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Theory of spherical functions on reductive algebraic groups over p-adic fields

Theory of spherical functions on reductive algebraic groups over p-adic fields ICHIRO SATAKE subgroup U, can be canonically identified with a quotient space of the form C'/W, where v denotes the " rank" of G, i.e. the dimension of a maximal vector part of a Cartan subgroup of G and W the restricted Weyl group of G. Recently the theory has been extended to the case of some classical groups over p-adic fields by Mautner [x7] , Tamagawa [23] and Bruhat [4], [5]- The main purpose of this paper is to show that the principal part of the theory, including the above-mentioned theorem of Harish- Chandra, holds for a wider class of reductive algebraic groups over p-adic fields, containing all simple classical groups without center. To be more precise, let k be a local field, G a Zariski-connected reductive algebraic subgroup of GL(n, k), A a maximal k-trivial torus in G and N a maximal k-closed unipotent subgroup of G, normalized by A (G, A, N, ... being understood as to represent the groups of k-rational points); the pair (A, N) is then unique up to inner automorphisms of G. Put dim A = v and denote by A" the unique maximal compact subgroup of A. Then the restricted Weyl group of G relative to A, W= N(A)/Z(A), operates in a natural way on A as a group of automorphisms, and hence also on the character group (in the algebraic sense) of A, Y=X(A)(~Z~), and on the group of quasi-characters (in the topological sense) of A/A", Hom (A/A", C*)=Y| Now let k=R or C, and let U be a maximal compact subgroup of G; the quotient space S = U\G is then the associated symmetric space. A C~~ co on G is called a zonal spherical function (or elementary spherical function) on G relative to U, if it satisfies the following conditions (i) r = co(g) for all g~G, u, u'~U, (ii) ~(I) = I, (iii) co, considered as a function on S, is an eigen-function for all invariant differential operators on S. As is well-known, the algebra (over C) of all invariant differential operators on S is canonically isomorphic to the algebra of all W-invariant polynomial functions on the dual of the Lie algebra of A, or, what is the same, on Y| so that it is actually a polynomial algebra of v variables over t21. (Thus the condition (iii) is reduced to the v conditions corresponding to the generators of this algebra. Furthermore, it is also known that the condition (iii) may be replaced by a similar condition for invariant integral operators.) These being said, the theorem of Harish-Chandra asserts that the set of all zonal spherical functions on G relative to U is identified with the quotient space YNC/W. The proof for this consists essentially in showing that, U, A, N being taken suitably so that we have the decompositions (.) G=U.A.U = U.AN, the Fourier transformation with respect to zonal spherical functions (parametrized by Y| gives actually an isomorphism from the algebra of the invariant differential operators onto that of the W-invariant polynomial functions on Y| The first difficulty, in translating these results from real to p-adic, arises from the 230 ICHIRO SATAKE subgroup U, can be canonically identified with a quotient space of the form C'/W, where v denotes the " rank" of G, i.e. the dimension of a maximal vector part of a Cartan subgroup of G and W the restricted Weyl group of G. Recently the theory has been extended to the case of some classical groups over p-adic fields by Mautner [x7] , Tamagawa [23] and Bruhat [4], [5]- The main purpose of this paper is to show that the principal part of the theory, including the above-mentioned theorem of Harish- Chandra, holds for a wider class of reductive algebraic groups over p-adic fields, containing all simple classical groups without center. To be more precise, let k be a local field, G a Zariski-connected reductive algebraic subgroup of GL(n, k), A a maximal k-trivial torus in G and N a maximal k-closed unipotent subgroup of G, normalized by A (G, A, N, ... being understood as to represent the groups of k-rational points); the pair (A, N) is then unique up to inner automorphisms of G. Put dim A = v and denote by A" the unique maximal compact subgroup of A. Then the restricted Weyl group of G relative to A, W= N(A)/Z(A), operates in a natural way on A as a group of automorphisms, and hence also on the character group (in the algebraic