Hence, applying the induction assumption on Fr, eS0~r, we conclude that F r is a poly- nomial in ~(~/'s with coefficients in 13. It follows then by Theorem 3 that s U) is generated over C by c/~) (I<i~</). 1 (By the same arguments, replacing F r by 8 2(r~r)F r and t3 by Z in the above proof, we can also conclude that, under the condition (5.2), Sr U)z is generated (over Z) by c (~) (i~<i~</).) As to Theorem 2, we have to prove the following two statements: io Every oe~2 can be: written in the form co=co~ with seX(H)| 2 ~ We have o~----o)s, , if and only if ( 2~i 19I) with weW. $' ---- WS rood. ]-0~gq To prove I ~ let co be any z.s.f, on G relative to U. Let {re(I),..., xn (~1} be a system of generators of M and put Xi=q mli)'~. Then, by Theorem 3, s can be identified with the subalgebra of C[X~ l, ..., X~ ~1] formed of W-invariant elements. Since 13[X~l, ..., X~ l] is integral over Lf, the homomorphism (~5 :.Lf-+C can be extended to a homomorphism, denoted again by ~, from C[X~I, ..., X~ l] onto 13 (Ix6], p. 8, Th. I, p. 12, Prop. 4)- Since G(X~.) + o, one can put ~(X~) ~-q~ with s~e13. Take s eX(H) | C such that na (i). s ---- s~ (I ~< i~< v). Then we have G ~, so that c0 ~ o)~. Proof of 2 ~ For s, s'eX(H)| we have o)s----o) s, if and only if ~(s)----~(s') for all (? ~.oq o, or, what amounts to the same by Theorem 3, (? (s) ---- ~ (s') for all r e C[q~'S] w. But this last condition is clearly equivalent to saying that s'----ws (mod. 2~i 1~I] log q ] with weW. w 7. Homomorphisms. 7-x. Let G, G' be two algebraic groups satisfying the assumption (I) with respect to U, H, N and to U', H', N', respectively. We suppose further that there are given WcWn, resp. W'cWIv, and their fundamental domains A, resp. A' (defined by (3.13)) and that G' satisfies assumption (II1) with respect to A'. Let X be a homomorphism from G into G' satisfying the following conditions (i), (ii), (iii). (In ----- (),r(~)Fr(~))"~ ICHIRO SATAKE 3~ this paragraph, it is enough to assume that X is continuous in the sense of the p-adic topology.) (i) X(H) cH', X(N) oN', X(U) cU'. Then, since k(H u) cH 'u, k induces a homomorphism M(~H/H u) -+ M'(~H'/H'"), which we denote again by X, such that X(lh)=lx(h) for all hell. (ii) X(A) cA', (iii) O is normal in X-t(U'). It follows from (i), (iii) that (7. i) ~k-I(U ') = O T~nU, m~Mk ~m U = U7~ m for all mEMx, More generally, one obtains where Mx denotes the kernel of X:M-+M'. LF.M~. 7. I. We have (7.2) X-t(U'X(g)U')=UgX-I(U')= U Ugz~mu m EM x =X-~(U')gU= O urcmgU (disjoint union). m EM k In other words, we have U'X(gl)U'=U'X(g)U" if and only if we have UgxU = Ug~m U(= U~mgU) with m eMx, where m is uniquely determined by g, gl. Proof. According to (3. I4), one puts g=ut~ru2, gt=u~7~r'u4, X(gl) =U~X(g)u 2 with u 1, ..., u4eU, u;, u~eU', r, r~eA; then one has ~(u3))~(~rI)X(u4) =uiX(ul)X(rd)?,(u2)us Hence from (i), (ii) and (II1) for G', one gets X(rl) =X(r), so that rt= r + m with meMx. Then gt=uarc'~mu4-----(uau~-t)g(u-~lr:mu4)eUgr~mU by (7.1)(and similarly gleUr~mgU). To show the uniqueness of m, let Ug~mU = UgzWU with m, m' E M x. Then we have Ug~mU~Ugr~murcm'-m, so that grcm+z(m'-m)~Ug~mU for all leZ. But this is impossible unless m=m', because Ug~mU is compact and {gr~'lmleMz} is discrete, q.e.d. It follows, in particular, that in the above notation, if g=gl, we have r=rl, i.e. G also satisfies (II1). Now, let .oq~ U), ~~ U') and, for 9~5f, define 9's.oq ~ by = (gv)dv= X meg). (7-3) =fz-,(u') ~(vg)dv= E p(~mg) if g'eU'X(g)U', m ~k o ff g'r 256 THEORY OF SPHERICAL FUNCTIONS 33 From Lemma 7.I it is clear that q~' is well-defined and belongs to .go,. We put ?' =X*(?). The mapping ),* can also be defined as a linear mapping s _+&a, such that (7.4) X*(c,) =%,) for all reA. We now assume further that X satisfies the condition (iv) X(G).U' = U'.Z(G). PROPOSITION 7" I. Let G, G' be two algebraic groups over k satisfying (I), (II,) and let X be a homomorphism from G :into G' satisfying (i), (ii), (iii), (iv). Then a) X* is a homomorphism (of algebras over C) from 5f(G, U) into s U'). b) For ~'ef2(G', U'), we have ~=~'oXen(G, U) and (7-5) ~3(~) =g'(X*(p)) for all ?e~'(G, U). Proof. a) If X*(%) = q~ (i-= I, 2), we have t I t t ~)2(gJ-I I--1 I )dg, (%, %)(X(g)) = fa,%(X(g)g 0 --_fx(a)fu,~',(X(g)X(gl)u')~s ' (by (iv)) --= fx(o) q~; (X (g&)) ?~(X(g,)- ') dX(g,) = fo,~-,(,) dx(g*) fz-,(u,)~(gg, v,)dv, fx-,(v,)%(vr'gr*v2)dv2 (by (7.3)), where the measure on X(G)~=G/X-~(I) is normalized in such a way that the total measure of the open subgroup ;~(G)nU'-~X-I(U')/X-I(I) is equal to i. Hence the last expression is equal to = far 1%(gg2)%(g;lv2)d&dv2 "k- (U' =x*(~** ~2)x(g). On the other hand, if g' ~X(G)U', we have clearly (~* v~)(g') = x*(~** ~) (g') = o, If X*(~)---- ~', we b) It is clear that co'oX satisfies the conditions (i), (ii) of z.s.f. get quite similarly as above (q/. co') (X(g)) = fa, q~'(x(g)g[) ~' (g~- 1)dg~ = fx(~) q~'(x(g&))~'(x(g*)-*)dx(gl) = fo/x-,(,)J'x-,(v')~(gglv)~'~ (gg=) o~'o~,(g-*)d = j~ 2 g2 = (~, (~'oX)) (g). 5 34 ICHIRO SATAKE Thus we have q~ 9 (r ok) (g) = (q~', r (X(g)) = ~'(~'). r (X(g)), which proves (iii) (in N ~ 5. I). At the same time, (7.5) is proved, q.e.d. 7.2. Isogenies. We first apply the above considerations to the case of a k-isogeny. Let, as in N ~ 7. r, G, G' be algebraic groups over k satisfying (I) and suppose that G' satisfies (II1) ; we suppose luther that G, G' are connected and that W=Wa, W'=WA, , where A, A' denote the unique maximal k-trivial toruses in H, H', respectively. Let X:G~G' be a k-isogeny satisfying (i). Then, A, N and A', N' being the subgroups of G, G', respectively, corresponding under X, we have (7.6) A--the connected component of X-t(A'), A'=cl(X(A)), N--=the connected component of )~-I(N'), N'----Z(N). From the maximality of U, we also have (7-7) U=X-I(U') 9 Thus (iii) is trivially satisfied. Here X : M-->M' is injective, and, by our assumption, W, W' may be identified with each other canonically. Therefore, taking a linear order in M induced from that in M', we have (7.8) A = X -1 (A'), or A--A'nM, when we consider that McM'. Thus (ii) is also satisfied. Therefore, as we stated in 7-I, G satisfies (II1). Furthermore, it is clear that if G' satisfies (II), so does also G. To proceed further, let us first note that X is not necessarily surjective. In fact, it is known that k(G) is a normal subgroup of G' and that G'/X(G) is finite, commutative. (This is true for any isogeny between connected algebraic groups over a p-adic field. See Ix5], Prop. 3-) It follows, in particular, that X satisfies the condition (iv). PROPOSmON 7.2. The assumptions being as above, G'/(X(G)U') is canonically isomorphic to M'/X(M). Proof. Since X(G)U' is a normal subgroup of G' containing N'=X(N), we have, by (3.I), G'=H'.X(G)U'. Next, we assert that H'n (X(G)U') ----X(H)H '~. In fact, let H'c~(X(G)U')gh':X(g)u' with geG, u'eU' and put g:ulhu 2 with hell, ul, u2eU. Replacing h by uhu -1 with ueN(H)nU, if necessary, we may assume that lz(h), lh,, belong to one and the same fundamental domain w'A' (w'eW'). Then, from h':X(ul)X(h)X(@u', we have r(h'):r(X(h)) and so /h,:lz(h), i.e. we obtain h'=X(h)u" with u"eH 'u. This proves the inclusion r the inverse inclusion is trivial. It follows that (7.8) G'/(X(G)U') ~ H'/(X(H)H'U). 