Proceedings of Symposia in
PuRE
MATHEMATics
Volume 33, Part 1
Automorphic Forms, Representations, and L-functions A. Borel W. Casselman
PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY
HELD AT OREGON STATE UNIVERSITY CORVALLIS, OREGON JULY II-AUGUST 5, 1977
Prepared by the American Mathematical Society with partial support from National Science Foundation Grant MCS 76-24539
Library of Congress Cataloging-in-Publication Data Symposium in Pure Mathematics, Oregon State University, 1977. Automorphic forms, representations, and £-functions. (Proceedings of symposia in pure mathematics; v. 33) Includes bibliographical references and index. 1. Automorphic forms - Congresses. groups- Congresses. 1941-
2. Lie groups - Congresses.
4. £-functions- Congresses.
III. American Mathematical Society.
QA33l.S937
5121.7
1977
3. Representations of
I. Borel, Armand.
IV. Title.
V. Series.
II. Casselman, W.,
78-21184
ISBN 0-8218-1435-4 (v.l) ISBN 0-8218-1437-0 (v.2)
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02 01 00 99 98 97
CONTENTS Foreword
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Part I.
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ix
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I
Reductive groups. Representations
Reductive groups . 3 By T. A. SPRINGER Reductive groups over local fields 29 By J. TITS Representations of reductive Lie groups 71 By NOLAN R. w ALLACH Representations of GL2(R) and GL2(C) ..................................87 By A. w. KNAPP Normalizing factors, tempered representations, and L-groups . 93 By A. W. KNAPP and GREGG ZucKERMAN Orbital integrals for GL2(R) . . 107 By D. SHELSTAD Representations of p-adic groups: A survey . . . Ill By P. CARTIER Cuspida! unramified series for central simple algebras over local fields . 157 By PAUL GE:RARDIN Some remarks on the supercuspidal representations of p-adic semisimple groups . . . .. . . . 171 By G. LUSZTIG .
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Automorphic forms and representations
II.
Decomposition of representations into tensor products By D. FLATH Classical and adelic automorphic forms. An introduction . By l. PIATETSK!-SHAPIRO Automorphic forms and automorphic representations . By A. BoREL and H. JACQUET On the notion of an automorphic representation. A supplement to the preceding paper . . . . . By R. P. LANGLANDS Multiplicity one theorems . . By I. PIATETSKI-SHAPIRO Forms of G L(2) from the analytic point of view By STEPHEN GELBART and HERVE JACQUET Eisenstein series and the trace formula By JAMES ARTHUR 0-series and invariant theory . . . .. .. By R. HOWE .
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. 185
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189
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203
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213
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253
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275
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v
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CONTENTS
VI
Examples of dual reductive pairs . . 287 By STEPHEN GELBART On a relation between SL2 cusp forms and automorphic forms on orthogonal groups 297 By S. RALLIS A counterexample to the "generalized Ramanujan conjecture" for (quasi-) split groups . . . . . 315 By R . HowE and 1.1. PIATETSKI-SHAPIRO .
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Part 2 III. Automorphic representations and £-functions
Number theoretic background .. . . . . 3 By J. TATE Automorphic £-functions . . .. . . . . . . 27 By A. BOREL Principal £-functions of the linear group .... . . . . . ... . . . 63 By HERVE JACQUET Automorphic £-functions for the symplectic group GSp4 .. .. . ..87 By MARK E. NOVODVORSKY On liftings of holomorphic cusp forms . . . . .. 97 By TAKURO S H INTANI Orbital integrals and base change . . . .. Ill By R. KOTTWITZ The solution of a base change problem for GL(2) (following Langlands. Saito, Shintani) . . . .. . .. . . . . . 115 By P. GERARDIN and J. -P. LABESSE 1 35 Report on the local Langlands conjecture for G L 2 By J. TUNNELL .
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Arithmetical algebraic geometry and automorphic £-functions
The Hasse-Weil �-function of some mod uli varieties of dimension greater than one . . .. .. . . .. . . . . 141 B y w . CASSELMAN Points on Shi mura varieties mod p . . .. .. . . . 1 65 By J. S. MILNE Combinatorics and Shi mura varieties mod p (based on lectures by Langlands) . . . . . . 185 By R. KOTTWITZ Notes on L-ind istingu ishabi lity (based on a lecture by R. P. Langlands) ....193 By D. SHELSTAD Automorphic representations, Shimura varieties. and motives. Ein Marchen . . . . .. . . . . 205 By R. P. LANGLANDS .
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CONTENTS
vii
Varietes de Shimura : Interpretation modulaire, et techniques de construction de modeles canoniques . 247 By PIERRE DELIGNE Congruence relations and Shimura curves . . . ... 29 1 By YASUTAKA IHARA Valeurs de fonctions L et periodes d'integrales . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1 3 By P. DELIGNE with an appendix : Algebraicity of some products of values of the F function . . . . . . . . . . . . . . . . . 343 By N. KOBLITZ and A. 0GUS An introduction to Drinfeld's "Shtuka" . . . 347 By D. A. KAZHDAN Automorphic forms on GL2 over function fields (after V. G. Drinfeld) . . . . . . 357 By G. HARDER and D. A. KAZHDAN .
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Index Author Index
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38 1
Foreword
The twenty-fifth AMS Summer Research Institute was devoted to automorphic forms, representations and L-functions. It was held at Oregon State University, Corvallis, from July 1 1 to August 5, 1 977, and was financed by a grant from the National Science Foundation. The Organizing Committee consisted of A. Borel, W. Casselman (cochairmen:), P. Deligne, H. Jacquet, R. P. Langlands, and J. Tate. The papers in this volume consist of the Notes of the Institute, mostly in revised form, and of a few papers written later. A main goal of the Institute was the discussion of the L-functions attached to automorphic forms on, or automorphic representations of, reductive groups, the local and global problems pertaining to them, and of their relations with the L functions of algebraic number theory and algebraic geometry, such as Artin L functions and Hasse-Wei! zeta functions. This broad topic, which goes back to E. Heeke, C. L. Siegel and others, has undergone in the last few years and is undergo ing even now a considerable development, in part through the systematic use of infinite dimensional representations, in the framework of adelic groups. This devel opment draws on techniques from several areas, some of rather difficult access. Therefore, besides seminars and lectures on recent and current work and open problems, the Institute also featured lectures (and even series of lectures) of a more introductory character, including background material on reductive groups, their representations, number theory, as well as an extensive treatment of some relatively simple cases. The papers in this volume are divided into four main sections, reflecting to some extent the nature of the prerequisites. I is devoted to the structure of reductive groups and infinite dimensional representations of reductive groups over local fields. Five of the papers supply some basic background material, while the others are concerned with recent developments. II is concerned with automorphic forms and automorphic representations, with emphasis on the analytic theory. The first four papers discuss some basic facts and definitions pertaining to tho se, and the passage from one to the other. Two papers are devoted to Eisenstein series and the trace formula, first for GL 2 and there in more general cases. In fact, the trace formula and orbital integrals turned out to be recurrent themes for the whole Institute and are featured in several papers in the other sections as well. The main theme of the last four papers is the restriction of the oscillator representation of the metaplectic group to dual reductive pairs of subgroups, first in general and then in more special cases. III begins with the background material on number theory, chiefly on Wei! groups and their L-functions. It then turns to the L-functions attached to automor ph ic representations, various ways to construct them. their (conjectured or proven) properties and local and global problems pertaining to them . The remaining papers are mostly devoted to the base change problem for GL2 and its applications to the proof of holomorphy of certain nonabelian Artin series. Finally. IV relates automorphic representations and arithmetical algebraic geometry. Over function fields. it gives an introduction to the work of Drinfeld for ix
X
FOREWORD
G L , which constructs systems of /-adic representations whose L-series is a given 2 automorphic L-function. Over number fields, it is mainly concerned with problems on Shimura varieties: canonical models, the point of their reductions modulo prime ideals, and Hasse-Weil zeta functions. This Institute emphasized representations so that, at least formally, the primary object of concern was an automorphic representation rather than an automorphic form. However, there is no substantial difference between the two, and this should not hide the fact that the theory is a direct outgrowth of the classical theory of automorphic forms. In order to give a comprehensive treatment of our subject matter and yet not produce too heavy a schedule, it was decided to omit a number of topics on automorphic forms which do not fit well at present into the chosen framework. For example, the Institute was planned to have little overlap with the Conference on Modular Functions of One Variable held in Bonn (1976). The reader is referred to the Proceedings of the latter (Springer Lecture Notes 601, 627) and to those of its predecessor (Springer Lecture Notes 320, 350, 476) for some of those topics and a more classical point of view. Also, some topics of considerable interest in themselves such as reductive groups, their infinite dimensional representations, or moduli varieties, were discussed chiefly in function of the needs of the main themes of the Institute. These Proceedings appear in two parts. the first one contains sections 1 and II, and the second one sections III and IV. A.
BoREL
W. CASSELMAN
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part I, pp. 3-27
REDUCTIVE GROUPS T. A.
SPRINGER
This contribution contains a review of the theory of reductive groups. Some knowledge of the theory of linear algebraic groups is assumed, to the extent covered in § § 1 -5 of Borel ' s report [2] in the 1 965 Boulder conference. § § I and 2 contain a discussion of notion of the "root datum" of a reductive group. This is quite important for the theory of L-groups. Since the relevant results are not too easily accessible in the literature (they are dealt with, in a more general context, in the latter part of the Grothendieck-Demazure seminar [17]), it is shown how one can deduce these results from the theory of semisimple groups (which is well covered in the literature). In §§3 and 4 we review facts about the relative theory of reductive groups. There is more overlap with [2, §6], which deals with the same material. §5 contains a discussion of a useful class of Lie groups (the "selfadjoint" ones). We indicate how the familiar properties of these groups can be established, assum ing the algebraic theory of reductive groups. I am grateful to A. Borel for valuable suggestions and to J. J. Duistermaat for comments on the material of §5. 1. Root data and root systems. The notion of root datum (introduced in [17, Expose XXI] under the name of "donnee radicielle") is a slight generalization of the notion of root system, which is quite useful for the theory of reductive groups. Below is a brief discussion of root data. For more details see [loc. cit.]. For the theory of root systems we refer to [7] . 1.1. Root data. A root datum is a quadruple 1/f = (X, r!J, xv, rJJV) where : X and xv are free abelian groups of finite type, in duality by a pairing X x xv --+ Z denoted by ( , ) , r/J and rpv are finite subsets of X and xv and there is a bijection a ...-. aV of r/J onto rpv, If a E r/J define endomorphisms sa and Sav of X, xv, respectively, by
Then the following two axioms are imposed: (RD I) For all a E r/J we have (a, aV ) = 2; ( RD 2 ) For all a E r/J we have sa(r!J) c: r/J, Sav (r!JV) c: rpv, It follows from (RD I ) that s� = id, su (a) = - a (and similarly for sav) . It is clear from the definition of a root datum that if 'IJ! = (X, r!J, xv, r!JV) is one, then 'I]I'V = (XV, rpv, X, r!J) is also one, the dual of 1/f. Let 'If! be as above. Let Q be the subgroup of X generated by r/J and denote by X0 AMS (MOS) subject classifications ( 1 970). Primary 20G I 5 , 22E l 5 . © 1979, American Mathematical Society
3
T. A. SPRINGER
4
the subgroup of X orthogonal to (/Jx. Put V = Q ® Q, V0 = X0 ® Q. Define sim ilarly subgroups Q v, X'i{ of xv and vector spaces vv, Vi/. We say that lJ! is semisimple if X0 = {0} and toral if(/) is empty.