sense) of A, Y=X(A)(~Z~), and on the group of quasi-characters (in the topological sense) of A/A", Hom (A/A", C*)=Y| Now let k=R or C, and let U be a maximal compact subgroup of G; the quotient space S = U\G is then the associated symmetric space. A C~~ co on G is called a zonal spherical function (or elementary spherical function) on G relative to U, if it satisfies the following conditions (i) r = co(g) for all g~G, u, u'~U, (ii) ~(I) = I, (iii) co, considered as a function on S, is an eigen-function for all invariant differential operators on S. As is well-known, the algebra (over C) of all invariant differential operators on S is canonically isomorphic to the algebra of all W-invariant polynomial functions on the dual of the Lie algebra of A, or, what is the same, on Y| so that it is actually a polynomial algebra of v variables over t21. (Thus the condition (iii) is reduced to the v conditions corresponding to the generators of this algebra. Furthermore, it is also known that the condition (iii) may be replaced by a similar condition for invariant integral operators.) These being said, the theorem of Harish-Chandra asserts that the set of all zonal spherical functions on G relative to U is identified with the quotient space YNC/W. The proof for this consists essentially in showing that, U, A, N being taken suitably so that we have the decompositions (.) G=U.A.U = U.AN, the Fourier transformation with respect to zonal spherical functions (parametrized by Y| gives actually an isomorphism from the algebra of the invariant differential operators onto that of the W-invariant polynomial functions on Y| The first difficulty, in translating these results from real to p-adic, arises from the 230 THEORY OF SPHERICAL FUNCTIONS difference of the nature of maximal compact subgroups. Whereas they are " algebraic " and mutually conjugate in the real case, they have no more such a property in the p-adic case. As a matter of fact, we have to say that, for the time being, our knowledge on this subject is still very poor. Therefore, in this paper, we will make certain assumptions on G, assuring the existence of a favorable maximal compact subgroup U (the assumptions (I), (II) in w 3). These assumptions are nothing but p-adic analogues of some well-known properties of semi-simple Lie groups; in particular, the condition (I) implies the possibility of a decomposition like (.), but A Should now be replaced by a bigger subgroup H, such that AcHcZ(A), and W-invariant. On the other hand, as we shall show in Chapter III, these conditions are satisfied by " all " known examples of classical groups, by virtue of the theory of elementary divisors. Thus one may hope to find a unified proof for these assumptions. Now, under these assumptions, let ~(G, U) denote the algebra (over C) of all invariant integral operators on U\G, whose kernel is given by a function on G with compact carrier. One defines a zonal spherical function as a function on G satisfying (i), (ii) and the condition (iii) stated in terms of ~(G, U) ; then a zonal spherical function determines a homomorphism (of algebras over C) from ~f'(G, U) onto C and vice versa. On the other hand, call H u the unique maximal compact subgroup of H and put M-=H/HU(~-Z~). Then our main theorem (Th. 3 in w 6) asserts that ~(G, U) is isomorphic to the algebra of i, ll W-invariant polynomial functions on Horn(M, C*)(=~ Cv), allowing this time negative powers in an obvious sense; thus ~4'(G, U) is an affine algebra of (algebraic) dimension v over C. From this follows immediately the analogue of the theorem of Harish-Chandra asserting that the totality of zonal spherical func- tions on G relative to U is canonically identified with a quotient space of the form (C*)'[W (Th. 2 in w 5). As examples, it will be shown that, in case G is a simple classical group without center and U a maximal compact subgroup of G defined by a " maximal lattice ", the algebra ~q~(G, U) is actually a polynomial algebra of v variables over C (Th. 7, 9 in w 8, 9)- More precisely, the so-called (local) " Hecke ring " .W(G, U)z is a polynomial ring of v variables over Z. These theorems are proved by the usual method of Fourier transformation and, in fact, rather simply, compared with the real case. But to determine the explicit form of zonal spherical functions and the Plancherel measure, it seems necessary to know the (infinite) matrix of this Fourier transformation more