258 THEORY OF SPHERICAL FUNCTIONS On the other hand, since M=~H/H u, M'~H'/H 'u and X(H ~) =X(H) nH '", we have (7.9) H'/(X(H) H'") ~ M'/X(M). From these, we conclude the Proposition, q.e.d. We denote by ~, the character group of G'/(X (G) U') and consider ~ ~ E as a function on G'. Then E operates on ~Sf'=.~f(G', U') by (7. io) (~.q/)(g') =~(g').q/(g') for ~E, q/sNe'. The correspondence ~'~q0' is clearly an automorphism of s (as algebra over C). We denote by ~'(G', U') -=" the subalgebra of s formed of all ~-invariant elements. Then we have PROPOSITION 7.3- The assumptions being as above, X* is an injective homomorphism (of algebras over C) .from s176 U) into ~(G', U') and its image is equal to .oq~ ', U') ~. Proof. Since X satisfies the conditions (i),~ (iv), it follows from Proposition 7-I that X* is a homomorphism. Since X : M -~M' is injective, it follows from (7.4) that ~* is injective. Finally, it is clear that an element q~' of &~ U') belongs to the image of?,*, if and only if its carrier is contained in U'),(G), and this latter condition is equivalent to saying that q~' is invariant under E, q.e.d. Remark. It is easy to see from the definition that X*(.Lf(G, U)z ) =X*(Se(G, U))n Lf(G', U')z= (Se(G', U')z) z. On the other hand, E operates also on ~'----~(G', U'). Namely, as is easily seen, for every ~EE, co'~f~', we have ~.o~'Ef~' ([22], Prop. 5) (in particular, putting to'=z (constant), we have ~f~'). From the definitions, it is clear that (7. I I) ~'(~') ~- ~--lO)'(q)'). Now, as stated in N ~ 5.3, 5.4, a part of f~' is parametrized by X(H')| [ ~log 2~i q 1VI') ; for s~X(H')| denote by % the corresponding z.s.f. on G'. Then we have the following 2~i 1~I'~ L~MMA 7.2. If ~]H'++S~ roOd. log q ] in the sense of N o 5.4, we have ~--]. i t (7- I2) ~ 9 %-- %+,~. Remark. This Lemma is valid for any quasi-character ~ of G' which is trivial on U' and N'. Proof. If ~z,++s, one has o~(g') = fu,+~(g'-tu')du ' Hence one obtains ~-~. co~(g') = ~(g'-l) fu, dA'(g'-lu')du' ,I. f-'-Xu'~du' JlJ,'~g ) ' 259 ICHIRO SATAKE 3 6 because ~(U') =~(N') = i. As ~<-+s~, one has 32~++s+s~. Hence the last expression is equal to o'.+~(g'), q.e.d. By (7.II), (7.I~2) is equivalent to (7.13) ~. r = ~'(s + s~). Next, let us consider the correspondence ~'~ c0'-+~o = o~'oX~fl more closely. Let kn : H ~H' be the restriction of X on H and tX. : X(H') ~X(H) its" dual ". Then *X u is injective, with finite cokernel, so that it can be extended canonically to an isomorphism X(H')| -+ X(H)| t 1QI' Since we have XH( ) clQI, it induces a homomorphism which we denote again by tXi~. from X(H')| ( 2~i onto X(H)| ( 27~i ]QI). We now assert that the q ) \log q following diagram is commutative: 4' (7.14) [ 2r:i 19I] +- [ 2~i 19I'] X(H)| ~l-~gq ] X(H')| ] Namely we shall prove the following L~.MMA 7.3. If seX(H)| s'eX(H')| are such that s=tXH(s'), we have ca) s = ca) s, o~,~ (7" 15) or equivalently, by (7.5), for all v e(G, U). (7. i6) Proof. In view of (6.3) and the relation m.s =X(m).s' (meM), it is enough to show that we have ~'(~m) ='~x(,)(~zz(m)) for all r, meM. By (6.8) and by the coincidence of ~ for G and G', this is equivalent to (U~(UT~ru N U=mN)) = ~ (U'~(U'7~)'(r) U'~ U'~Z(m)i')), which can be proved as follows. From Lemma 7. I, we have X-I(U'~:~(r)u ') = UnrU. Since N'=X(N) O,(G), it follows that (X(rzm)N ') n U'~ x(') U' -----x(.mN r Ur:r U). 260 THEORY OF SPHERICAL FUNCTIONS This shows that the mapping U'\(U='U n U=mN) --> U'\(U'= x(r) U' n U'7~ zlm) N') defined by Ug-+U'X(g) (gz==NnU='U) is surjecfive. It is also injective because of the relation (),(~m)N')nU'=X(="NnU). This proves our assertion, q.e.d. The formula (7. I6) implies the commutativity of the diagram -~(G, U) z*, .~(G', U') (7.17) ] c[q q w , w', where the second horizontal homomorphism is defined by the correspondence qm.S _+qX(m).~. Finally, let us suppose that G' (and hence also G) satisfies the assumption (II). Then it follows that the mapping Y~+-~' in (7. I4) is surjective and that we have o~o).=to~oX for to~, coW, f2' if and only if to~ =~.c% with ~eE. In fact, the" if" part being trivial, suppose that to~loX = co~oX with sl, sz~X(H')| Then, by Lemma 7.3, we have tXa(sl)~t),H(s~)(mod. W 2~i 1V[) and so sl---s~ (mod. W'. 2yi ,X~(l~) " logqg q ' log q ]" On the other hand, by virtue of Proposition 7.2, we have ^ " ! l~176 I" (7. I8) 'Xa~(M)/M'= t Hence it follows that s2=slq-s~ ( mod. w' .1-~alVI' ) with ~E, and so, by Lemma 7.2, ~, = 4-1. co~,, which proves the " only if" part. Thus f2 can be identified with fV/E. Summing up, we obtain TrrF.OR~.M 4" Let G, G' be two connected algebraic groups over k satisfying assumption (I) with respect to U, H, N and to U', H', N', respectively, and let X be a k-isogeny from G to G' such that X(H)r X(N)cN', X(U)cU'; suppose further that G' satisfies (II) (with respect to W'=WA, ). Let E be the character group of G'/X(G)U' (which is a finite commutative group). Then, G satisfies also (II) with respect to W=W A and the linear order in M induced from that in M'; and a) .~ operates on .~~ U') by (7. xo), and ~F(G, U) can be identified with .s U') z by the mapping X* defined by (73). Moreover the diagram (7.17) is commutative. b) E operates on f2(G', U') similarly, and f~(G, U) can be identified with t~(G', U')/E by the mapping ~(G', U')~ e0'-+o~'oXef~(G, U). Moreover the diagram (7. I4) is commutative. Remark. Supposing only that (II~) and Theorem 3 (hence also Theorem ~) hold for G', U', instead of assuming the assumption (II) for G', in the above theorem, we 261 3 8 ICHIRO SATAKE can conclude that Theorem 3 for G, U, as well as the statements a), b), hold. In fact, in view of the diagram (7. I7) and Proposition 7-3, it is enough to show that ~(G', U') "= and C[qX(M)'s] w' are corresponding under the isomorphism .LP(G', U')~C[qM"s] w'. But this follows immediately from (7. I3) and (7-I8). 7.3. As a second application of N ~ 7- i, we consider the case where X is injective. Namely, let G be a k-closed subgroup of G' (both satisfying (I)), and suppose that X=identity satisfies the conditions (i), (ii). Then, again from the maximality of A, N, U, we get (7-I9) A = the connected component of A'nG, N=N'oG, U=U'nG. Thus (iii) is trivially satisfied. It follows that, if G' satisfies (II1) , so does also G. On the other hand, since H" =Hr~H'", X : M ~M' is injective, and by what we have stated in N ~ 3.4, 3 ~ it can readily be seen that (7.20) A--A'nM. Therefore, it is clear that, if G' satisfies (II), so does also G (with respect to the induced linear order in M). Now, it is known (cf. No 8.2) that G' =GL(n, k) satisfies the conditions (I), (II) with respect to H'=A'=D(n, k), N'----TU(n,k), U' --= GL(n, o), W'=~n (symmetric group of n letters); in this case, M' is canonically identified with Z n, and, taking the lexicographical linear order in M', we have A'={m---- (m,) eg'lml>~...>/m,}. For GcGL(n,k), put U=GnGL(n, o). Then, the conditions (I) for G, U and (i), (ii) for ?,=identity can be stated as follows: (I*) There exist, in G, a connected k-closed subgroup A contained in D(n, k) and a k-closed subgroup N contained in T~(n, k), normalized by A, such that we have G=U.AN=U.A.U. (II*) There exists a subgroup W of W A such that every w~W can be written in the .form w=w~, with usN(A)r~U, and that, for m=(m~)eM(cZ n) with wm<~m for all weW (with respect to the lexicographical order in Zn), we have ma>>. .. . >~m,. Therefore, from what we have stated above, we obtain the following: TI-IEORF.i 5 [xg]" Let G be a (connected) k-closed subgroup of GL(n,k), U = Gn GL(n, o), and suppose that G, U satisfy the conditions (I*), (II*). Then Theorems I, ~, 3 hold for G, U. As is seen from (7.2o) and N ~ 6.2, the connectedness assumption on G is unnecessary. Theorem 5 can be applied, for instance, to SL(n, k), Sp(n, k), SO(n, k, S), taken in a suitable matricial expression. 