1 .2. LEMMA. Q n Xo p:
=
{0} and Q + Xo hasfinite index in X.
This is contained in [17]. We sketch a proof. Define a homomorphism X-+ xv by p(x) = I; <x, aV)av aEI/J
Ker p. Since <x, p(x)) = I:aEI/J <x, aV)2 we have X0 Next observe that if aE (/) we have p(a ) = 1-
=
This shows that p ® id is a surjection V-+ vv, whence dim vv � dim V. By sym metry we then have dim V = dim vv, whence Q n Ker p = {0}. The assertion now follows readily. 1 . 3 . Root systems. It follows from the proof of 1 .2 that vv can be identified with the dual of the vector space V. We write again< , ) for the pairing. Also identify (/) with (/) ® I c V and assume that (/) =1- 0. We then see that (/) is a root system in V in the sense of [7]. Recall that this means that the following conditions are satisfied: (RS I) (/) is finite and generates V, moreover 0 �(/) ; (RS2) for al l aE (/) there is aVE yv such that
=
X + ( <x, 1Sa(f3Y)) - <x, S a ( f3) V)) sa(f3) ,
where 1sa is the transpose of sa. Since <sa(f3) 1sa(f3Y)) - <sa(f3), sa(f3)V)
=
REDUCTIVE GROUPS
5
1 . 5 . Properties of root systems. Let f/J c V be a root system. Proofs of the pro perties reviewed below can be found in [7]. (a) If exEf/Jand.AexEf/Jthen ).= ±1 , ±t, ±2. The root system f/J is called reduced if for all exEf/Jthe only multiples of ex in f/J are ±a. To every root system, there belong two reduced root systems, obtained by removing for every exEf/J the longer (or shorter) multiples of ex. (b) f/J is the direct sum of root systems f/J' c V' and f/J" c V" if V= V' $ V" and f/J= f/J' U f/J". A root system i s irreducible i f it is not the direct sum of two subsystems. Every root system is a direct sum of irreducible ones. (c) The only reduced irreducible root systems are the usual ones: An (n � 1 ), Bn (n � 2 ), en (n � 3), Dn (n � 4 ), Es, E7, Es, F4, Gz. (d) For each dimension n there exists one irreducible nonreduced root system, denoted by Ben (see below). ExAMPLES. (i) Bn. Take V = Qn with standard basis { e l > · · · , en}. Then vv = n Q , with dual basis {ei', . .. , ei{}. We have Bn = { ±e;±ei (i < j) and ±e; (1 � i, j � n)}. If ex= ±e;±ei then exv = ±e('±eY; if ex= ±e; then exv = ±2e;. The Weyl group W(Bn) consists of the linear transformations which permute the coordinates and change their signs in all possible ways. The exVE vv form an irreducible root system of type en. We have ween) � W(Bn). (ii) With the same notations we have Ben= { ±e;±ei (i < j), ±e;, and ±2e; (1 � i, j � n)}. Then W(Ben) = W(Bn). 1 .6. Weyl chambers. Let f/J c V be a root system. We now view it as a subset of VR = V ®" R. A hyperplane H of VR is singular if it is orthogonal to an exv. A Weyl chamber C in VR i s a connected component of the complement of the union of the singular hyperplanes. To a Weyl chamber one associates an ordering of the roots : ex > 0¢> (x, exv) > 0 for all x E C. ex Ef/J is simple (for this ordering) if it is not the sum of two positive roots. The set of simple roots L1 is called a basis of f/J. We have the following properties. (a) The Weyl group W(f/J) acts simply transitively on the set of Weyl chambers. (b) The sa (exEL1) generate W(f/J) . More precisely, ( W, (sa)aELI) is a Coxeter system (see [7]) . (c) Every root is an integral linear combination of simple roots, with coefficients all of the same sign . (d) Say that L1 is connected if it cannot be written as a disjoint union L1= L1' u L1", where (L1' + L1") n L1 = 0. Then we have: f/J is irreducible¢> L1 is connected . A connected L1 leads to a connected Dynkin graph. These are described in [7]. 1.7. We collect a few facts about root data to be needed later. First, there is the notion of direct sum of root data. This is clear and we skip the definition. Next we have to say something about morphisms of root data. The following suffices. For more general cases see [7] ; see also 2.ll(ii) and 2. 1 2. Let 1/f= (X, f/J, xv, r!JV) and 1/f' (X', f/J' , (X')V, (f/J')V) be two root data. A homomorphism/: X' --+ X is called an isogeny ofl/!' into 1/f if : (a)/i s injective and I m f has finite index i n X, =
T.
6
A. SPRINGER
(b) .f induces a bijection of C/J' onto C/J and its transpose �{"induces a bijection of rpv onto (C/J')v. Notice that then if is also an injection xv � (X')V with finite cokernel. Also, coker f and coker(�{) are in duality. EXAMPLE. Given If!, we shall construct a If!' and an isogeny of X into X', which we shall call the canonical isogeny associated to If!. If L is a subgroup of X we denote by L the largest subgroup containing L such that Lj Lis finite. Then L = L if and only if Lis a direct summand. Let X0 and Q be as in 1.1. By 1 . 2 we can view (/J as a subset of X/ X0. It follows that lf!i = (XI X0, C/J, Qv , (/JV) is a semisimple root datum. Likewise, If!� = (X/Q, 0, X0, 0) is a toral root datum. Put If!' = If!; EB If!�. Then the canonical isogeny f: X � (XJX0) EB (X/Q) is the canonical homomorphism of X into the right-hand side. 1.8. Let (/J be a root system in the Q-vector space V. The av (aE C/J) in the dual V"' of V also form a root system rpv (the dual or inverse root system). There are finitely many semisimple root data (X, (/J, xv, rpv) where Xc V, xv c vv. In fact, let Q and QV be the lattices in V and VV generated by (/J and rpv, respectively, and define P {xE Vl(x, aV)E Zfor all aE C/J } . pvc vv is defined similarly. Then Q c P and P/Q is a finite group, in duality with pv/Qv. An X as above is then contained between P and Q and for each such X there is a unique xv between pv and Qv such that (X, C/J, xv, rpv) is a root datum. 1.9. let vr = (X, (/J, xv. (/JV) be a root datum. Assume that its root system (/J c v is reduced. Let L1 be a basis of C/J. Then JV = {aVIaE .J} is a basis of the dual root system rpv c vv. We caJl based root datum a sextuple lf!0 = (X, (/J, .J, xv, rpv, JV), where (X, C/J, xv, (/JV) is a root datum with reduced root system C/J and where J is a basis of C/J. However, since J and .JV determine C/J and C/J'' uniquely, it also makes sense to view a based root system as a quadruple 1]!0 = (X, J, xv, JV). This we shall do. =
2. Reductive groups (absolute theory).
2.1 . Let G be a connected reductive linear algebraic group. In this section we consider the absolute case, where fields of definition do not come in. So we can view G as a subgroup of some GL(n, Q), Q an algebraically closed field (see [2]). LetS be a subtorus of G. We define the root system C/J(G, S) of G with respect toS to be the set of nontrivial characters of S which appear when one diagonalizes the represen tation of Sin the Lie algebra g of G, S operating via the adjoint representation. 2.2. The root datum of G. Fix a maximal torus of G. We shall associate to the pair (G, T) a root datum cjJ(G, T) = (X, C/J, xv, rpv) (also denoted by y(G)) . X is the group of rational characters X* (T ) of T. This is a free abelian group of finite rank. xv is the group X*(T) of !-parameter multiplicative subgroups ofT, i.e., the group of homomorphisms (of algebraic groups) GL1 � T. Then xv can be put in duality with X by a pairing ( , ) defined as follows: if x E X*(T), uE X* ( T), then x(u(t)) = t<x, u> (tE Q*). We take (/J = C/J(G, T), the root system of G with respect to T. To complete the definition we have to describe rpv_ If aE C/J, let Trx be the identity component of the kernel of a. This is a subtorus of codimension I. The centralizer Zrx of Trx in G is a connected reductive group with maximal torus T, whose derived group Grx is semi simple of rank 1 , i.e., is isomorphic to either SL(2) or PSL(2). There is a unique
T
REDUCTIVE GROUPS
7
homomorphism av: GL1--> Ga such that T = (Im aV)Ta , (a, aV) = 2. These aV make up (/JV. The axiom (RD I ) is built into the definition of av. We use 1 .4 to establish (RD2). Let na EGa - Ta normalize Ta. Then n� E Ta and, sa being as in 1 .1, we have, for XEX, tET, In fact, working in Ga one shows that there is u E xv such that the left-hand side equals tx-<x,u>a. One then shows that (a, u) = 2 and that (x, u ) = 0 if (x , av) = 0. It follows that (RD2') holds. So C/J(G) is a root datum. The root system (/J is reduced (for all these facts see [3] or [14]). 2. 3 . To each a E (/J there is associated a unique homomorphism of algebraic groups Xa : Ga --> Ga such that
(t E T, u E 0).