explicitly. This has been done by Mautner [i7] for PL(2, k), but is still an open problem for the general case. As a partial result in this direction, we will calculate in Appendix I (local) Hecke series and especially ~-functions attached to GL(n, R), where 2R is a central division algebra over k, and to the group of symplectic similitudes. Besides these, we will analyze in w 7 the behavior of zonal spherical functions under a homomorphism ~ : G-+G', and especially under a k-isogeny (Th. 4)- Here we do not assume a priori the conditions (I), (II) on G, G' in full, and will see how (parts of) tile conditions on the one of G, G' imply the corresponding conditions on the 231 ICHIRO SATAKE other. As another application, we will determine in Appendix II all zonal spherical functions of positive type, or, what amounts to the same, all unitary equivalence classes of irreducible unitary representations of the first kind, of PL(2, ~), obtaining again a result quite analogous to the real case. A part of results of this paper (N O 7-3) has been announced in a short note [xg] , which will also serve as an introduction to this paper. The author has much profited by seminars on Spherical functions, organized during the period of 196o-62, by Professor Y. Akizuki, to whom this paper is dedicated with sincere gratitude and respects. Notations and Conventions. Throughout this paper, k denotes a p-adic number field, i.e. a finite extension of the p-adic number field O p. The valuation-ring in k and its (unique) prime ideal are denoted by o, p = (r~), respectively, r~ denoting a prime element. We denote by I Ip (or simply by I ]) the normalized valuation of k, i.e. ]~[p=q-Or%~ for ~ak, q denoting the number of elements in the residue class field o/p. All algebraic groups we consider are supposed to be a.fine, so that they are realized as groups of matrices. Thanks to a result of Rosenlicht (Annali di Matematica, vol. 43 (I957), P. 44), in any such group, defined over k, the subgroup formed of k-rational points is everywhere dense in the sense of the Zarisld topology, provided all the connected components contain a k-rational point. Hence, in this paper, we will understand by an " algebraic group over k " the group formed of k-rational points of an algebraic group (in the sense of algebraic geometry) defined over k. If G is a (Zarisld-) connected algebraic group over k and if K is an overfield of k, the group formed of K-rational points in the same algebraic group will be denoted by G K. In an algebraic group G over k, one can consider two kinds of topologies, i.e. the p-adic topology and the Zariski topology. Without any specific reference, the words " closed" (or" k-closed")," connected" will be used exclusively in the sense of the Zariski topology, while the words " open ", " compact " will always be understood in the sense of the p-adic topology. " Closure" (in the sense of the Zaristd topology) of M is denoted by cl(M). When G, G' are algebraic groups over k and ~ a k-morphism (i.e. a rational homomorphism defined over k) from G into G', the symbols Im q0 = ~ (G), Ker qo = ~- 1 ( I ) are used in the set-theoretical sense; thus V-I(I) is k-closed but q0 (G) is not, and (in case G is connected) the image of q~ in the algebraic sense is cl(q0(G)). (More generally, the similar convention is made for any rational map defined over k.) In particular, for a k-closed normal subgroup H of G, we denote (by abuse of notation) the factor group in the algebraic sense by el(G/H). As usual, for any ring R with the unit element I, R* stands for the multiplicative group of regular elements in R, and M~(R) for the ring of all n � n matrices with coefficients in R. The unit matrix of degree n is denoted by i n (or simply by i). 232 THEORY OF SPHERICAL FUNCTIONS For Xi~Mn~(R ) (I<<.i<~r), the symbol diag. (X1, ..., Xr) will represent a matrix of degree n=~n i of the following form: o "Xr/" Z, Q, R, C denote, respectively, the ring of rational integers, the rational number field, the real number field and the complex number field; the real and imaginary parts of s~C are denoted by Re s, Im s, respectively. For a finite set M, the symbol # M represents the number of elements in M. For a map ? defined on a set M and for a subset M~ of M, the symbol q~nM1 stands for the restriction of ~ to M1. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Publications mathématiques de l'IHÉS Springer Journals