262 THEORY OF SPHERICAL FUNCTIONS Example. SO(n, k, S). By definition, SO(n, k, S) ={geGL(n, k) l'gSg = S, det g =~ }, where S is a non-singular symmetric matrix of degree n. By a theorem of Witt, S can be transformed into the following form S= S0 o~}n0, n ~- n o .ql- 2V, L~ 9 , S o = symmetric matrix of degree n o corresponding to an " anisotropic " "'o quadratic form (i.e. for xek ~~ txS0x=o implies x=o). Then, from the theory of maximal lattices (cf. w 9), it can be verified that G=SO(n, k, S) satisfies (I*), (II*). But, if we consider the group of similitudes with respect to S instead, it appears that (I*) is not always satisfied. Remark. Proposition 7-I cannot be applied, unless the condition (iv) is satisfied. It can surely be applied to SL(n, k), and more generally to the situation considered in Proposition 3.2. In these cases, Lemma 7.3 (i.e. commutativity of the diagram (7.14)) can also be proved. 7.4. Finally we apply our considerations in N ~ 7. I to the situation considered in Proposition 3.3- Namely, let G be an algebraic group over k satisfying (I), (II1) with respect to U, H, N, WcW H, let Z be a k-trivial torus contained in the center of G and put (7.2i) G-~G/Z, U=(HoU)/Z, fl=H/Z, I~=ZN/Z, H o being defined by (3-7)- Suppose that H o normalizes U. Then G satisfies (I) (Proposition 3.3)- We now prove that G satisfies also (II1). In fact, put (7.22) M0={meM]� ~ for some � zeZ}. Then, M0-~H0/H u and so M= - "~H / H u~'~ _~H/ H o= ~M / M 0. Since W operates trivially on M0, the fundamental domain A consists of cosets modulo M0, and A = AIM o is clearly a fundamental domain, in 1~I, of the group ~rcWn (Vr determined by W in a natural way. (More precisely, we may assume, without any loss of generality, that the linear order in M defining A is " adapted " to M0, i.e. satisfies the condition that, if m, m'~M, mr re>o, m___-m'(mod. M0) , then m'>o. Then, the linear order in M induces in a natural way that in 1VI= M/M0, and the fundamental domain of ~/ defined by the latter is precisely A.) Then, it is clear that = LI UnrU (disjoint union), rEX which proves our assertion. It is also clear that, if G satisfies (II) (with respect to a linear order in M adapted to M0) , so does also G (with respect to the " induced " linear order in/VI). 263 ICHIRO SATAKE 4 ~ Now, call X the canonical homomorphism G-+G = G/Z. Then, by our assumptions, X satisfies the condition (i)~ (iv) (U', H', N', A' being replaced by U, H, N, A, respec- tively). We note that, in the notation ofN ~ 7.1, we have X-I(U ') =HoU , Mx=M 0. PROPOSmON 7.4- The notations and assumptions being as above, X* is a surjective homomorphism (of algebra over C) from ~~ U) onto ~~ U), and its kernel is equal to the ideal generated by Co---c ~ (meMo) , or, what amounts to the same, by co--%(i) (I~i~<vo), { re(t}, ..., m(~~ (~0 = rank Mo) being a system of generators of M 0. Remark. It is easy to see that X*(~(G, U)z ) =oLP(G, U)z. Therefore, by this proposition, ~(G, U)z can be identified with the factor ring of ~~ U)z by the ideal generated by Co--C., (mEMo). Proof. The first statement is a direct consequence of Proposition 7-I, (7-4) and what we have stated above. To prove the second, choose a system of representatives {r} ofA/M 0. Then every ~e.~qP(G, U) can be written uniquely in the form ~= E E x,,,,e,+m (7.23) r:A/Mo mEMo with Xr,,eC, and we have ~k*(q)) = r A/M. E (In'M0 ~r'na) Cx(r)" Therefore, we have X*(~) =% if and only if (7.24) mEMo On the other hand, from our assumption on Ho, we have for each r. Cr*Cm~Cm*er~--Cr+ m for all reM, meMo. Hence, putting c (i) = em(~), we have, for m = ~ nim (i), er+ m = (g(1))n',... * (g(vo))nvo,er, (C(r denoting the ni-th power of c (~) with respect to the convolution. Thus our assertion on the kernel of k* is reduccd to the following, easy, purely algebraic lemma: LEMMA 7- 4- Let ~(X) = ,~ X, ....... %X? ... X~C[X~', .... X~]. Then, wehave Z X ......... =o, if and only if ,p belongs to the ideal generated by I--Xi(I~<i<,~ ). nl, ...,n v Now assume that G is connected and satisfies (II) and consider the relation between f~----f~(G, U) and ~ =~(G, U). By Proposition 7.4, ~ can be identified with the subset of fl formed of all co ell satisfying the condition G (c 0- Can) = o for all m e M0, or equivalently, if we write co = % with seX(H)| ^ (s) for all meM0. C m = I 264 THEORY OF SPHERICAL FUNCTIONS Now, from (6.3), we have 2,.(s) =q-'~" for m~M o. In fact, from the definitions, we have ~(~m)= I for meMo; and if r~m'NnU~mU+o for memo, m'eM, we have', (~m'-mN) oU~eo and so m'=m (N ~ 3.4, I~ 9 Hence by (6.7) we have I if m'=m, )= Cm(~lDfl' i, o if m'+m, which proves our assertion. It follows that, for o~f~, we have c%e~, if and only if (7.25) m.s -- o mod. l-ogq Z for all meM o. This result can also be obtained from the commutativity of the following diagram: (7.26) X(H)| [ 2rd 1VII +- X(H)| [ 2r~i I~i~ \logq ] !,logq ]' which can be proved quite similarly as Lemma 7.3. Thus we obtain PROPOSITIO~ 7"5" Let the notations and assumptions be as stated at the beginning of the paragraph, assume further that G is connected and satisfies (II) with respect to W r n and a linear order in M adapted to M o. Then, G = G/Z satisfies also (II) with respect to ~r = Wi, and the induced linear order in i~I= M/M0; and ~ can be identified with the subset of f~ formed of all co, such that m.s--o mOd. l-ogqZ for all meM 0. Moreover, the diagram (7.26) is commutative. 2,65 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Theory of spherical functions

Publications mathématiques de l'IHÉS , Volume 18 (1) – Aug 7, 2007
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Springer Journals
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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/BF02684783
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Abstract

CHAPTER II w 5" Zonal spherical functions (1). 5. x. The algebra ~(G, U). Let G be a unimodular locally compact group and U a compact subgroup of G. We denote by ~=~-r U) the algebra over C formed of all complex-valued continuous functions q~ on G with compact carrier and satisfying the condition (5. i) 9(ugu') =9(g) for all geG, u, u'~U, the product in ~ being defined by the " convolution " 9 ~b (g) = fa ~ (gg~- ~) + (gl) dgl. If U is an open subgroup and if the Haar measure of G is normalized in such way that _ (vdg= i, the characteristic function c o of U is the unit element of the algebra 5r Moreover, in this case, all Z-valued functions in 5r forms a subring, which we denote by ~q~(G, U)z. From the arithmetical point of view, it is important to consider this ring. The following theorem will be proved: THEOREM I. Let G be a connected algebraic group over a p-adic number field k and U an open compact subgroup of G, satisfying the assumptions (I), (II) (see w 3), and let v be the dimension of a maximal k-trivial torus in G. Then the algebra ~~ U) is an a/fine algebra of (algebraic) dimension v over C, i.e. a (commutative) integral domain with unit element, which is finitely generated over C and of transcendence degree v over C. Moreover, if A (see n o 3-4) is generated (as semi-group) by r(~), ..., r (~1 and if c (~) denotes the characterist# function of UT~r(ilU, ~(G, U) is generated over C by c (~) (I~i~<l). Remark. If, besides the above assumptions, an additional condition that (5.2) UT:'NoUrc'U=Ur~' for all rEA is satisfied, we shall see that ~Cp(G, U)z is generated (over Z) by c (~1 (I<~i<<.l). (a) For the fundamental concepts on spherical functions, see [11], [22]. 