Put Ua = Im(xa) and let Xa E 9 be a nonzero tangent vector to Ua. Then 9 = Lie(T) E9 2: OX a. aE�
Let B be a Borel subgroup containing T. There is a unique ordering of (/J (as in 1 .6) such that B is generated by T and the Ua with a > 0, and any B ::J T is so ob tained. It follows that we can associate to the triple (G, B, T) a based root system cp0(G, B, T) = (X''(T), Ll, X*(T), LJV) (or cp0(G)) , where L1 is the basis of (/J deter mined by the ordering associated to B. 2.4. lsogenies. An isogeny ¢: G --> G' of algebraic groups is a surjective rational homomorphism with finite kernel. ExAMPLES. (i) The canonical homomorphism SL(2) --. PSL(2) {PSL(2) is to be viewed as the group of linear transformations of the space of 2 x 2-matrices of the form x ,__. gxg-1, whereg E SL(2)). If char 0 = 2 this is an i somorphism of abstract groups, but not of algebraic groups. (ii) Let G be defined over the finite field Fq. The Frobenius i sogeny G --> G raises all coordinates to the qth power. It is again an isomorphism of groups, but not of algebraic groups. let G and G' be connected reductive and let T be a maximal torus of G. central isogeny ¢ : G--> G' i s an i sogeny which (with the notations of 2.3) induces an iso morphism in the sense of algebraic groups of Ua onto its i mage, for all a E C/J. Equivalently, drp(Xa) i= 0 for all a E (/J (where d¢ i s the induced Lie algebra ho momorphism). The i mage T' = rp(T) is a maximal torus of G'. We shall say that ¢ is a central isogeny of (G, T) onto (G', Let/(¢) be the homomorphism X*(T') --> X*(T) defined by ¢.
A
T' ) .
2. 5 . PROPOSITION. (i) If¢ is a central isogeny then f(¢) is an isogeny of cp(G', T') into cp(G, T) ; (ii) if¢ and¢' are central isogenies of (G, T) onto (G', such that/(¢) = /(¢/) then there is t E T with ¢' = ¢ Int(t). a
1.7
T' )
That/(¢) has property (a) of i s equivalent to the fact that ¢ induces a sur jection T --. T' with finite kernel. There is a bijection a ,__. a' of root systems such
8
T. A. SPRINGER
that ifJ( Ua) = Ua' or that difJ(Xa) Xa' (choosing Xa' properly). We then have Ad(if;(t))Xa' a(t)Xa'• whence f(rp)(a') = a. Then tj(rp)(a') (a')v, as follows for example from the equality = Sav established in the proof of I.8. This proves (i). Let L1 be a basis of I (i .e., such that p is an isomorphism of groups). Lett be the Lie algebra of we have Ker(difJ) c t. Now t can be identified with xv ®z Q. It follows that Ker(difJ) is isomorphic to the kernel of .f(ifJ)V ® id : xv ®z Q--> (X')V ®z Q, which is isomorphic to ((X')Vj p(X')V + Im.f(ifJ)V) ® FpQ. Hence Ker(drp) = 0 if and only if Coker{!(¢;)) (which is dual to Coker(tj(rp))) has order prime to p. If p > then Ker(drp) is a central restricted subalgebra of g, which is stable under Ad(G). Let GfKer(difJ) be the quotient of G by Ker difJ (see [3, p. 376].) It follows that we can factor if;, G.!.... G/Ker (drp) __:.., G', where a is the canonical (central) isogeny of [loc. cit.]. These remarks imply that we can factorize if; as follows : =
=
=
1 sa
TT
T--> T', T)
T').p
T.
=
p T;
I
where if; = n,_1 = G;/Ker(drp;) Also, if
o
• . .
o
n0 . Put if;;
(I � i � s
-
=
1l:s-I
.
o
. on; (i � I). Then G1 •
=
I) and the 11:; are canonical isogenies.
G/Ker if;, G;H
G _.i___. G '
1
G
_.i___.
1
G'
is a commutative diagram of isogenies, we can arrange the factorizations of if; and ;p such that there is a diagram with commuting squares (2)
G
=
G0----+ G1
----+
..
G
=
Go ----+ G,
----+
..·
1
1
·
----+
G,_,
----+
G.
=
G'
----+
G._,
----+
G.
=
G'
1
1
Notice that the vertical arrows are uniquely determined once the first one is given.
Let andthat be central isogeniesThen(G' of (G, T), T'onto) and(Gi, (G', T')T{)andare(Giisomor , Ti), respecti v el y . Assume phic. 2.7. LEMMA.
if;
¢;1 Imf(rp)
=
Im/(¢;1).
REDUCTIVE GROUPS
9
Let G" c: G' x G� be the image of the homomorphism g ,_. (tjJ(g) , tjJ1(g) ). Let c): G -+ G" be the induced homomorphism . Then ¢ is a central isogeny of con nected reductive groups and so are the projections :n:1: G" -+ G', :n:2: G "-+ G�. A straightforward check (working in tori) shows that Ker :n:; = {e} , Ker d:n:; = {0} (i = l. 2). Hence :n:1 and :n:2 are isomorphisms, whence the lemma. 2 . 8 . LEMMA. Let If!' be a root datum and let fbe an isogeny oflf!' into <jJ(G, T). Then there exist a pair (G', T') and a central isogeny tjJ of (G, T) onto (G', T') such If' = <jJ(G', T'), f(tjJ) = f
that
From the knowledge of fwe can recover successively the groups G; figuring in
(I). This allows one to define G'. We omit the details.
2.9. THEOREM. (i) For an.v root datum If! with reduced root system there exist connected reducti1•e group G and a maximal torus Tin G such that If! = <jJ(G, T). The pair (G, T) is unique up to isomorphism ; (ii) let lJT= ¢( G, T), If!' = cj;(G', T'). Iffis an isogeny of If!' into If! there exists a central isogeny tjJ of (G, T) onto (G', T') with f(t/J) = . Two such tjJ differ by an automorphism Int(t) (t E T) of G.
a
f
1.7.
Let fbe the canonical isogeny lJT-+ lJT' of Using 2 . 8 we see that it suffices to prove the existence statement of (i) for the two cases that lJT is semisimple or toral. The second case ((/J= tjJ) is easily dealt with : take for G the torus T= Hom(X, Q*) . In the semisimple case the statement follows from the existence theorem of the theory of semisimple groups which can be dealt with using the theory of Chevalley groups. (See [18] or [6, part A]. The uniqueness statement of (i) is part of (ii). To prove (ii) one first reduces to the case that f is an isomorphism (using 2.7 and 2.8).) In the case that G is semisimple the statement of (ii) is Chevalley's fun damental isomorphism theorem, proved, e.g., in [10, Expose 24], or in [14, Chapter XI]. The case of a torus G is easy. In the general case, there are central i sogenies G1 x S-+ G, Gi. x S' -+ G', where G1 and Gi. are the derived groups of G and G', and where S and S' are tori , such that the corresponding isogenies of root data are just the canonical ones of 1.7 (see also 2.1 5). Now /defines an i sogeny/1 of cj;(Gi. x S') into cj;( G x S) and we may assume that there exists a central i sogeny ¢1 : G1 x S -+ G� x S' with f(t/J1) = A We can then complete the diagram like (2), with G1 x S, Gi. x S', G, G' instead of G, G, G', G', respectively, and with ¢1 as first arrow. The right-hand arrow, which i s uniquely determined by t/J1o i s then the required i somorphism. The last point of (ii) follows from 2 . 5 (ii). 2.10. REMARKS. (i) In the semisimple case the existence statement of 2.9(i) is due to Chevalley [Seminaire Bourbaki, Expose 2 1 9, I 960--196 1 ]. He constructs a group scheme G0 over Z such that G = G0 x z k. This construction is also discussed in [6, p art A] . A generalization of 2.9, where the field k is replaced by a base scheme, is con tained in [17, Expose XXIV]. (ii) The result on the existence of central isogenies of red uctive groups contained in 2.9(ii) is a special case of one on arbitrary isogenies, which we shall briefly indi cate. Let tjJ: G -+ G' be an isogeny of connected reductive groups with tjJ(T) = T'. Let / be the induced homomorphism X*(T') E X*(T). Let Xa, Xa' (a E (/J, a' E (/J')
T.
10
A. SPRINGER
be as in 2.3. One shows that there is a bijection a >-+ a' of f/J onto f/J ' and a function q : f/J ..... {pnln EN} (p the characteristic exponent) such that the Xa and Xa' can be so normalized that cj> (xa( t)) = xa-(tqis a central i sogeny then all q(a) are 1 . If¢> is the Frobenius isogeny of 2.4 then fis multiplication by q and all q(a) are q. Such anf is called ap- morphism in [17, Expose XXI]. The analogue in question of 2. 9(ii) is obtained by assuming f to be a p-morphism and admitting in the conclusion an arbitrary i sogeny cj>. The proof can be given along similar lines, reducing to the case of an isomorphism . For semisimple G the result is due to Chevalley [10, Expose 23]. EXAMPLE OF A p-MORPHISM. p = 2, G is semisimple of type B 2 and, with the notations of the example in 1 . 5, we have f (e1 ) = e1 + e2, f (e2) = e1 - e2• A classification of the possible p-morphisms can be found, e . g . , in [17, Expose XXI, p . 7 1 ] . 2. 1 1 . Let ¢>: G ..... G' b e a homomorphism of connected reductive algebraic groups. Let Tand T ' be maximal tori in G, G' with cj>(T) c T'. Assume that Im ¢>is a normal subgroup of G'. We shall briefly describe the relation between the root data cp(G, T ) = (X, f/J , xv, rpv) and cjJ (G ', T') = ( X ' , f/J , (X')V , (f/J ')V ) . Letf : X'-+ X be the dual of¢>: T ..... T'. In general, f i s neither injective nor surjective. Put f/J 1 = f/J n Im J, f/J 2 = f/J - f/J 1 . Then f/J = f/J 1 U f/J 2 is a decomposition into orthogonal subsets (i.e., (f/Jh f/J>j) = (f/J z , f/J�) = 0). Likewise, i f f/J2 = f/J ' n Ker J, f/Ji = f/J ' - f/J?_ , then f/J ' = rp;_ U f/Jz is a decompo sition into orthogonal subsets. There is a bijection a >-+ a ' of f/J 1 onto f/Ji and a function q : f/J 1 ..... {pnln EN}, such that for a Ef/J 1 we have
f(a') = q(a)a, Moreoverf(a) = 0 if a E f/J2 and tj (a) = 0 if a Ef/J 2• 2. 1 2. It follows readily from 2.9(i) that if W0 is a based root datum with reduced root system there exists a triple (G, B, T) as in 2 . 3 with ¢0(G, B, T) = '1Jf0, which is unique up to isomorphism. There is no canonical isomorphism of one such triple onto another one. In fact, based root systems have nontrivial automorphisms . In this connection the following results should be mentioned. Let ¢0(G, B, T) = ¢0(G). For each a E L1 fix an element ua i= e in the group Ua. The following i s then an easy consequence of 2. 5(ii). 2. 1 3 . PROPOSITION. Aut ¢0(G) is isomorphic to the group Aut(G, B, T , {ua } a<J ) of automorphisms of G which stabil iz e B, T and the set of ua. 2. 1 4. CoROLLARY. There is a spl it exact sequence {1}
_____.
Int(G)-----> Aut(G) _____.Aut ¢0(G )
_____.
{ 1 }.