Theory of spherical functions on reductive algebraic groups over p-adic fields

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

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References (16)

Publisher
Springer Journals
Copyright
Copyright © 1963 by Publications mathématiques de l’I.H.É.S
Subject
Mathematics; Mathematics, general; Algebra; Analysis; Geometry; Number Theory
ISSN
0073-8301
eISSN
1618-1913
DOI
10.1007/BF02684781
Publisher site
See Article on Publisher Site

Abstract

ICHIRO SATAKE subgroup U, can be canonically identified with a quotient space of the form C'/W, where v denotes the " rank" of G, i.e. the dimension of a maximal vector part of a Cartan subgroup of G and W the restricted Weyl group of G. Recently the theory has been extended to the case of some classical groups over p-adic fields by Mautner [x7] , Tamagawa [23] and Bruhat [4], [5]- The main purpose of this paper is to show that the principal part of the theory, including the above-mentioned theorem of Harish- Chandra, holds for a wider class of reductive algebraic groups over p-adic fields, containing all simple classical groups without center. To be more precise, let k be a local field, G a Zariski-connected reductive algebraic subgroup of GL(n, k), A a maximal k-trivial torus in G and N a maximal k-closed unipotent subgroup of G, normalized by A (G, A, N, ... being understood as to represent the groups of k-rational points); the pair (A, N) is then unique up to inner automorphisms of G. Put dim A = v and denote by A" the unique maximal compact subgroup of A. Then the restricted Weyl group of G relative to A, W= N(A)/Z(A), operates in a natural way on A as a group of automorphisms, and hence also on the character group (in the algebraic sense) of A, Y=X(A)(~Z~), and on the group of quasi-characters (in the topological sense) of A/A", Hom (A/A", C*)=Y| Now let k=R or C, and let U be a maximal compact subgroup of G; the quotient space S = U\G is then the associated symmetric space. A C~~ co on G is called a zonal spherical function (or elementary spherical function) on G relative to U, if it satisfies the following conditions (i) r = co(g) for all g~G, u, u'~U, (ii) ~(I) = I, (iii) co, considered as a function on S, is an eigen-function for all invariant differential operators on S. As is well-known, the algebra (over C) of all invariant differential operators on S is canonically isomorphic to the algebra of all W-invariant polynomial functions on the dual of the Lie algebra of A, or, what is the same, on Y| so that it is actually a polynomial algebra of v variables over t21. (Thus the condition (iii) is reduced to the v conditions corresponding to the generators of this algebra. Furthermore, it is also known that the condition (iii) may be replaced by a similar condition for invariant integral operators.) These being said, the theorem of Harish-Chandra asserts that the set of all zonal spherical functions on G relative to U is identified with the quotient space YNC/W. The proof for this consists essentially in showing that, U, A, N being taken suitably so that we have the decompositions (.) G=U.A.U = U.AN, the Fourier transformation with respect to zonal spherical functions (parametrized by Y| gives actually an isomorphism from the algebra of the invariant differential operators onto that of the W-invariant polynomial functions on Y| The first difficulty, in translating these results from real to p-adic, arises from the 230 ICHIRO SATAKE subgroup U, can be canonically identified with a quotient space of the form C'/W, where v denotes the " rank" of G, i.e. the dimension of a maximal vector part of a Cartan subgroup of G and W the restricted Weyl group of G. Recently the theory has been extended to the case of some classical groups over p-adic fields by Mautner [x7] , Tamagawa [23] and Bruhat [4], [5]- The main purpose of this paper is to show that the principal part of the theory, including the above-mentioned theorem of Harish- Chandra, holds for a wider class of reductive algebraic groups over p-adic fields, containing all simple classical groups without center. To be more precise, let k be a local field, G a Zariski-connected reductive algebraic subgroup of GL(n, k), A a maximal k-trivial torus in G and N a maximal k-closed unipotent subgroup of G, normalized by A (G, A, N, ... being understood as to represent the groups of k-rational points); the pair (A, N) is then unique up to inner automorphisms of G. Put dim A = v and denote by A" the unique maximal compact subgroup of