247 ICHIRO SATAKE 5.2. G, U being as above, a complex-valued continuous function co on G in called a zonal spherical function (abbreviated in the following as z.s.f.) on G relative to U, if the following conditions are satisfied: (i) co(ugu') = o~(g) for all gEG, u, u'eU. (ii) c0(i) =- I. (iii) For every Os~q', co is an eigen-funetion of the integral operator defined by ~, i.e. we have (5.3) with X~eC. We denote by a=f~(6, u) the totality of z.s.f, on G relative to U. For coef~, we denote X, in (5.3) by ~(q0), i.e. we put (5.4) Co (~) = f o ~? (g) o~ (g- 1)dg. Then it is clear that Co is a homomorphism (of algebras over C) from ~ onto C. Conversely, if U is an open subgroup of G, it is easy to see that any non-trivial homomorphism o~:~ v ~C comes in this way from a uniquely determined z.s.f, co ([22], pp. 366-367). Thus, under the assumptions of Theorem I, f~ may be viewed as a model of the affine algebra ~v. More precisely, the correspondence gives a bijection of f~ onto an affine variety associated with o~ ~ C[c (t~, ..., c(zl]. 5.3. Construction of z.s.f. From now on, until the end of w 6, we assume that G is an algebraic group over k satisfying the assumption (I) with respect to the subgroups U, H, N and that the Haar measures are normalized as stated in N ~ 4. I. In order to construct z.s.f, depending on complex parameters, we make use of the representations of the " principal series " of G. Namely, let e be a quasi-character of H (i.e. a continuous homomorphism of H into C* with respect to the p-adic topology) and call 3r '~ the Hilbert space formed of all complex-valued measurable functions f on G satisfying the following conditions for all geG, hell, heN, (5.5) f (ghn) = o~( h) f (g) (5.6) llf]l ~-- folf(.)[ 2du< oo, the inner product in 3f ~ being defined by For go e G, fe o~ ~ we put (5.7) (To~of) (g) =f(go lg). Then we have: 2~8 THEORY OF SPHERICAL FUNCTIONS PROPOSITION 5" I. In the above notations, for every g0eG, feJf ~, we have T~of~Of TM, and T ~go is a bounded operator on ~. The correspondence go ~ T~o is a strongly continuous representation of G (with respect to the p-adic topologr) by bounded operators on 3r ~. Proof. It is clear that T~,,fsatisfies the condition (5.5). If we put U=U/(Un HN), it follows from Lemma 4. I that II T~'0f 11 ~ =- frj [J (go lu) I s da = f lf(u'h'n')I = f~lf(u')1~ [ ~(h')]2 a(h')dff'. Since I~(h')12a(h ') is a continuous function of u(UnHN)eU, we have I~(h')I~a(h')<C with a positive constant C. Therefore we have llW flI <ClIfll , i.e. T ~ is a bounded operator on ~. Moreover, if g0eU, we may take as u'=go~U, g* h'=n'=I, so that T ~ (g0eU) is a unitary operator. Now the correspondence g0~T~o g~ is clearly a representation; to prove that it is strongly continuous, it is enough to show that, for a fixed fe~', go~T~.f is strongly continuous at go= i. Since f can be approximated by a continuous function in 3/Z ~ as closely as we wish (with respect to the norm in a%Z~), we may assume that f is continuous. Then f is uniformly continuous on U and our assertion is obvious, q.e.d. From the above proof, we obtain. COROLLARY I. The representation T ~' is unitary, if and only if (5.8) The representation T ~ is called " of the first kind " (relative to U), if there exists an element d/4~o in W~ such that T~qb=d/ for all uaU. COROLLARY (2. The representation T ~ is of the first kind, if and only if (5.9) a(H") = i. Proof. If there exists + as stated above, one has +(uhn)=oc(h)~b(i), so that +(I)+O, ~(H")=I. Conversely if 0r one can define +~ satisfying the above condition by putting (5- xo) ~,(uhn) = ~r q.e.d. In case cr satisfies condition (5-9), one sees readily that (5. I I) ~(g) = <+~, T;+~> = f +=(g-lu)du is a z.s.f, on G relative to U ([2z], p. 37o). In particular, if e is a restriction on H of a quasi-character of G, denoted also by ~, which is trivial on U and N, we have (5.I 2) o~'(g) = ~-t(g). 5.4. Parametrization ofz.s.f. Now we introduce complex parameters in {r as follows. If ~ is a quasi-character of H satisfying (5.9), e is uniquely determined 249 ICHIRO S~ATAKE by ~(~m) (meM) (see N ~ 2.3), and the correspondence m~logq~(r#) is a homomorphism from M into C. Therefore one can find s~X(H)| such that (5- = q- % m.s denoting the natural pairing of meMoir(H) and seX(H)| such s is uniquely determined modulo 2rd 1VI, where log q (5.14) /~/I={seX(H)| for all meM}. Thus we obtain a canonical isomorphism Hom(H/H", C*)~ X(H)| (5. I5) t \tog q ] If {Zl, ..-, X~} is a system of generators of X(H), our correspondence can also be defined by the relation ~(h) =I] [ z~(h)[;i <=~s = Esd( ~ with s~C. When (5.13) holds, we write ~++s. When S2~++s, we put co~---=%. Now the operation of w aW =W H on X(H) can be canonically extended to a C-linear transformation of X(H)| leaving 1VI invariant. Then we have PROVOSmON 5" 2. We have (5.16) o~_s(g) = %(g-1), (5.17) o~=% for all w~W n. Proof. AsinLemma4.i, put gotu=u'h'n '. Then gou'=u(h'n')-t=uh'-an '' with n"eN. Hence from the definition and Lemma 4. I, we have %(g~ = f~5 +~'(g~ = f ~J ~(h')-a ~(h')-a du ---- f 6 +~_, ~-~(goau) d ff = o)_~(go), 1 1 because, if 32~+-~s, one has 3~(3-1~-1)+-~--s. This proves (5.16)- The proof of (5.17) (depending on Lemma 4.3) will be given in N ~ 6. I, q.e.d. In the following, we denote by W. 1~1o~ ] the group of" affine transformations 2":i lql. of X(H)| generated by W and by the group of translations defined by log q ( 2,:i ) Since W leaves 19i invariant, W. \l-0-~g q l~I is actually a semi-direct product of these two groups. From (5.x7) it follows that two parameters s, s'eX(H)| which are ( 2hi l~I) give one and the same z.s.f. Actually we equivalent with respect to W. \1o-0-~ will prove the following. 250 THEORY OF SPHERICAL FUNCTIONS THEOREM 2. The assumptions being as in Theorem i, all z.s.f, o~ on G relative to U can be written in the form o~=% with sEX(H)| and we have %=%, for s, s'eX(H)| [ 27:i ) if and only if s, s' are equivalent with respect to W. ~ ~ . Thus f~ = f~(G, U) is analytically isomorphic to the quotient space X(H)| \l-~g ( 27:i q l~I). w 6. Proofs of the main results. For ~oE.Lf, we define 6.t. Fourier transform. We keep the notations in w 5. its Fourier transform ~ by (6.I) ~(s) = ~,(~o) =f~(g)%(g-~)dg. Then we have, if 3~+-',s, (by (5.1 I)) ~(s) = f~f~(g)+=(gu)dg du (by (g, u) -+(gu -t, u)) =: f ~ ~(g)+~(g)dg (by (4-4)) = Un (by (5.1o)). Hence, putting V(h) = ~(h) fNv(hn)dn, (6.2) we have which depends actually only on the class of h (rood. HU), (6.a) ~(s) = m~Z ~,(~m) ~(~m) - ~- 2: ~,~m,q- ~.,j . Since q~ has a compact carrier, one has ~(r~ m) =~o for only finitely many meM. Incidentally we notice that (5.17) is equivalent to the fact that the Fourier coefficients '~(7: m) (mEM) are invariant under the operation of the Weyl group W, which was already established by Lemma 4.3. This remark completes the proof of Proposition 5.2. Now by virtue of (5.17), we can further transform (6.3) in the form (6.4) ~ (s) = 2] '~(~') y, _-wr., re A "to :W/Wr q ' the second summation being taken over a complete set of representatives of W/Wr, W r denoting the subgroup of W consisting of all the w EW leaving r invariant. Thus, if we denote by {m (t), . .., m 0)} a system of generators of M, ~ is a Fourier polynomial (allowing negative powers) in q m(i/'s (I ~<i<v) invariant under W. We denote the totality of such Fourier polynomials by C[qMa] w. 251 28 ICHIRO SATAKE 6.2. Up to now, we have not used the connectedness of G and the assumption (II). Now we shall show that, under these assumptions, the correspondence ~-+~ gives actually an isomorphism of .