In fact, an isomorphism as in 2. 1 3 defines a splitting. Any two such splittings differ by an automorphism Int(t) (t E T ). 2. 1 5. Let G be a connected reductive group, with a maximal torus T. Put cjJ(G, T) = (X, f/J , xv, f/JV ).
REDUCTIVE GROUPS
11
We have a decomposition G = G' · S, where G' is the derived group of G (which is semisimple) and where S is a torus, viz. the identity component of the center C of G. We have T = T' · S, where T' is a maximal torus of G'. We use the notations of 1 . 1 and 1 .7. The following facts can be checked without difficulty. (a) rp(G', T') = (XfX0, (/J, QV, (/JV) (we view (/J as a subset of XfX0, as we may by 1.2). (b) The character group Homalg.gr. (C, Q*) is X/Q, and we have X * (S) = X/ Q , xv (S) = X'rf. (c) The isogeny G' x S-+ G defines the canonical isogeny of (G, T) (see 1 .7). This fact was already used in the proof of 2. 5(ii). It follows that G is semisimple if and only if rp(G, T) is semisimple. In that case we say that G is if X = Q and if X = P (notation of 1 .8). From 2.5(ii), using what was said in 1 .8, we see that a semisimple group G is adjoint (resp. simply connected) if and only if a central isogeny ,P: G -+ G' (resp. rp : G' -+ G) is an isomorphism. In the case of a general reductive G we have the following facts. (d) The derived group G' is adjoint- X = Q EB X0- xv = pv EB X(/. (e) G' is simply connected- P c X + (Xo ® Q) - QV = QV. (f) The center of G is connected- Q = Q- pv c xv + (X(/ ® Q).
adjoint
simply connected
3. Reductive groups (relative theory). Here we let a ground field k c Q come into play. We denote by k the algebraic closure of k in Q and by k, its separable closure. A linear algebraic group G which is defined over k will be called a We then denote by G(k) the group of its k-rational points (and not by Gk> as in [2]). If A is a k-algebra, we denote by G(A) the group Homk(k[G], A) (see 2]). 3. 1 . III, § 1 ]. Let G and G' be k-groups. G' is said to be a k-form of G if G and G' are isomorphic over Q. EXAMPLE. k = R. Then U(n) is an R-form of GL(n). To describe k-forms one proceeds as follows. The k-group G is completely de termined by the group G(k,) of k,-rational points. This means the following: if G -+ GL(n) is an isomorphism of G onto a closed subgroup of GL(n), everything being defined over k, then the subgroup G(k,) of GL(n, k,) determines G, up to k-iso morphism. The fact that G is defined over k is reflected in an action of the Galois group rk = Gal(k,/k) on G(k,). The k-forms G' of G can be described as follows (up to k-isomorphism). We have G'(k,) = G(k,) and there is a continuous func tion >-+ of rk to the group of k,-automorphisms of G (the Galois group being provided with the Krull topology and the second group with the discrete topol ogy), satisfying
Forms of algebraic groups [16,
[
k-group.
c: s c.
such that the action of rk on G'(k.) (denoted by >-+ is obtained by "twisting" the original action with (H g) = ·g). G' is k-isomorphic to G if and only if there exists an automorphism such that = c- 1 We say that G' is an of G if all are inner automorphisms. If c is a group on which rk acts, the continuous functions f-+ Cs of rk to c which satisfy ( * ) are called 1 -cocycles of rk with values in C. The equivalence classes of
c: c c.(s(s, g)c5 s * g)·sc. innerform c, s
T. A . SPRINGER
12
these cocycles for the relation: (c,) (c�) if and only if there is c E C such that c� = c-1 c, · (sc), form the !-cohomology set H 1 (k, C). It has a privileged ele ment 1 , coming from the constant function c, = e. 3.2. Reductive k-groups. Now let G be a connected reductive k-group. It is said to be quasi-split if it contains a Borel subgroup which is defined over k (this is a very restrictive property). G is split (over k) if it has a maximal torus which is defined over k and k-split. In this case G is quasi-split. ExAMPLE. G = SO(F) (see [2, pp. 1 5 - 1 6]). This is quasi-split but not split if and only if the dimension n of the underlying vector space is even and the index equals -!-n- 1 . From the splitting of 2. 1 4 one concludes that G i s an inner form of a quasi-split group. Now let B be a Borel subgroup of G and T c B a maximal torus, both defined over k,. Let cj;0(G) = (X, LJ, xv, LJV) be the based root datum defined by (G, B, T). If s E rk there is g, E G(k,) such that �
•
int(g,)(sB) = B, int(g,)(sT) = T. Then int(g ,) o s defines an automorphism of T depending only on s (since the coset Tg, is uni quely determined). This automorphism determines an automorphism fJ.c(s ) of X, permuting the elements of L1 (since int(g,) o s fixes B) . It is easy to check that fJ.G defines a homomorphism fJ.c: Fk-+ Aut cj;0(G). Let G' be a k-form of G. Then fJ.c = fJ.G' if and only if G and G' are inner forms of each other. 3.3. Restriction of the base field [22, 1.3]. Let l c k, be a finite separable exten sion of k. Let G be an /-group. Then there exists a k-group H = R11kG charac terized by the following property [2, 1 .4] : for any k-algebra A we have H(A) = G(A ®k /). In particular, H(k) = G(l ). Let}; be the set of k-isomorphisms I-+ k,. We then have H(k,) = G(k,r:. The action of rk = Gal(k,/k) on H(k,) is as follows. If ¢ E G(k,)Z is a function on }; with values in G(k,), and s E rk, then, for (J E .2,
(s · ¢)(o) = ¢(s (J). ·
R11kG is obtained from G by restriction of the ground field from l to k. If G is connected or reductive then so is R11k G. Now let G be connected and reductive. Fix B and T (defined over k,) as in 3 . 2 and let cj;0(G) b e the based root datum defined b y (G, B , T). Then H = R11kG con tains the Borel subgroup B1 = Bz and the maximal torus T1 = rz. The based root datum cj;0(R11kG) (relative to B1 and T1 ) is then cj;0(G)z. The action of rk on the lattice X2' is like before : if s E Fk,
REDUCTIVE GROUPS
13
3.5. Properties of reductive k-groups. We next review the properties o f reductive groups. The reference for these is [5]. Let G be a connected reductive k-group. Let S be a maximal k-split torus of G, i.e., a k-subtorus of G which is k-split and maximal for these properties. Any two such tori are conjugate over k, i.e., by an element of G(k). Their dimension is called the k-rank of G. The root system f/J(G, S) of G with respect to S (see 2. 1 ) is called the relative root system of G (notation krp or kf/J(G)) . This is indeed a root system in the sense of [7], lying in the subspace V of X*(S) ® Q spanned by kf/J. Its Weyl group is the relative Weyl group of G (notation k W or k W(G)) . Let N(S) and Z(S) denote nor malizer and centralizer of S in G ; these are k-subgroups. Then N(S)/Z(S) operates on kf/J and in V. In fact it can be identified with k W. Any coset of N(S)/Z(S) can be represented by an element in N(S) (k). Z(S) is a connected reductive k-group. Its derived group Z(S)' is a semisimple k-group which is anisotropic. To a certain extent G can be recovered from Z(S)' and the relative root system kf/J (for details see [19]). There is a decomposition of the Lie algebra g of G : where for a E X*(S) we have defined 9a = {X E g i Ad(s)X = s"'X, s E S} . Then g0 is the Lie algebra of Z(S). If a E kf/J there is a unique unipotent k-subgroup Ua of G normalized by S, such that its Lie algebra is Sa · In the absolute case (k = Q) S is a maximal torus, f/J is the ordinary root system and the Ua are as in 2.3. If G is split over k then S is a maximal torus of G and kf/J coincides with the absolute root system f/J. In the general case kf/J need not be reduced, nor is dim Sa = dim Ua always 1 . 3.6. Parabolic subgroups. Recall that a parabolic subgroup P of an algebraic group G is a closed subgroup such that G/P is a projective variety. Equivalently, P is parabolic if P contains a Borel subgroup of G. Now let G be as in 3 . 5 . Then the minimal parabolic k-subgroups of G are con jugate over k. If P is one, there is a maximal k-split torus of G such that P is the semidirect product of k-groups P = Z(S) · Ru(P) (Ru(P) denotes the unipotent radical). There is an ordering of krp such that P is generated by Z(S) and the Ua of 3 . 5 with a > 0. The minimal parabolic k-subgroups containing a given S cor respond to the Weyl chambers of kf/J. They are permuted simply transitively by the relative Weyl group. Fix an ordering of krp and let kLI be the basis of krp defined by it. For any subset 0 c kJ denote by P 0 the subgroup of G generated by Z(S) and the U where a E krp is a linear combination of the roots of kJ in which all roots not in 0 occur with a coefficient � 0. Then P Jk = G, P ,p = P and P 0 � P. The P 0 are the standard parabolic subgroups of G containing P. Any parabolic k-subgroup Q of G is k-conjugate to a unique P 0• If S 0 is the identity component of n ac::o(Ker 0) then S 0 is a k-split torus of G and we have P 0 = Z(So) · Ru(Po), a semidirect product of k-groups. The unipotent radical Ru(P 0) is generated by the Ua where a is a positive root which is not a linear combination of elements of 0 . Let Q b e any parabolic k-subgroup o f G, with unipotent radical V (which i s defined over k). A Lel'i subgroup of Q is a k-subgroup such that Q is the semi-
L
14
T.