A. Then the restricted Weyl group of G relative to A, W= N(A)/Z(A), operates in a natural way on A as a group of automorphisms, and hence also on the character group (in the algebraic sense) of A, Y=X(A)(~Z~), and on the group of quasi-characters (in the topological sense) of A/A", Hom (A/A", C*)=Y| Now let k=R or C, and let U be a maximal compact subgroup of G; the quotient space S = U\G is then the associated symmetric space. A C~~ co on G is called a zonal spherical function (or elementary spherical function) on G relative to U, if it satisfies the following conditions (i) r = co(g) for all g~G, u, u'~U, (ii) ~(I) = I, (iii) co, considered as a function on S, is an eigen-function for all invariant differential operators on S. As is well-known, the algebra (over C) of all invariant differential operators on S is canonically isomorphic to the algebra of all W-invariant polynomial functions on the dual of the Lie algebra of A, or, what is the same, on Y| so that it is actually a polynomial algebra of v variables over t21. (Thus the condition (iii) is reduced to the v conditions corresponding to the generators of this algebra. Furthermore, it is also known that the condition (iii) may be replaced by a similar condition for invariant integral operators.) These being said, the theorem of Harish-Chandra asserts that the set of all zonal spherical functions on G relative to U is identified with the quotient space YNC/W. The proof for this consists essentially in showing that, U, A, N being taken suitably so that we have the decompositions (.) G=U.A.U = U.AN, the Fourier transformation with respect to zonal spherical functions (parametrized by Y| gives actually an isomorphism from the algebra of the invariant differential operators onto that of the W-invariant polynomial functions on Y| The first difficulty, in translating these results from real to p-adic, arises from the 230 THEORY OF SPHERICAL FUNCTIONS difference of the nature of maximal compact subgroups. Whereas they are " algebraic " and mutually conjugate in the real case, they have no more such a property in the p-adic case. As a matter of fact, we have to say that, for the time being, our knowledge on this subject is still very poor. Therefore, in this paper, we will make certain assumptions on G, assuring the existence of a favorable maximal compact subgroup U (the assumptions (I), (II) in w 3). These assumptions are nothing but p-adic analogues of some well-known properties of semi-simple Lie groups; in particular, the condition (I) implies the possibility of a decomposition like (.), but A Should now be replaced by a bigger subgroup H, such that AcHcZ(A), and W-invariant. On the other hand, as we shall show in Chapter III, these conditions are satisfied by " all " known examples of classical groups, by virtue of the theory of elementary divisors. Thus one may hope to find a unified proof for these assumptions. Now, under these assumptions, let ~(G, U) denote the algebra (over C) of all invariant integral operators on U\G, whose kernel is given by a function on G with compact carrier. One defines a zonal spherical function as a function on G satisfying (i), (ii) and the condition (iii) stated in terms of ~(G, U) ; then a zonal spherical function determines a homomorphism (of algebras over C) from ~f'(G, U) onto C and vice versa. On the other hand, call H u the unique maximal compact subgroup of H and put M-=H/HU(~-Z~). Then our main theorem (Th. 3 in w 6) asserts that ~(G, U) is isomorphic to the algebra of i, ll W-invariant polynomial functions on Horn(M, C*)(=~ Cv), allowing this time negative powers in an obvious sense; thus ~4'(G, U) is an affine algebra of (algebraic) dimension v over C. From this follows immediately the analogue of the theorem of Harish-Chandra asserting that the totality of zonal spherical func- tions on G relative to U is canonically identified with a quotient space of the form (C*)'[W (Th. 2 in w 5). As examples, it will be shown that, in case G is a simple classical group without center and U a maximal compact subgroup of G defined by a " maximal lattice ", the algebra ~q~(G, U) is actually a polynomial algebra of v variables over C (Th. 7, 9 in w 8, 9)- More precisely, the so-called (local) " Hecke ring " .W(G, U)z is a polynomial ring of v variables over Z. These theorems are proved by the usual method of Fourier transformation and, in fact, rather simply, compared with the real case. But to determine the explicit form of zonal spherical functions and the Plancherel measure, it seems