~q~ U) onto C[qM'] w. For reM, we denote by c r the characteristic function of the double coset Un'U. Then, by assumption (II1) , {c r (reA)} forms a basis of ~(G, U) (over C). On the other hand, if we put (6.5) F,(s)= ~--~"' ,o :W IWr q {F r (teA)} forms a basis of C[q~"] w. By (6.4) , we have (6.6) ~,= Z Z(~")F,,. r'eA Here we have, for every r, meM, (6.7) =~Ta(7~m) fNCr(T~mn) dn = 3~(nm). meas. of ((=-mUn'U) n N). Hence it follows from (II) that we have ~',(~e)=o for r, r'eA, r<r', and it is clear from the definition that cr(n)+ o for all teA. These mean that the infinite matrix ( c,(n )) with the indices r, r'eA arranged in the linear order < is of the lower triangular form with non-zero diagonal elements. r' We shall now show that the matrix (c,(=)) has actually an inverse matrix. For that purpose, let T (resp. S) be the torus (resp. semi-simple) part of G, let A' (resp. A") be a maximal k-trivial torus in T (resp. S) such that A=cl(A'.A"), and put YQ= X(A)| Y~=X(A')|174 and YQ=X(A)| then YQ is identified with the direct sum of Y~ and Y~', and YQ with the dual space of YQ over Q. Moreover, call M" the intersection of M cYQ with the annihilator of Y~. From what we have stated in n ~ 2.2, M" can be also defined as the submodule of M formed of all meM such that ]x(~m) lp=I for all zeX(G). Then we have LEMMA 5.I. If rcmNnU~'U:~o with m, reM, we have m=r(mod. M"). Proof. Let X be any k-rational character of G. Then, since N has no non-trivial k-rational character, we have z(N) = I. On the other hand, we have clearly IX(u)Ip = I for all ueU. Therefore, if genmNnU='U, we have ]x(g)[, = IX(~) I,--IX(n') I~ and so [Z(n m-') [p= I. As this holds for all ~(eX(G), we have m--reM", q.e.d. LEMMA 5.2. For every reA, the set of r'eA such that r=r'(mod. M") and r'<r is finite in number. Proof. We extend the linear order in M to that in #/Q in a natural manner. Then there exists a Q-linear form L on YQ, not identically zero, such that, for xeYQ, L(x)>o implies x>o (and hence that x>o implies L(x)~>o). For each weW, w+ 1, we can define an order in YQ by x>o (i--w)x>o, ("t'~ar(TI~m) THEORY OF SPHERICAL FUNCTIONS 29 which induces a linear order on the factor space of YQ modulo the subspace formed of all x eYQ such that wx =x. Hence, similarly as above, there exists a non-trivial linear form L w on ~'Q having the property that Lw(x)>o implies x>wx (and hence that x>wx implies L~0(x)>~o and that x=wx implies Lw(x) =o). Therefore, if we put AQ={X~XZQI x>>.wx for all w~W}, A~={xeY, ILw(x)>~o for all weW), AQ is contained in A~ and the interior of A~ is contained in AQ. Since W = W A is the Weyl group of the restricted root system of G relative to A, we can conclude from this that we have AQ = A~ and that AQ is a (closed) " Weyl chamber " of W. Thus AQ is actually a closed cone defined by v"(= rank Y") linear inequalities Li(x) = Lwi(x ) >I o (I < i<, ~"). Now consider first the case where X(G) = I, i.e. YQ=Y~'. The Li's being linearly independent, we can write as L = EX~L~ with X ieQ. Here we assert that all Xi are >o. Clearly, it is enough to show that, if L~(x)>~o for all i (i.e. x~AQ) and x4=o, then L(x)>o. From the assumption, it follows that x>~wx and so L(x)>lL(wx) for all weW. But, since Y~={o}, we have ~ wx=o. Hence, if L(x)~<o, we would have L(wx)=o wEW forall wsW. But this is impossible, because forany x4:o, the set {wxlweW } contains always v" linear independent vectors. It follows that, for any reA, the set {xeAQlx<r } is bounded, and therefore that {r'EAIr'<r} is finite. In the general case, we have, from what we have proved above, L--Y~NLi (mod. Y~) with ?,~>o. Hence, if x--r is in the annihilator of Y~ and x~<r, we have L(r--x)=XXiLi(r--x)>~o. Therefore the set formed of all xeA~ satisfying these conditions is bounded, and so the intersection of it with A is finite, q.e.d. From Lemmas 5-i, 2, we see that the matrix (~',(~*'))(r, r'eA) can be decomposed FP into the direct sum of the (countably many) matrices (c,(r~))(r, r'eA, r_ r'(mod. M")), each of which is of the lower triangular form with respect to a set of indices isomorphic to {i, 2, ...}. Hence (c,(~)) has an inverse matrix of the same form. Thus we conclude that the mapping q0-+~ is an isomorphism (of vector spaces over 13) from .WiG , U) onto C[qg'~] w. Since it is also a homomorphism of algebras over 13, we have proved the following theorem: THEOREM 3" The assumptions being as in Theorem I, the Fourier transformation r gives an isomorphism (of algebras over 13) from -~a(G, U) onto 13[qM.~]w, the algebra of all W-invariant Fourier polynomials in q~:m(% with coedficients in C, {m/~t, ..., m ~'/} denoting a system of (independenO generators of M. Remark 1. The connectedness of G was needed essentially only in the proof of Lemma 5-I. (We used it also in the proof of Lemma 5.2, but this is not indis- pensable.) As is seen from that proof, this condition may be weakened, without changing the conclusion, into any one of the following conditions, where G O denotes the connected component of the neutral element of G : COt) The rank of the character module X(G) is equal to that of X(G~ 253 ICHIRO SATAKE 3o (02) In the notation of the assumption (I), we have G~176176 On the other hand, it follows from Lemma 5. I that, if the condition (II) is satisfied for a linear order in M, the same is also true for any linear order in M inducing the same order in M". Remark 2. The formula (6.7) can be further transformed in the following form: (6.8) m = r ) =~-i(v:m)u ym # (U\(U~:mNnUCU)). In fact, one first observes that by (4.3) meas. of (Nnv:-~Uv: m) = ~(nan) -1. On the other hand, it is easy to see that the cosets in (NnT:-mu~m)\(Nn~-mu~ru) are in one-to-one correspondence with those in U\(UC"NnU~rU). From these follows (6.8). Now, if the additional condition (5.2) is satisfied, we have Yrr= I for all reA. Therefore, in that case, the matrix (y,,) (r, r'eA) and its inverse arc integral, so that the Fourier transformation q~-+(~ gives actually an isomorphism (of rings) of the subring Lf(G, U)z= Z crZ of ~q~(G, U) onto the subring  (8-~(zcr)Fr)Z of rCA r6A C[q~ '] w . Remark 3. From the definitions, we have %(s) = # (U\U~'U)~,(~-'). Therefore we have (6.9) %(,-') = # (U\U~'U) -t Z y~,.8-~(~")Fr ,, r'EA rt~r which shows that c%(7: -r) is a W-invariant Fourier polynomial in q*m(~)'t But it seems far more difficult to describe how c%(~ -r) depends on reA. In any case, it is one of fundamental problems in our theory to obtain a handy expression (analogous to the " character formula "?) for (%(v: -r) or for Y,e- 6.3. Proofs of Theorems 1, 2. The first half of Theorem I follows from Theorem 3 immediately. To prove the second half, let {r (1), ...,r (~)} be a set of generators of A (as semi-group) and put c (~) = Or(i). (Note that A, being defined by a finite number of linear inequalities with integral coefficients, has clearly a finite set of generators.) The notations being as in the preceding paragraph, let A. denote an intersection of A with a coset modulo M"; by Lemma 5.2, A. is isomorphic (as ordered set) with the set of natural numbers. We prove, by induction on reA., that F r can be expressed as polynomial in ~(t), ..., ~(*) with coefficients in C. For that purpose, denote, for every r~A, by ~Jt r the vector space over C generated by F r, with r'eA, r'<r, r=r'(mod. M"). Then it is clear that we have (6. Io) Fr.F ,, --- Fr+ r, (mod. 93tr + r, ) for every r, r'eA. It follows that (6.1I) F,. ~r, C931,+,,, ~01,. 