A. SPRINGER
direct product of k-groups Q = V. It follows from the above that such exist. Two Levi subgroups of Q are k-conjugate. If is a maximal k-split torus in the cen tre of then = Z ) If is any k-split subtorus of then there is a parabolic k-subgroup Q of with Levi subgroup L. Two such Q are not necessarily k conjugate (as they are when is a maximal k-split torus). Two parabolic k-sub groups Q1 and Q2 are if they have Levi subgroups which are k-conjugate. This defines an equivalence relation on the set of parabolic k-subgroups. If QI and Q2 are two parabolic k-subgroups, then (QI n Q2) · RiQI) is also a parabolic k-subgroup, contained in Q1. It is equal to Q1 if and only if there is a Levi subgroup of Q1 containing a Levi subgroup of Q2. Q1 and Q2 are called if Q1 Q2 is a Levi subgroup of Q1 and Q2. 3.7. Let and S be as in 3 . 5 and put = Ru(P). If we, denote by a representation in N(S)(k). The of asserts that is the disjoint union of the double cosets
L, L G (A . A associateAd
L
L
·
A
G
n Bruhat decomposition of G(k). P G(k),W).W G(k)nw
(we
W
opposite U tion Bruhat decomposi U(k)nwP(k)
One can phrase this in a more precise way. If w e, there exist two k-sub groups U�, u; of u such that u = u� X u� (product of k-varieties) and that the map u� X p ..... sending onto is an isomorphism. We then have
UnwP (x, y) xnwy (G/P)(k) G( /P(k) G ..... GjP. GjP. W0 W P0 (k)\G(k)jP0, (k) W(O)\,W/W(O'). W G(k), The Tits building. G Tits building of(G, P (PPII>n ··, Pn)Pn P. P' G . P. chambers. wall. adjacent i s). G(k) =
= U 1r ( U�(k)),
k)
W"'kW
where 1r is the projection If k = {) this gives a cellular decomposition of the projective variety If () e,LJ let be the subgroup of , generated by the reflections defined by the a e,LJ. If (), ()' e,LJ there is a bijection of double cosets
�
Let .2'be the set of generators of, defined by,LJ. The above assertions (except for the algebra-geometric ones) then all follow from the fact that ( P(k), Z(S )(k), .2') is a Tits system in the sense of [7]. Let be the connected reductive k-group. We define a 3.8. simplicial complex 81, the (simplicial) k), as follows. The vertices of 81 are the maximal nontrivial parabolic k-subgroups of G. A set of distinct vertices determines a simplex of 81 if and only if = .. . n is parabolic. In that case, the P; are uniquely determined by P. It fol lows that the simplices of 81 correspond to the nont rivial parabolic k-subgroups of Let ap be the simplex defined by Then ap is a face of ar if and only if c The maximal simplices correspond to minimal k-parabolics. These sim A codimension 1 face of a chamber is a Two plices are called chambers are if they are distinct and have a wall in common. One shows that any two chambers a, a' can be joined by a gallery, i.e., a set of chambers a = a0, a1, · · · , 75 =a', such that a; and a;+I are adjacent (0 � < It is clear that operates on 81. One can show (using a concrete geometric realization of the abstract simplicial complex 81) that 81 has the homotopy type of a bouquet of spheres. For more details about buildings see [20].
REDUCTI V E G ROUPS
15
3 .9. ExAMPLES. (i) The preceding results apply when k = Q, the absolute case. In particular, we then have the properties of parabolics of 3.6 and the Bruhat decomposition of 3. 7. (ii) G = GL(n) (k arbitrary) . This is indeed a reductive k-group. Its Lie algebra g is the Lie algebra of all n x n-matrices. Let S be the subgroup of d i agonal matrices. Thi s is a maximal k-split torus which i s also a maximal torus of G (in the absolute sense) . Let e,. E X = X *(S) map s E S onto its ith diagonal element. The e, form a basi s of X. The root system (/) = C/J(G, S), which coincides with the relative root system kC/J, consists of the e,. - ej E X with i 1= j. One checks that the root datum of G is given by X = Zn , xv = zn , C/J = {e1 - ej }i" j ' C/J " {e/ - ej }1=j• where (e� ) is the basis of X dual to (e1). The subgroup B of all upper triangular matrices i s a minimal parabol ic k subgroup . It is a Borel subgroup. Its unipotent radical U is the group of all upper triangular matrices with ones in the diagonal . The basis J of (/) defined by B is (e1 - e1 +1 ) 1, 1 2 n - l · The Weyl group W (which coincides with the relative Weyl group k W) is isomorphic to the symmetric group Zm viewed as the group of per mutations of the basis (e1). The parabolic subgroups P ::J B are the groups of block matrices =
( t:.��;· : ·:. <: ) 0
0
· · ·
Ass
where A,.j is an n1 x nrmatrix with n1 + · · · + ns n, the A ,.,. being nonsingular. Its unipotent radical consists of these matrices where A ,.,. = I (1 � i � s ). The sub group of P of matrices with A ij = 0 for j > i is a Levi subgroup of P. The center of consists of those elements of S at which the elements of L1 different from one of the en ,. - en,+I (1 � i � s ) are trivial . Hence with the notations of 3 .6, we have P = P0, where () E i s the complement of the set of these roots. A more geometric description of parabolic subgroups i s as follows. Let V = Qn . A .flag i n V is a sequence 0 = V0 c V1 c · · · c Vs = V of distinct subspaces of V. A k-flag is one where all V, are defined over k, i . e . , have a basis consisting of vectors i n k n . G operates on the set of all flags. The parabolic subgroups of G are then the isotropy groups of flags . One sees that there is a bijection of the set of all parabol ic subgroups of G onto the set of all flags, under which k-subgroups correspond to k-flags. If P is a parabolic subgroup, then the points of G/P can be viewed as the flags of the same type as P (i . e . , such that the subspaces of the flags have a constant dimension) . The Tits building of (G, k) can then also be described in terms of flags : The sim plices correspond to the nontrivial k-flags (i . e . , those with s > 1 ). If rJ1 is the simplex defined by the flag /, then rJ1 i s a face of rJf' if and only iff' refines /(in the obvious sense) . The chambers correspond to the maximal flags (s = n , dim V,. = i) and the vertices of S?J are described by the nontrivial k-subspaces of V. We see that the com binatorial structure of S?J pictures the incidences in the projective space Pn_1(k). =
L
L1
T . A. SPRINGER
16
The smallest nontrivial special case is n = 3, k = F2• Here !?a is a graph with 1 4 vertices and 2 1 edges (drawn in [20, p. 2 1 0]). (iii) Let char k '# 2 . Let V be a vector space over k (in the sense of algebraic geometry) . Let F be a nondegenerate quadratic form on V which is defined over k. With respect to a suitable basis of V(k) we have
F(X J ,
·
•
· ,
Xn) = XJXn + XzXn-l +
·· ·
+ XqXn -q+l + Fo (Xq+b
·• · ,
X n-q) ,
where F0 is anisotropic over k (i . e . , does not represent zero nontrivially). The index q of F is the dimension of the maximal isotopic subspaces of V(k) . Let G = SO(F) be the special orthogonal group of F. It is a connected semi simple k-group . A maximal k-split torus S in G is given by the matrices of the form diag(th
··· ,
tq ,
I,
··· ,
I,
t!\
·· · ,
r;1) .
Then Z(S) i s the direct product of S and t h e anisotropic k-group SO(F0) . For a description of a mini mal parabolic k-subgroup and the determination of the relative root system //J we refer to [2, p . 1 6] . The latter is of type Bq i f 2q '# n and of type D q otherwise. If q < [n/2] there are always subgroups Ua of dimen sion > 1 (notations of 3 . 5) . A geometric description o f parabolic k-subgroups sim ilar to the one for GL(n) can be given. They are in this case the i sotropy groups of isotropic k-fl ags in V, i . e . , fl ags a l l o f whose subspaces are isotropic with respect t o F.
4. Special fields. Let G be a k-group. In this section we discuss some special fea tures for particular k. 4. 1 . R and C. If G is a C-group then G(C) has a canonical structure of complex Lie groups. The latter i s connected if and only if G i s Zarisk i-connected (this can be deduced from Bruhat's lem ma, compare 4.2) . Now let k = R . Then G(R) is canonically a Lie group. 4.2. LEMMA. (i) G(R) is compact (f and only if the identity component G0 is a reduc tive anisotropic R-group ; (ii) G(R) has finitely many connected components. (i) is easily established. As to (ii), it suffices to prove this if G is connected reduc tive. In that case one reduces the statement, via Bruhat's lemma, to the case that G i s either compact or a torus. In these cases the assertion i s clear. G(R) need not be connected if G is Zariski-connected , as one sees in sim ple cases (e . g . , G = GL(n)) . If G is a C-group then the real Lie group Rc1iG)(R) (see 3 . 3) is that defined by the complex Lie group G(C). 4.3. Finite fields. Let k = Fq and let k be an algebraic closure. F denotes the Froben ius automorphism x ...... xq of kfk. The basic result here i s Lang' s theorem [6, p . 1 7 1 ] .
4.4. THEOREM. If G is a connected k-group then g ...... g - ( Fg) is a surjective map of G(k) onto itself.
1
Using that G is an inner form of a quasi-spl it k-grou p (see 3 . 1 ) one deduces that connected reductive k-group is quasi-split. A complete classification of si mple k-groups can then be given .
a
REDUCTIVE GROUPS
17
Before continuing with local and global fields, we must say a little about group schemes over rings. 4.5. Groups over rings. If G is a k-group, the product and inversion are described (see [2, p. 4]) by morphisms of algebraic varieties f-1. : G x G --+ G, p : G --+ G, which in turn are given by homomorphisms of k-algebras f-1. * : k[G] --+ k[G] ® k k[G] and p * : k[G] --+ k[G]. These have a number of properties (which we will not write down) expressing the group axioms. We thus obtain a description of the notion of linear algebraic group in terms of the coordinate algebra. The fact that k is a field does not play any role in this description. Replacing k by a commutative ring o, we get a notion of "linear algebraic group G over o", which is habitually called "affine group scheme G over o " , which we abbreviate to o-group. It can be viewed as a functor, cf. [2, p. 4]. We write G(o) for the group of o-points of G (i.e., the value of the functor at o ). Let o[G] be its algebra. If o' is an o-algebra we have, by base extension, an o' group G x 0 o', with algebra o[G] ® o o'. Let m be a maximal ideal of o and put k(m) o/m ; this is an o-algebra. DEFINITION. The o-group G has good reduction at m if G x o k(m) is a k(m)-group. G is smooth if it has good reduction at all maximal ideals m . EXAMPLE O F BAD REDUCTION. 0 Z, G i s the group o f matrices =
=
with a 2 - 2b 2 = 1 . Then Z[G] Z[X, Y]/(X 2 - 2 Y 2 - 1 ) and Z[G] ® F2 � F2 [X, Y]!(X 2), which cannot be the coordinate ring of a linear algebraic group over F2 , since it contains nilpotent elements. Now let G be a k-group and let o be a subring of k. We shall say that G i s de finable o ver o if there exists a smooth o-group G0 such that G � G0 x o k. By abuse of notation, we sometimes write, if o' is an o-algebra, G(o') for G0 (o'). One can also define when an algebraic variety over k is definable over o ; it is clear how to do this. EXAMPLE. By a theorem of Chevalley a complex connected semisimple group is definable over Z [6, A, § 4]. 4.6. Local .fields. Let k be a local field . We denote by o its ring of integers and by m the maximal ideal of o . The residue field o/m is denoted by F. A profound study o f reductive groups over local field has been made by Bruhat and Tits . So far, only part of this has been published [9]. For a resume see [8]. In Tits ' contribution [21] in these PROCEEDINGS more details are given about the Bruhat-Tits theory. In particular, he discusses the building of a reductive k group and the theory of maximal compact subgroups. Here we mention only a few results. =
4.7. LEMMA. Let G be a connected reductive k-group. Then there is an unramified extension I of k such that G is quasi-split o ver I.