necessary to know the (infinite) matrix of this Fourier transformation more explicitly. This has been done by Mautner [i7] for PL(2, k), but is still an open problem for the general case. As a partial result in this direction, we will calculate in Appendix I (local) Hecke series and especially ~-functions attached to GL(n, R), where 2R is a central division algebra over k, and to the group of symplectic similitudes. Besides these, we will analyze in w 7 the behavior of zonal spherical functions under a homomorphism ~ : G-+G', and especially under a k-isogeny (Th. 4)- Here we do not assume a priori the conditions (I), (II) on G, G' in full, and will see how (parts of) tile conditions on the one of G, G' imply the corresponding conditions on the 231 ICHIRO SATAKE other. As another application, we will determine in Appendix II all zonal spherical functions of positive type, or, what amounts to the same, all unitary equivalence classes of irreducible unitary representations of the first kind, of PL(2, ~), obtaining again a result quite analogous to the real case. A part of results of this paper (N O 7-3) has been announced in a short note [xg] , which will also serve as an introduction to this paper. The author has much profited by seminars on Spherical functions, organized during the period of 196o-62, by Professor Y. Akizuki, to whom this paper is dedicated with sincere gratitude and respects. Notations and Conventions. Throughout this paper, k denotes a p-adic number field, i.e. a finite extension of the p-adic number field O p. The valuation-ring in k and its (unique) prime ideal are denoted by o, p = (r~), respectively, r~ denoting a prime element. We denote by I Ip (or simply by I ]) the normalized valuation of k, i.e. ]~[p=q-Or%~ for ~ak, q denoting the number of elements in the residue class field o/p. All algebraic groups we consider are supposed to be a.fine, so that they are realized as groups of matrices. Thanks to a result of Rosenlicht (Annali di Matematica, vol. 43 (I957), P. 44), in any such group, defined over k, the subgroup formed of k-rational points is everywhere dense in the sense of the Zarisld topology, provided all the connected components contain a k-rational point. Hence, in this paper, we will understand by an " algebraic group over k " the group formed of k-rational points of an algebraic group (in the sense of algebraic geometry) defined over k. If G is a (Zarisld-) connected algebraic group over k and if K is an overfield of k, the group formed of K-rational points in the same algebraic group will be denoted by G K. In an algebraic group G over k, one can consider two kinds of topologies, i.e. the p-adic topology and the Zariski topology. Without any specific reference, the words " closed" (or" k-closed")," connected" will be used exclusively in the sense of the Zariski topology, while the words " open ", " compact " will always be understood in the sense of the p-adic topology. " Closure" (in the sense of the Zaristd topology) of M is denoted by cl(M). When G, G' are algebraic groups over k and ~ a k-morphism (i.e. a rational homomorphism defined over k) from G into G', the symbols Im q0 = ~ (G), Ker qo = ~- 1 ( I ) are used in the set-theoretical sense; thus V-I(I) is k-closed but q0 (G) is not, and (in case G is connected) the image of q~ in the algebraic sense is cl(q0(G)). (More generally, the similar convention is made for any rational map defined over k.) In particular, for a k-closed normal subgroup H of G, we denote (by abuse of notation) the factor group in the algebraic sense by el(G/H). As usual, for any ring R with the unit element I, R* stands for the multiplicative group of regular elements in R, and M~(R) for the ring of all n � n matrices with coefficients in R. The unit matrix of degree n is denoted by i n (or simply by i). 232 THEORY OF SPHERICAL FUNCTIONS For Xi~Mn~(R ) (I<<.i<~r), the symbol diag. (X1, ..., Xr) will represent a matrix of degree n=~n i of the following form: o "Xr/" Z, Q, R, C denote, respectively, the ring of rational integers, the rational number field, the real number field and the complex number field; the real and imaginary parts of s~C are denoted by Re s, Im s, respectively. For a finite set M, the symbol # M represents the number of elements in M. For a map ? defined on a set M and for a subset M~ of M, the symbol q~nM1 stands for the restriction of ~ to M1.

Journal

Publications mathématiques de l'IHÉSSpringer Journals

Published: Aug 7, 2007

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