99~, Cg~r +,, 9 254 THEORY OF SPHERICAL FUNCTIONS 3~ On the other hand, by n ~ 5.2, we have (6.12) ~r -%rFr (rood. 9J~) with ?~---=y,,3 ~(r~')~=o. Now let teA. and suppose that our assertion is true for all r'eA., r'<r. Since {r (1/, ... r (~)} is a set of generators of A, r can be expressed in the form r = Y, nir (~) ' i=1 with n~Z, ni>>-o. Then from (6.1o), (6.11), (6.12)we have ni I1 ~(~)"i _= 1-I = (IIxr(il)Fr (mod. ~J'~). i i i Hence, applying the induction assumption on Fr, eS0~r, we conclude that F r is a poly- nomial in ~(~/'s with coefficients in 13. It follows then by Theorem 3 that s U) is generated over C by c/~) (I<i~</). 1 (By the same arguments, replacing F r by 8 2(r~r)F r and t3 by Z in the above proof, we can also conclude that, under the condition (5.2), Sr U)z is generated (over Z) by c (~) (i~<i~</).) As to Theorem 2, we have to prove the following two statements: io Every oe~2 can be: written in the form co=co~ with seX(H)| 2 ~ We have o~----o)s, , if and only if ( 2~i 19I) with weW. $' ---- WS rood. ]-0~gq To prove I ~ let co be any z.s.f, on G relative to U. Let {re(I),..., xn (~1} be a system of generators of M and put Xi=q mli)'~. Then, by Theorem 3, s can be identified with the subalgebra of C[X~ l, ..., X~ ~1] formed of W-invariant elements. Since 13[X~l, ..., X~ l] is integral over Lf, the homomorphism (~5 :.Lf-+C can be extended to a homomorphism, denoted again by ~, from C[X~I, ..., X~ l] onto 13 (Ix6], p. 8, Th. I, p. 12, Prop. 4)- Since G(X~.) + o, one can put ~(X~) ~-q~ with s~e13. Take s eX(H) | C such that na (i). s ---- s~ (I ~< i~< v). Then we have G ~, so that c0 ~ o)~. Proof of 2 ~ For s, s'eX(H)| we have o)s----o) s, if and only if ~(s)----~(s') for all (? ~.oq o, or, what amounts to the same by Theorem 3, (? (s) ---- ~ (s') for all r e C[q~'S] w. But this last condition is clearly equivalent to saying that s'----ws (mod. 2~i 1~I] log q ] with weW. w 7. Homomorphisms. 7-x. Let G, G' be two algebraic groups satisfying the assumption (I) with respect to U, H, N and to U', H', N', respectively. We suppose further that there are given WcWn, resp. W'cWIv, and their fundamental domains A, resp. A' (defined by (3.13)) and that G' satisfies assumption (II1) with respect to A'. Let X be a homomorphism from G into G' satisfying the following conditions (i), (ii), (iii). (In ----- (),r(~)Fr(~))"~ ICHIRO SATAKE 3~ this paragraph, it is enough to assume that X is continuous in the sense of the p-adic topology.) (i) X(H) cH', X(N) oN', X(U) cU'. Then, since k(H u) cH 'u, k induces a homomorphism M(~H/H u) -+ M'(~H'/H'"), which we denote again by X, such that X(lh)=lx(h) for all hell. (ii) X(A) cA', (iii) O is normal in X-t(U'). It follows from (i), (iii) that (7. i) ~k-I(U ') = O T~nU, m~Mk ~m U = U7~ m for all mEMx, More generally, one obtains where Mx denotes the kernel of X:M-+M'. LF.M~. 7. I. We have (7.2) X-t(U'X(g)U')=UgX-I(U')= U Ugz~mu m EM x =X-~(U')gU= O urcmgU (disjoint union). m EM k In other words, we have U'X(gl)U'=U'X(g)U" if and only if we have UgxU = Ug~m U(= U~mgU) with m eMx, where m is uniquely determined by g, gl. Proof. According to (3. I4), one puts g=ut~ru2, gt=u~7~r'u4, X(gl) =U~X(g)u 2 with u 1, ..., u4eU, u;, u~eU', r, r~eA; then one has ~(u3))~(~rI)X(u4) =uiX(ul)X(rd)?,(u2)us Hence from (i), (ii) and (II1) for G', one gets X(rl) =X(r), so that rt= r + m with meMx. Then gt=uarc'~mu4-----(uau~-t)g(u-~lr:mu4)eUgr~mU by (7.1)(and similarly gleUr~mgU). To show the uniqueness of m, let Ug~mU = UgzWU with m, m' E M x. Then we have Ug~mU~Ugr~murcm'-m, so that grcm+z(m'-m)~Ug~mU for all leZ. But this is impossible unless m=m', because Ug~mU is compact and {gr~'lmleMz} is discrete, q.e.d. It follows, in particular, that in the above notation, if g=gl, we have r=rl, i.e. G also satisfies (II1). Now, let .oq~ U), ~~ U') and, for 9~5f, define 9's.oq ~ by = (gv)dv= X meg). (7-3) =fz-,(u') ~(vg)dv= E p(~mg) if g'eU'X(g)U', m ~k o ff g'r 256 THEORY OF SPHERICAL FUNCTIONS 33 From Lemma 7.I it is clear that q~' is well-defined and belongs to .go,. We put ?' =X*(?). The mapping ),* can also be defined as a linear mapping s _+&a, such that (7.4) X*(c,) =%,) for all reA. We now assume further that X satisfies the condition (iv) X(G).U' = U'.Z(G). PROPOSITION 7" I. Let G, G' be two algebraic groups over k satisfying (I), (II,) and let X be a homomorphism from G :into G' satisfying (i), (ii), (iii), (iv). Then a) X* is a homomorphism (of algebras over C) from 5f(G, U) into s U'). b) For ~'ef2(G', U'), we have ~=~'oXen(G, U) and (7-5) ~3(~) =g'(X*(p)) for all ?e~'(G, U). Proof. a) If X*(%) = q~ (i-= I, 2), we have t I t t ~)2(gJ-I I--1 I )dg, (%, %)(X(g)) = fa,%(X(g)g 0 --_fx(a)fu,~',(X(g)X(gl)u')~s ' (by (iv)) --= fx(o) q~; (X (g&)) ?~(X(g,)- ') dX(g,) = fo,~-,(,) dx(g*) fz-,(u,)~(gg, v,)dv, fx-,(v,)%(vr'gr*v2)dv2 (by (7.3)), where the measure on X(G)~=G/X-~(I) is normalized in such a way that the total measure of the open subgroup ;~(G)nU'-~X-I(U')/X-I(I) is equal to i. Hence the last expression is equal to = far 1%(gg2)%(g;lv2)d&dv2 "k- (U' =x*(~** ~2)x(g). On the other hand, if g' ~X(G)U', we have clearly (~* v~)(g') = x*(~** ~) (g') = o, If X*(~)---- ~', we b) It is clear that co'oX satisfies the conditions (i), (ii) of z.s.f. get quite similarly as above (q/. co') (X(g)) = fa, q~'(x(g)g[) ~' (g~- 1)dg~ = fx(~) q~'(x(g&))~'(x(g*)-*)dx(gl) = fo/x-,(,)J'x-,(v')~(gglv)~'~ (gg=) o~'o~,(g-*)d = j~ 2 g2 = (~, (~'oX)) (g). 