This can be deduced from the fact that a maximal unramified extension of k is a field of dimension � 1 (see [16, p. 11- 1 1 ]). The group G(k) of k-rational points is a locally compact topological group (even a Lie group over k). It is a compact group if and only if the identity com-
T. A. SPRINGER
18
ponent GO is a reductive anisotropic k-group. It is shown in the Bruhat-Tits theory that if G is connected and simply connected simple k-group, there is a finite-dimen sional division algebra D with center k such that G(k) � SL(I, D). If G is connected and reductive and is definable over o (which is always the case if G is k-split, see [8, p.3 1 ]), then G(o) is a compact subgroup of G(k). There is a re duction map G( o) ..... G(F), which is surjective. 4.8. Global.fields. Now let k be a global field. If v is a valuation of k, let k. be the corresponding completion of k. If v is nonarchimedean, o. denotes the ring of integers of k •. If S is a nonempty finite set of valuations of k, containing all the archimedean ones, denote by os the ring of elements of k which are integral out side S. It is a Dedekind ring. Let G be a connected reductive k-group. 4.9. LEMMA. (i) There is an S such that G is definable over os ; (ii) G x k k. is quasi-split for almost all v. (i) is easily established . Let C be the group of inner automorphisms of G; it is a semisimple k-group. We identify it with its group of k.-rational points. There is r e fll(k, C) such that G, twisted by a cocycle c from r (see 3. 1), is quasi-split. This is another way of saying that G is an inner form of a quasi-split group. Now c defines a principal homogeneous space Cc of C over k, i.e., an algebraic variety over k, on which C acts simply transitively, the action being defined over k (see [16, p. 1-58]). We have r = 1 if and only if Cc has a k-rational point. To prove (ii) it now suf fices to show that the image of r in H l (k., C x k k.) is trivial for almost all v, or that Cc has a k.-rational point for almost all v. Let v be nonarchimedean such that C x k k. and Cc x k k. are definable over o., say C x k k. = C0 x "• k., Cc x k k. = Cc .o x "• k •. Assume further more that the reduced group C0 x "• F., over the residue field F., is a connected F.-group. These conditions are satisfied for almost all v. By 4.4, it follows that Cc,o x " • F. has an F.-rational point. A version of Hensel's lemma then gives that Cc,o has an o.-rational point, which shows that C has a k.-rational point. This implies (ii), as we have seen . 4. 1 0. Adelization. Let A be the adele ring of k. It is a k-algebra, so the group of A-points G(A) of G is defined. Let G 4 GLn be an embedding over k. Then ,_. maps G(A) bijectively onto a closed subset of A n2 + 1 EB A n2+I. Endowed with the induced topology, G(A) is a locally compact group, the adele group of G. It has G(k) as a discrete subgroup. The topology on G(A) is independent of the choice of the embedding G 4 GL(n) . An alternative way to define G(A) is as fol lows. Let S0 have the property of 4.9(i). For each finite set of valuations S :J S0, the group
g
(g, g-1)
G(A s)
=
n G(o.)
v <Jc S
X
n G(k.)
V'-
S
is a locally compact group. If S c S' then G(A s) c G(As, ) G(A) can also be de fined as the limit group G(A) i nj li ms => so G(As) (this is independent of the choice of S0). For each v, we have an injection G(k.) ..... G(A). Ex AMPLES. (a) G = GL( I ) . T he n G(A) is the group of ideles (the units of A ) . (b) k Q, G = SL(n). One checks that .
=
=
REDUCTIVE GROUPS
G(A)
=
SL(n, R)
·
( � SL(n, Zp))
19 ·
G0 ,
from which one sees that there is a surjective continuous map G(A)/G(Q) --+ SL(n, R)/SL(n, Z). More precisely, if one defines, for a positive integer the con sequence subgroup F(N ) of SL(n, Z) by F(N) = { r e SL(n, Z) I r = 1 (mod then G(A)/G(Q)
=
proj lim SL(n, R)/ F(N) .
N, N)},
(c) Let G be a Q-group and let G 4 GL(n) be an embedding (over Q). Fix a lattice L in Q n and let r be the subgroup of G(Q) of elements stabilizing L. There exists a connection, similar to that of the previous examples, between G(A)/G(Q) and G(R)/r (see [1]). The main results about G(A)/G(k) are as follows (G a connected reductive k-group ). Let X be the group of k-rational characters of G. For each X e X define a character I x I : G(A) --+ R* by I x I ((gv) ) = TI v I x(gv) l v , where l l v is an absolute value, normalized so as to satisfy the product formula. Let G(A) O be the intersection of kernels of the l x l , for x e X. The product formula shows that G(k) c G(A) O . 4. 1 1 . THEOREM. (i) G(A) OfG(k) (ii) (G G(A)0/G(k)
semisimple)
finite iinfvariandaonlnt voly ifume;G is anisotropic over ishascompact
k.
This is a consequence of reduction theory, due to Borel and Harish-Chandra for number fields and to Harder for function fields (see [1] and [11]). Notice, that by restriction of the ground field, it suffices to prove this for k = Q or k = Fq( T). 5. A class of Lie groups. In this section we discuss a class of Lie groups close to the groups of real points of reductive R-groups. This is the class of groups occurring in Wallach's paper in these PROCEEDINGS (see also [12]). We shall indicate briefly how the properties of these groups can be deduced from the algebraic properties of reductive groups, discussed above. We shall say that an algebraic group G defined over a field of characteristic zero is reductive if its identity component GO (in the Zariski topology) is so. 5 . 1 . Let G be a Lie group, with Lie algebra g . Its identity component is denoted by We denote by the intersection of the kernels of all continuous homomor phisms G --+ Rt. Then O G is a closed normal subgroup and Gf O G is a vector group. A of G is a vector subgroup V of such that G = · V,
0G G0 • G 0G split component {e}. G and a morphism G There i s a reducti v e Rroup g whose image is an open subgroup of G0(2) The image of in the automorphism group of of the identity component of n V
=
We assume henceforth that (1)
0G with finite kernel
possesses the following properties : G IJ : --+ G(R) G(R). It follows that IJ induces an isomorphism of g onto the set of real points of Lie G. We shall often identify g and IJ(g). It also follows that IJ(G) O = G(R) O and that has finite index in G (since this is so for G(R), see 4.2(ii), and ker IJ is finite). gc = g ® R C G G0 G. The main reason to allow for finite coverings of linear groups is to include the metaplectic group and all connected semisimple groups with finite center. The
lies in the image
20
T. A. SPRINGER
main use of (2) is to insure that G acts trivially on the center of the universal en veloping algebra of g. 5.2. Let ()0 be the automorphism of GL(n, R) which sends g to tg- 1 • Let s (resp. S0) be t he se t of real symmetric (resp. a n d positive nondegenerate) n x n-matrices. Then exp : x ,_. exp x = I + x + x2/2 ! + · · · i s an isomorphism of s onto S0• Given s E S0, there i s a unique analytic subgroup of GL(n, R), i somorphic to R, contained i n S0, and passing through s. It i s contained in any 00-stable Lie subgroup of GL(n, R) with finitely many connected components which contains s. 5 . 3 . LEMMA . Let G c GL(n, C) be an R-subgroup stable under 00 . (i) Let s E S0 n G(R). Then there exists a 00-stable R-split torus S of G such that 0 s E S( R) ; (ii) let X E s n Lie(G ) . There is a 00-stable R-sp/it subtorus of G whose Lie algebra contains X; (i i i) G is reductive.
The element s generates an i nfinite subgroup of G, whose Zariski closure is a torus with the required properties. (ii) follows from (i), applied to exp x. Let U be the unipotent radical of G, let s E G ( R ) . Then s and (00 s ) s 1 are unipotent. By (i) the last element i s also semisimple, which implies s = I . Hence U= { I } . This proves (iii) . 5.4. By definition, a Cartan i nvolution of GL(n, R) is an automorphism conj u gate to ()0 by an i nner automorphism. Let G and G be as in 5 . 1 . Let G c GL(n, C) be an embedding over R. Then v(G) i s stable under some Cartan involution 0 of GL(n, R) . In other words, we may assume, after conj ugation, that v(G) is stable under 00 (in which case i t is said to be selfadj oint) [1], [15] . Let f (resp. s) be the fixed point (resp. - 1 eigenspace) of 0 in g, and K the i nverse image in G of the fixed point set of () in v( G) . -
5 . 5 . PROPOSITION. The automorphism 0 of !J extends uniquely t o a n automorphism of G whose fixed point set is K. The map f-t : (k, x) ,_. k · exp x is an isomorphism of analytic manifolds of K x s onto G.
The automorphisms of G thus defined are the Cartan involutions of G. They form one conj ugacy class with respect to inner automorphisms by elements of G0 . The decomposition G = K · S (S = exp s) i s a Cartan decomposition of G. After conj ugation, we may assume that 0 = ()0. If G = GL(n, R), then K O(n), S = S0 and our assertion follows from the polar decomposition of real matrices. Assume now that v i s the identity. Write g E G as a product g = k · s where k E O(n), s E S0. Then s2 = (()0 g)- 1 g E G, and the unique ! -parameter subgroup i n S0 through sz (see 5 . 2) i s contained i n G. In this group, there is a uni que element with square s2, which must then be equal to s. Thus s E G, hence also k E G. This i mplies that f-t is surj ective. Injectivity follows from the uniqueness of the polar decomposition. The decomposition g = f + s i mplies that the tangent map at any point is bij ective ; hence f-t is an analytic i somorphism. Thus G is the direct product of K and a euclidean space. This proves the proposition when v i s the identity. Let G be the si mply connected group with Lie algebra g, K the analytic subgroup of G with Lie algebra f and 11: : G --+ GO the natural projection. Since the fundamental group of K is that of GO, the group K is the universal covering of K; hence ker 11: c K. The automorphism () =
•
REDUCTIVE GROUPS
21
of g extends to one of G, which fixes K pointwise, hence acts trivially on ker 11: , and goes down to an automorphism of G0 . The result for ])(G) implies that G = K · S = K · GO. Since ker ]) acts trivially on g, hence on GO, () : GO --> GO extends obviously to an automorphism of G which fixes K pointwise. The remaining asser tions are then obvious. 5.6. COROLLARY. (i) K is a maximal compact subgroup of G ; (ii) K meets every connected component of G. G is the topological product of K by a connected space S, whence (ii). The first assertion follows from the fact that every s i= 1 in S generates an infinite discrete subgroup. Fix a Cartan involution () of GL(n, R) stabilizing ])(G). We also denote by () the Cartan involution of G defined in 5 . 5 . 5. 7. Let C b e the center o f G. I t has again the properties ( 1 ), (2) and it is {}-stable. The group corresponding to G is the center of G. The subset corresponding to S is now a vector group V. It is, in fact, the maximal {}-stable vector subgroup contained in C. Let G1 be the derived group of G. 5 . 8 . LEM MA . (i) 0 G = KG 1 and V is a split component of G ; (ii) O G has the properties ( I ), (2) and is {}-stable. KG 1 is a {}-stable closed normal subgroup of G, contained in °G. The Lie algebra is the direct sum of those of KG 1 and of V, which implies (i). As to (ii), for the algebraic group of ( 1 ) we take the Zariski closure of ])(OG) in G. Its identity com ponent differs from GO only in its center. This implies (2), and the final assertion is clear. 5.9. Parabolic subgroups. A parabolic subalgebra p of g is a subalgebra such that Pc is the Lie algebra of a parabolic R-subgroup of GO. A parabolic subgroup P of G is the normalizer in G of a parabolic subalgebra (which then is the Lie algebra of P). The parabolic subgroups of G correspond to the parabolic R-subgroups of G0 . Let P be a parabolic R-subgroup of GO. Let N be its unipotent radical. Put L = P n OP.