5 34 ICHIRO SATAKE Thus we have q~ 9 (r ok) (g) = (q~', r (X(g)) = ~'(~'). r (X(g)), which proves (iii) (in N ~ 5. I). At the same time, (7.5) is proved, q.e.d. 7.2. Isogenies. We first apply the above considerations to the case of a k-isogeny. Let, as in N ~ 7. r, G, G' be algebraic groups over k satisfying (I) and suppose that G' satisfies (II1) ; we suppose luther that G, G' are connected and that W=Wa, W'=WA, , where A, A' denote the unique maximal k-trivial toruses in H, H', respectively. Let X:G~G' be a k-isogeny satisfying (i). Then, A, N and A', N' being the subgroups of G, G', respectively, corresponding under X, we have (7.6) A--the connected component of X-t(A'), A'=cl(X(A)), N--=the connected component of )~-I(N'), N'----Z(N). From the maximality of U, we also have (7-7) U=X-I(U') 9 Thus (iii) is trivially satisfied. Here X : M-->M' is injective, and, by our assumption, W, W' may be identified with each other canonically. Therefore, taking a linear order in M induced from that in M', we have (7.8) A = X -1 (A'), or A--A'nM, when we consider that McM'. Thus (ii) is also satisfied. Therefore, as we stated in 7-I, G satisfies (II1). Furthermore, it is clear that if G' satisfies (II), so does also G. To proceed further, let us first note that X is not necessarily surjective. In fact, it is known that k(G) is a normal subgroup of G' and that G'/X(G) is finite, commutative. (This is true for any isogeny between connected algebraic groups over a p-adic field. See Ix5], Prop. 3-) It follows, in particular, that X satisfies the condition (iv). PROPOSmON 7.2. The assumptions being as above, G'/(X(G)U') is canonically isomorphic to M'/X(M). Proof. Since X(G)U' is a normal subgroup of G' containing N'=X(N), we have, by (3.I), G'=H'.X(G)U'. Next, we assert that H'n (X(G)U') ----X(H)H '~. In fact, let H'c~(X(G)U')gh':X(g)u' with geG, u'eU' and put g:ulhu 2 with hell, ul, u2eU. Replacing h by uhu -1 with ueN(H)nU, if necessary, we may assume that lz(h), lh,, belong to one and the same fundamental domain w'A' (w'eW'). Then, from h':X(ul)X(h)X(@u', we have r(h'):r(X(h)) and so /h,:lz(h), i.e. we obtain h'=X(h)u" with u"eH 'u. This proves the inclusion r the inverse inclusion is trivial. It follows that (7.8) G'/(X(G)U') ~ H'/(X(H)H'U). 258 THEORY OF SPHERICAL FUNCTIONS On the other hand, since M=~H/H u, M'~H'/H 'u and X(H ~) =X(H) nH '", we have (7.9) H'/(X(H) H'") ~ M'/X(M). From these, we conclude the Proposition, q.e.d. We denote by ~, the character group of G'/(X (G) U') and consider ~ ~ E as a function on G'. Then E operates on ~Sf'=.~f(G', U') by (7. io) (~.q/)(g') =~(g').q/(g') for ~E, q/sNe'. The correspondence ~'~q0' is clearly an automorphism of s (as algebra over C). We denote by ~'(G', U') -=" the subalgebra of s formed of all ~-invariant elements. Then we have PROPOSITION 7.3- The assumptions being as above, X* is an injective homomorphism (of algebras over C) .from s176 U) into ~(G', U') and its image is equal to .oq~ ', U') ~. Proof. Since X satisfies the conditions (i),~ (iv), it follows from Proposition 7-I that X* is a homomorphism. Since X : M -~M' is injective, it follows from (7.4) that ~* is injective. Finally, it is clear that an element q~' of &~ U') belongs to the image of?,*, if and only if its carrier is contained in U'),(G), and this latter condition is equivalent to saying that q~' is invariant under E, q.e.d. Remark. It is easy to see from the definition that X*(.Lf(G, U)z ) =X*(Se(G, U))n Lf(G', U')z= (Se(G', U')z) z. On the other hand, E operates also on ~'----~(G', U'). Namely, as is easily seen, for every ~EE, co'~f~', we have ~.o~'Ef~' ([22], Prop. 5) (in particular, putting to'=z (constant), we have ~f~'). From the definitions, it is clear that (7. I I) ~'(~') ~- ~--lO)'(q)'). Now, as stated in N ~ 5.3, 5.4, a part of f~' is parametrized by X(H')| [ ~log 2~i q 1VI') ; for s~X(H')| denote by % the corresponding z.s.f. on G'. Then we have the following 2~i 1~I'~ L~MMA 7.2. If ~]H'++S~ roOd. log q ] in the sense of N o 5.4, we have ~--]. i t (7- I2) ~ 9 %-- %+,~. Remark. This Lemma is valid for any quasi-character ~ of G' which is trivial on U' and N'. Proof. If ~z,++s, one has o~(g') = fu,+~(g'-tu')du ' Hence one obtains ~-~. co~(g') = ~(g'-l) fu, dA'(g'-lu')du' ,I. f-'-Xu'~du' JlJ,'~g ) ' 259 ICHIRO SATAKE 3 6 because ~(U') =~(N') = i. As ~<-+s~, one has 32~++s+s~. Hence the last expression is equal to o'.+~(g'), q.e.d. By (7.II), (7.I~2) is equivalent to (7.13) ~. r = ~'(s + s~). Next, let us consider the correspondence ~'~ c0'-+~o = o~'oX~fl more closely. Let kn : H ~H' be the restriction of X on H and tX. : X(H') ~X(H) its" dual ". Then *X u is injective, with finite cokernel, so that it can be extended canonically to an isomorphism X(H')| -+ X(H)| t 1QI' Since we have XH( ) clQI, it induces a homomorphism which we denote again by tXi~. from X(H')| ( 2~i onto X(H)| ( 27~i ]QI). We now assert that the q ) \log q following diagram is commutative: 4' (7.14) [ 2r:i 19I] +- [ 2~i 19I'] X(H)| ~l-~gq ] X(H')| ] Namely we shall prove the following L~.MMA 7.3. If seX(H)| s'eX(H')| are such that s=tXH(s'), we have ca) s = ca) s, o~,~ (7" 15) or equivalently, by (7.5), for all v e(G, U). (7. i6) Proof. In view of (6.3) and the relation m.s =X(m).s' (meM), it is enough to show that we have ~'(~m) ='~x(,)(~zz(m)) for all r, meM. By (6.8) and by the coincidence of ~ for G and G', this is equivalent to (U~(UT~ru N U=mN)) = ~ (U'~(U'7~)'(r) U'~ U'~Z(m)i')), which can be proved as follows. From Lemma 7. I, we have X-I(U'~:~(r)u ') = UnrU. Since N'=X(N) O,(G), it follows that (X(rzm)N ') n U'~ x(') U' -----x(.mN r Ur:r U). 260 THEORY OF SPHERICAL FUNCTIONS This shows that the mapping U'\(U='U n U=mN) --> U'\(U'= x(r) U' n U'7~ zlm) N') defined by Ug-+U'X(g) (gz==NnU='U) is surjecfive. It is also injective because of the relation (),(~m)N')nU'=X(="NnU). This proves our assertion, q.e.d. The formula (7. I6) implies the commutativity of the diagram -~(G, U) z*, .~(G', U') (7.17) ] c[q q w , w', where the second horizontal homomorphism is defined by the correspondence qm.S _+qX(m).~. Finally, let us suppose that G' (and hence also G) satisfies the assumption (II). Then it follows that the mapping Y~+-~' in (7. I4) is surjective and that we have o~o).=to~oX for to~, coW, f2' if and only if to~ =~.c% with ~eE. In fact, the" if" part being trivial, suppose that to~loX = co~oX with sl, sz~X(H')| Then, by Lemma 7.3, we have tXa(sl)~t),H(s~)(mod. W 2~i 1V[) and so sl---s~ (mod. W'. 2yi ,X~(l~) " logqg q ' log q ]" On the other hand, by virtue of Proposition 7.2, we have ^ " ! l~176 I" (7. I8) 'Xa~(M)/M'= t Hence it follows that s2=slq-s~ ( mod. w' .1-~alVI' ) with ~E, and so, by Lemma 7.2, ~, = 4-1. co~,, which proves the " only if" part. Thus f2 can be identified with fV/E. Summing up, we obtain TrrF.OR~.M 4" Let G, G' be two connected algebraic groups over k satisfying assumption (I) with respect to U, H, N and to U', H', N', respectively, and let X be a k-isogeny from G to G' such that X(H)r X(N)cN', X(U)cU'; suppose further that G' satisfies (II) (with respect to W'=WA, ). Let E be the character group of G'/X(G)U' (which is a finite commutative group). Then, G satisfies also (II) with respect to W=W A and the linear order in M induced from that in M'; and a) .~ operates on .~~ U') by (7. xo), and ~F(G, U) can be identified with .s U') z by the mapping X* defined by (73). Moreover the diagram (7.17) is commutative. b) E operates on f2(G', U') similarly, and f~(G, U) can be identified with t~(G', U')/E by the mapping ~(G', U')~ e0'-+o~'oXef~(G, U). Moreover the diagram (7. I4) is commutative. Remark. Supposing only that (II~) and Theorem 3 (hence also Theorem ~) hold for G', U', instead of assuming the assumption (II) for G', in the above theorem, we 261 3 8 ICHIRO SATAKE can conclude that Theorem 3 for G, U, as well as the statements a), b), hold. In fact, in view of the diagram (7. I7) and Proposition 7-3, it is enough to show that ~(G', U') "= and C[qX(M)'s] w' are corresponding under the isomorphism .LP(G', U')~C[qM"s] w'. But this follows immediately from (7. I3) and (7-I8). 