g
5. 10. LEMMA. (i) L is a Levi subgroup of P ; (ii) the Lie algebra of G is the direct sum of those of N, ON and L . 5.3(iii) shows that L is reductive. L N i s a parabolic R-subgroup o f G 0 contained in P (see 3.6) with unipotent radical N, hence equal to P. This proves (i) . Then (ii) follows by using that P and ()P are opposite parabolics. 5. 1 1 . Let S be the maximal R-split torus in the center of L. Put A = ])- 1 (S(R)) 0 , N = ]) - 1 (N(R)) 0 . These are subgroups of G. Since A is a {}-stable vector group, we have ()a = a - 1 for all a E A . Let P be the parabolic subgroup of G defined by P and put L = P n ()P, M = OL. Then L is the centralizer of A in G. Also, L and M are {}-stable. 5. 1 2 . PROPOSITION. (i) L satisfies ( 1), (2) of 5. ! . A is a split component of L and of P ; (ii) (m, a, n) ,.... man defines an analytic diffeomorphism of M x A x N onto P. Let H be the centralizer of S in G. Then ])(L) c H. Moreover, the identity component n o is reductive and is equal to H n GO (the last point because centralizers
T. A. SPRINGER
22
of tori in Zariski-connected algebraic groups are connected, see [3, p. 27 1 ]). It fol lows that ( 1 ) and (2) hold for L. The last point of (i) is easy. From P = LN we conclude that P = LN = MAN. Now (ii) readily follows. The decomposition of 5. 1 2(ii) is called the Langlands decomposition of P. There is a similar decomposition of the Lie algebra of P : p = 111 + o + n . A parabolic pair in G is a pair (G, A ) where P is a parabolic subgroup of G and A is as above. 5. 1 3. Minimal parabolic subgroups. N ow assume that P is a minimal parabolic subgroup. Then P is a minimal parabolic R-subgroup of G0 . In that case the derived group of L is an anisotropic semisimple R-group. It follows that M is compact. We then must have M c K (recall that M is 8-stable), so M = K n P. Let r/J be the root system of (G, S) (see 3 . 5). If a E r/J put 9a = {X E g I Ad(a) X = a" X a E A } . Then we also have 9a
,
= {X E g I [H, X] = da(H)X, H E a}
(a being the Lie algebra of A). Also, there is an ordering of r/J such that n .E a>O 9a• On = .Ea
5. 14. LEMMA . (i) We have direct sum decompositions g = a +
(ii)
a
111
+
n
+ On,
g =
r
(1
+
+
n;
is a maximal commutative subalgebra of 5.
The first decomposition follows by using 5. 1 O(ii). It then follows that t is the direct sum of m and the space of all X + ex (X E n). Hence t n 11 = {0}, t + 11 = m + n + On. This gives the second decomposition . We also get that 5 is the direct sum of a and the space of all X - OX (X E n). Since o commutes with no nonzero element of n + On, the assertion of (ii) follows. From the above we see that r/J is also the root system of the symmetric pair (G, K) (see [13, Chapter VII]). 5. 1 5. PROPOSITION (IWASAWA DECOMPOSITION). (k, a, n) diffeomorphism of K x A x N onto G.
f-+
k
an is an analytic
Let ¢ be the map of the statement. (a) im ¢ is closed. AN is a closed subgroup of G and G/AN is compact (because G/P and M are compact). Let n: be the projection G --> G/AN. Since K is compact, im(n: ¢) is closed . Hence so is im ¢ = n:- 1 im(n: ¢). (b) im ¢ meets all components of G, since k does (see 4.6(ii)) . (c) The tangent map d¢ is bijective at any point a, n). This follows from the direct sum decomposition g = t + a + n . (a), (b) and (c) imply that im ¢ is open and closed and meets all components. Hence ¢ is surjective. To finish the proof, it suffices to show that ¢ is injective. It is enough to prove that kan = a 1 implies a = ab k = n = e . Now if this is so we have O n = a2na12• The image under )) of the last element is unipotent. It then follows that a2 = ar, a = ab whence O n E N n O N = e } . o
o
(k,
{
5 . 1 6. CoROLLARY. For any parabolic subgroup P of G we have G = KP.
REDUCTIVE GROUPS
23
Let W be the Weyl group of the root system f/J. This is the relative Weyl group of (GO, S) (see 3 . 5), i .e. ,
W
=
N0ocRJ (S)/Z0o cRJ (S).
5. 1 7. LEMMA . M meets all components of G. It suffices to prove this for the case that v(G) G0 (R). In that case it follows from Bruhat's lemma that the connected components of GO(R) all meet N = N0o cRJ (S). Let a E (/) be a simple root (for the order defined by P), U < a > the uni potent subgroup whose Lie algebra is Co a)c + (oza)c (where the second term is zero if 2a rt f/J), and U< -a> 8( U< a> ) . It is known that U< a > (R) · U< -a> (R) U< a > (R) contains an element of N representing the reflection in W defined by a. It follows that N c Z0o cRJ (S)G0, which implies the assertion. It follows from the lemma that W Nc(A)/Zc(A). =
·
=
=
5. 1 8 . LEMMA. W � (K n Nc ( A ) )J M. Let g kan E Nc(A) . Then also (0n)- 1 a2n E Nc(A ). Let w0 E N0o cnJ ( S ) represent the element of maximal length of w. Then on-1 Woll l WiJI, for some 11 1 E N. We then have 11 1 w01a211 E w0 1 Nc(A). The uniqueness statement of Bruhat's lemma then implies that n 1 = e, whence n e. This implies the assertion. From 5. 1 8 we see that W is the Weyl group of the symmetric spaces GfK ([13, p. 244]). =
=
=
5. 1 9. PROPOSITION (CARTAN DECOMPOSITION ) . We hal'e G
=
KA K.
This follows from 5.5 and the following lemma. Here S i s as in 5.3 (observe that K normalizes S). 5.20. LEMMA . S
=
uk,c K kA k -1 •
It suffices to prove this for v(G), i.e., when v id. Let s E S. Then s lies in a 8stable R-split torus sl c GO. Since s is a maximal R-split torus in GO (because p is a minimal parabolic R-subgroup of GO, see 3 . 5) we have that S1 is conjugate to a subtorus of S by an element of G0 (R). By 5. 1 7 we may take this element to be in GO(R)O, hence in G. So there is g E G with a 1 g- 1 sg E A . Writing g = ka11 we obtain =
=
Using again the uniqueness statement of Bruhat's lemma, as in the proof of 5. 1 8, we see that a1 commutes with 11. It follows that s is conjugate to an element of A via K, which is what had to be proved. 5 . 2 1 . We finally give a brief elementary discussion of the geometric properties of the symmetric space G/K. We identify it with S (cf. 5.5). lt is a homogeneous space for G, the action being given by (x , s) ,_. x · s = xs(Ox)-1 • If x E 5 define I I X II 2 Tr(XZ). This defines a K-invariant Euclidean distance on 5. The exponential map exp defines a diffeomorphism of 5 onto S. Its inverse is denoted by log. Define a Euclidean metric d( , ) on A by d(a, b) ll log a - log b ll · This deter mines a structure of Euclidean affine space on A . W e may and shall assume that v = id. =
=
T. A. SPRINGER
24
LetE s,sucht E that x · s and x · t lie in A; There i s x A suchiftx'hatisx-anot1nx'herfixeselement sandwit. th the property of then there is n E normalizing x· s = a, x·t = b. st-1 ab-1 a-1 12xs1 12 b-x'1 12xt= 1e.12 s, ,tEA. If x E x · A = A then x normalizes A in = e, s 1 = xsx-1 = s. A, t' = e. x x E x'xs(Ox)A. x ·A A apartments A -+ a ...... x·a s,s t Et, t, i n dependent geodesics geodesic segment [st], [st]. d A A s,a clt,osedu E sphered(s, t)xdi=Esd(s,a did(x,su)tancea)d(u,on r}t) itshencompact u lies onin the segment [st] ; = GL{n, there iises:a unique involutorial analytic diffeomorphismis anofisometwirtyhFort(forhe foleachd),lowis nEg propert s issttheabilonlizesyallfixedgeodesi pointcsofthrough s. We have (}",.. , = x x-1 . x = e. d(e, e E t ...... t-1 . Lets, s' be distinct points of let m be the midpoint [ss']. Let t be 5.22. LEMMA. (i) (ii)
S.
G
(i)
G
(i) follows from 5.20. To prove (ii) it is sufficient to assume Then Put and lie in K from It now follows that which one concludes that and are conjugate in G. The uniqueness part of Bruhat's lemma then implies that these elements are conjugate by an element of the Weyl group W, from which (ii) follows. 5.23. LEMMA.
G,
G.
Apply 5.22(ii), taking a regular element of It follows that we may assume K and Since s is regular, centralizes The assertion follows. 5.24. The translates of in S are called in S. It follows from 5.23 that for any apartment d there i s a unique structure of Euclidean affine space on d such that any bijection d of the form is an isomorphism of such spaces. 5.22(i) shows that for any two elements S there is an apartment d con taining them . It follows from 5.22(ii) that, if =f. the line in d containing s and together with its structure of ! -dimensional affine space, is of the choice of d. We call such lines in S. It now also makes sense to speak of the and of the midpoint of It also follows that there is a unique G-invariant function on S x S whose restriction to x is the function of 5.2 1 . 5.25. PROPOSITION. (i) (ii) if S, (iii) {
XI
+
S;
�
X.