7.3. As a second application of N ~ 7- i, we consider the case where X is injective. Namely, let G be a k-closed subgroup of G' (both satisfying (I)), and suppose that X=identity satisfies the conditions (i), (ii). Then, again from the maximality of A, N, U, we get (7-I9) A = the connected component of A'nG, N=N'oG, U=U'nG. Thus (iii) is trivially satisfied. It follows that, if G' satisfies (II1) , so does also G. On the other hand, since H" =Hr~H'", X : M ~M' is injective, and by what we have stated in N ~ 3.4, 3 ~ it can readily be seen that (7.20) A--A'nM. Therefore, it is clear that, if G' satisfies (II), so does also G (with respect to the induced linear order in M). Now, it is known (cf. No 8.2) that G' =GL(n, k) satisfies the conditions (I), (II) with respect to H'=A'=D(n, k), N'----TU(n,k), U' --= GL(n, o), W'=~n (symmetric group of n letters); in this case, M' is canonically identified with Z n, and, taking the lexicographical linear order in M', we have A'={m---- (m,) eg'lml>~...>/m,}. For GcGL(n,k), put U=GnGL(n, o). Then, the conditions (I) for G, U and (i), (ii) for ?,=identity can be stated as follows: (I*) There exist, in G, a connected k-closed subgroup A contained in D(n, k) and a k-closed subgroup N contained in T~(n, k), normalized by A, such that we have G=U.AN=U.A.U. (II*) There exists a subgroup W of W A such that every w~W can be written in the .form w=w~, with usN(A)r~U, and that, for m=(m~)eM(cZ n) with wm<~m for all weW (with respect to the lexicographical order in Zn), we have ma>>. .. . >~m,. Therefore, from what we have stated above, we obtain the following: TI-IEORF.i 5 [xg]" Let G be a (connected) k-closed subgroup of GL(n,k), U = Gn GL(n, o), and suppose that G, U satisfy the conditions (I*), (II*). Then Theorems I, ~, 3 hold for G, U. As is seen from (7.2o) and N ~ 6.2, the connectedness assumption on G is unnecessary. Theorem 5 can be applied, for instance, to SL(n, k), Sp(n, k), SO(n, k, S), taken in a suitable matricial expression. 262 THEORY OF SPHERICAL FUNCTIONS Example. SO(n, k, S). By definition, SO(n, k, S) ={geGL(n, k) l'gSg = S, det g =~ }, where S is a non-singular symmetric matrix of degree n. By a theorem of Witt, S can be transformed into the following form S= S0 o~}n0, n ~- n o .ql- 2V, L~ 9 , S o = symmetric matrix of degree n o corresponding to an " anisotropic " "'o quadratic form (i.e. for xek ~~ txS0x=o implies x=o). Then, from the theory of maximal lattices (cf. w 9), it can be verified that G=SO(n, k, S) satisfies (I*), (II*). But, if we consider the group of similitudes with respect to S instead, it appears that (I*) is not always satisfied. Remark. Proposition 7-I cannot be applied, unless the condition (iv) is satisfied. It can surely be applied to SL(n, k), and more generally to the situation considered in Proposition 3.2. In these cases, Lemma 7.3 (i.e. commutativity of the diagram (7.14)) can also be proved. 7.4. Finally we apply our considerations in N ~ 7. I to the situation considered in Proposition 3.3- Namely, let G be an algebraic group over k satisfying (I), (II1) with respect to U, H, N, WcW H, let Z be a k-trivial torus contained in the center of G and put (7.2i) G-~G/Z, U=(HoU)/Z, fl=H/Z, I~=ZN/Z, H o being defined by (3-7)- Suppose that H o normalizes U. Then G satisfies (I) (Proposition 3.3)- We now prove that G satisfies also (II1). In fact, put (7.22) M0={meM]� ~ for some � zeZ}. Then, M0-~H0/H u and so M= - "~H / H u~'~ _~H/ H o= ~M / M 0. Since W operates trivially on M0, the fundamental domain A consists of cosets modulo M0, and A = AIM o is clearly a fundamental domain, in 1~I, of the group ~rcWn (Vr determined by W in a natural way. (More precisely, we may assume, without any loss of generality, that the linear order in M defining A is " adapted " to M0, i.e. satisfies the condition that, if m, m'~M, mr re>o, m___-m'(mod. M0) , then m'>o. Then, the linear order in M induces in a natural way that in 1VI= M/M0, and the fundamental domain of ~/ defined by the latter is precisely A.) Then, it is clear that = LI UnrU (disjoint union), rEX which proves our assertion. It is also clear that, if G satisfies (II) (with respect to a linear order in M adapted to M0) , so does also G (with respect to the " induced " linear order in/VI). 263 ICHIRO SATAKE 4 ~ Now, call X the canonical homomorphism G-+G = G/Z. Then, by our assumptions, X satisfies the condition (i)~ (iv) (U', H', N', A' being replaced by U, H, N, A, respec- tively). We note that, in the notation ofN ~ 7.1, we have X-I(U ') =HoU , Mx=M 0. PROPOSmON 7.4- The notations and assumptions being as above, X* is a surjective homomorphism (of algebra over C) from ~~ U) onto ~~ U), and its kernel is equal to the ideal generated by Co---c ~ (meMo) , or, what amounts to the same, by co--%(i) (I~i~<vo), { re(t}, ..., m(~~ (~0 = rank Mo) being a system of generators of M 0. Remark. It is easy to see that X*(~(G, U)z ) =oLP(G, U)z. Therefore, by this proposition, ~(G, U)z can be identified with the factor ring of ~~ U)z by the ideal generated by Co--C., (mEMo). Proof. The first statement is a direct consequence of Proposition 7-I, (7-4) and what we have stated above. To prove the second, choose a system of representatives {r} ofA/M 0. Then every ~e.~qP(G, U) can be written uniquely in the form ~= E E x,,,,e,+m (7.23) r:A/Mo mEMo with Xr,,eC, and we have ~k*(q)) = r A/M. E (In'M0 ~r'na) Cx(r)" Therefore, we have X*(~) =% if and only if (7.24) mEMo On the other hand, from our assumption on Ho, we have for each r. Cr*Cm~Cm*er~--Cr+ m for all reM, meMo. Hence, putting c (i) = em(~), we have, for m = ~ nim (i), er+ m = (g(1))n',... * (g(vo))nvo,er, (C(r denoting the ni-th power of c (~) with respect to the convolution. Thus our assertion on the kernel of k* is reduccd to the following, easy, purely algebraic lemma: LEMMA 7- 4- Let ~(X) = ,~ X, ....... %X? ... X~C[X~', .... X~]. Then, wehave Z X ......... =o, if and only if ,p belongs to the ideal generated by I--Xi(I~<i<,~ ). nl, ...,n v Now assume that G is connected and satisfies (II) and consider the relation between f~----f~(G, U) and ~ =~(G, U). By Proposition 7.4, ~ can be identified with the subset of fl formed of all co ell satisfying the condition G (c 0- Can) = o for all m e M0, or equivalently, if we write co = % with seX(H)| ^ (s) for all meM0. C m = I 264 THEORY OF SPHERICAL FUNCTIONS Now, from (6.3), we have 2,.(s) =q-'~" for m~M o. In fact, from the definitions, we have ~(~m)= I for meMo; and if r~m'NnU~mU+o for memo, m'eM, we have', (~m'-mN) oU~eo and so m'=m (N ~ 3.4, I~ 9 Hence by (6.7) we have I if m'=m, )= Cm(~lDfl' i, o if m'+m, which proves our assertion. It follows that, for o~f~, we have c%e~, if and only if (7.25) m.s -- o mod. l-ogq Z for all meM o. This result can also be obtained from the commutativity of the following diagram: (7.26) X(H)| [ 2rd 1VII +- X(H)| [ 2r~i I~i~ \logq ] !,logq ]' which can be proved quite similarly as Lemma 7.3. Thus we obtain PROPOSITIO~ 7"5" Let the notations and assumptions be as stated at the beginning of the paragraph, assume further that G is connected and satisfies (II) with respect to W r n and a linear order in M adapted to M o. Then, G = G/Z satisfies also (II) with respect to ~r = Wi, and the induced linear order in i~I= M/M0; and ~ can be identified with the subset of f~ formed of all co, such that m.s--o mOd. l-ogqZ for all meM 0. Moreover, the diagram (7.26) is commutative. 2,65

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Publications mathématiques de l'IHÉS – Springer Journals

Published: Aug 7, 2007

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