It suffices to prove this for the case G R). A proof of (i) and (ii) is given in the appendix to this section. The proof of (iii) is easy. 5.26. PROPOSITION. (Js S (a) (18 (b) (c) (15 o
(15
S
(J.,
o
We may take The geodesic through and exp X consists of the exp(�X) (� R). Observing that exp(�X)) is proportional to 1 � 1 it follows that the only possibility for (J, is the map That this satisfies our requirements is clear. The final statement follows from the rest. 5.27. LEMMA.
S,
REDUCTIVE GROUPS
25
<
a!d(t,poins't)of. not lying on the geodesic through s and s'. Then d(t, m) !d(t, s) a as 2d(t, m) d(t, a(t)) � d(t, s) d(a(t), s) d(t, s) d(t, s'). s [t, s' as, s, s', t I( If is compact the group I( has afixed point in r y). I d(x,d(x,y)y) �r}r 1/n} (n a, yb E d(m, a y)b !d(a,my) !d(b, y) r, I( Let be a compact subgroup of Then fixes a point of is conjugate to a subgroup of All maximal compact subgroups of are conjugate. S
Let
= O"m ,
' = s .
then
+
We have
+
=
+
=
If the extreme terms are equal we have, by 5.25(ii) that lies on a(t)]. Then so does = and lie on a geodesic, which is contrary to the assumption. The inequality follows. If C is a subset of X we denote by C) the subgroup of the group of isometries of S whose elements stabilize C. 5.28. LEMMA.
C
C)
S.
Let = inf..Ex supy Ec d(x, Then F = {x E S sup y E C = is the intersection of the decreasing family of sets Fn = {x E S I sup yE C + = 1, 2, ) Since these are nonempty and compact {by 5.25 (iii)) it follows that C is nonempty. Suppose F, =/= and let be the midpoint of [ab]. We then have by 5.27, for each E C, < + = which is impossible. Hence F consists of only one point. It is clearly fixed by C). · · ·
.
5.29. THEOREM.
M
G.
M
S.
This follows by applying 5.28 to an orbit of M in S. 5.30. CoROLLARY. M
K.
5.3 1 . CoROLLARY.
G
5.32. REMARKS. ( 1 ) In our discussion of the symmetric space S we wanted to stress, more than is usually done, the analogy with the Bruhat-Tits building (7JJ of a p-adic reductive group. We mention a few features of this analogy. (a) It is clear from our discussion that S can be obtained, like (JJJ , by gluing together apartments (see [21, 2. 1]). (b) We have introduced metric and geodesics in S in the same way as is done in the case of (7JJ (see [9, 2.5] and [21, 2.3]). In the case of (7JJ an important role is played in the discussion of the metric, by the retractions onto an apartment [loc. cit., 2.2] . Such retractions can also b e introduced in S (an example is the map p used in the appendix). (c) The fixed point Theorem 5.29 has a counterpart for (7JJ [8, 3.2.4]. (d) I owe the proof of 5.29, using the strong convexity property 5.27, to J . J. Duistermaat. 5.29 can also be proved via the argument used in [8] (see also [21, 2.3]) to prove its counterpart for (JJJ . This requires the inequality is the mid point of [x,
y])
d(x, z)Z d(y, z)Z 2d(m, z)Z td(x, y)2, +
;;;;
(m
+
which can also be established in our situation (e.g. by using that the exponential map 5 - S increases distances) .
T. A. SPRINGER
26
Appendix. Proof of 5.25 for G R). A is now the group of all diagonal matrices with positive entries. It suffices to show : if if if A , E S, � +
GL(n, lies ona, bthEe sesgmentth[aenb].s d(a, b) d(a, s) d(s, b) equality holp(s)ding and only s u(s) s 1u(s)p(s)u(s). a p(asa) p(a)2p(s). d(p(s), e) d(s, e), equality holding ifand only ifs d(a, b) � d(a, p(s)) d(p(s), � d(a, s) d(as, b), d(p(s), a) d(s, a). s b1 b2 ·· bn n p(s). a1 na2 · · · an � b,-)2 � a;)2. g p(s)1 12 u(s), s 1g · g. n gv bl 12 b1(v, sv) a1( b1 a1b• 1b2 ·· · bp � a1a2 ·· · ap b1b2 · ·· bn a1a2 ··· an. a1a2 ···-an e; -ei)e;+J ( I i i,nj -� I, -(e;) a" a2, · an), y b" b2, ·, eb1n).+ · · · + e;. (y, n - y, (y, y) - y, - y) (y - yy). y (y, y) (y, - y) u(s)(y, y) Tr(s) (p(s) Tr(tu(s)p(s)u(s)) p(s). (s p(s) p(s) s. =
As is well known, E S being given there is a unique upper triangular unipotent matrix such that have = We shall prove the following.
E A and a unique If E A, we
=
�
LEMMA.
E A.
From the lemma it also follows that, for all E A, � Hence b) + proving the triangular in + equality. The case of equality is easily dealt with. It remains to prove the lemma. Let � � � be the eigenvalues of and � The lemma asserts that � · � those of i=I
� (log
(log
i:::::= l
Results of this kind are known, they can be found, e.g., in H . Weyl, Ges. Abh. Bd . There is a so = n v. Then ( , ) denoting the standard inner product in R , vector v E R with IV, p. 390. We use Weyl's method . Let
=
=
v)
=
( v,
�
v, v) ,
whence � Applying a similar argument, working in the exterior powers of Rn, we see that (1 � p � n 1 ) , and, of course, = We may, and shall, assume that 1. Let V E Rn be the subspace of vectors with coordinate sum 0. Let a; = � � is the canonical basis) , and define = o,i (1 � n 1 ) . Then w; is the projection onto V w; E V by (w;, ·· , J og (log Let x (log of J og J og log We then know that =
·
=
(x, a;) � 0,
a ,.
) � 0, (x
=
·
w ,.) � 0
(1 � i �
- 1),
+ x � (x, x) . Now (x, x) and we have to prove that (x 2 , x Since is a linear combination, with positive coefficients, of the w; and x is a similar combination of the a;, we have that x � 0. The inequality which we have to prove now becomes obvious. It is clear that we can only have (x, x) if x y. In that case we have of. e, we must have Tr ) . But it is immediate that if =
-
=
=
=
So Tr )
> Tr
=
Tr
implies
=
This finishes the proof of the lemma.
REFEREN CES
I. A. Borel, Some finiteness properties of adele groups over number fields, Inst. Hautes Etudes Sci . Publ. Math. 16 ( 1 963), 5-30. 2. , Linear algebraic groups , Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math . , vol. 9, Amer. Mat h . Soc . , Providence, R . I . , 1 966, pp. 3-20. -
REDUCTIVE GROUPS 3. A. Borel,
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Linear algebraic groups, Benjamin, New York, 1 969. Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 ( 1 962), 485-5 3 5 . 5. A. Borel a n d J . Tits, Groupes reduct(fs, Inst. Hautes E tudes Sci . Pub!. Math. 2 7 ( 1 975), 5 51 50 . 6. A. Borel et al . , Seminar in algebraic groups and related finite groups, Lecture Notes in Math., vol . 1 3 1 , Springer-Verlag, Berlin, 1 970. 7. N. Bourbaki , Groupes et algebres de Lie, Hermann, Paris, 1 968, Chapters IV, V, VI. 8. F. Bruhat and J. Tits, Groupes a/gebriques simples sur un corps local, Proc. Driebergen Con ference on Local Fields, Springer-Verlag, Berlin, 1 972, pp. 23-36. , Groupes reductlfs sur un corps local. I, Inst. Hautes E tudes Sci . Publ. Math . 41 ( 1 972), 9. 5-252. 10. C. Chevalley, Seminaire sur Ia classification des groupes de Lie algebriques, Paris, 1 95 6-58. 11. G . Harder, Minkowskische Reduktionstheorie uber Funktionenkorpern, Invent. Math. 7 ( 1 969), 33-54. 12. Barish-Chandra, Harmonic analysis on real reductive groups. I, J. Functional Analysis 1 9 ( 1 975), 1 04-204. 13. S. Helgason , Differential geometry and symmetric spaces, Academic Press, New York, 1 969 . 14. J. E. Humphreys, Linear algebraic groups, Grad. Texts in Math . , vol . 2 1 , Springer-Verlag, Berlin, 1 975. 1 5 . G . D. Mostow, Self-adjoint groups, Ann. of Math. (2) 62 (1 955), 44-5 5 . 1 6 . J -P. Serre, Cohomologie galoisienne, Lecture Notes in Math . , vol. 5, Springer-Verlag, Berlin, 1 964. 17. SGA 3 : Schemas en groupes (seminaire dirige par A. Grothendieck), Lecture Notes i n Math . , vol. 1 5 3 , Springer-Verlag, Berlin, 1 970. 18. R . Steinberg, Lectures on Chevalley groups, Yale University, 1 968. 19. J. Tits, Classification of algebraic semi-simple groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math . , vol. 9, Amer. Math . Soc. , Providence, R. I., 1 966, pp. 3 362. 20. , On buildings and their applications, Proc. Internal. Congr. Math. (Vancouver, 1 974), vol. I, Canad. Math . Soc. , 1 975, pp. 209-22 1 . 21. ---, Reductive groups o ver local fields, these PROCEEDINGS, part 1 , pp. 29-69. 22. A. Wei!, Adeles and algebraic groups, Institute for Advanced Study, Pri nceton, N. J., 1 96 1 . 4. A. Borel and Barish-Chandra,
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M ATHEMATISCH INSTITUUT DER RIJKSUNIVERSITEIT, UTRECHT
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part 1 , pp. 29-69
REDUCTIVE GROUPS OVER LOCAL FIELDS J.
TITS
TABLE OF CONTENTS 0. Preliminaries 0. 1 . Introduction 0.2. Notations .
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1 . The apartment of a maximal split torus and the affine root system 31 31 1 . 1 . The split case . . 1 .2. The apartment A(G, S, K) . . . . . . . .31 32 1 . 3. Remark . . .. . . 32 1 .4. Filtration of the groups Ua (K) . . . . . 33 1 . 5. About proofs, references and generalizations 33 1 .6. The affine root system War 33 1 .7. Half-apartments, chambers, affine Weyl group 1 . 8 . Bases, local Dynkin diagram, characteristic dimensions 34 1 .9. Root system attached to a point of A and special points . . . . . 35 1 . 10. Behaviour under field extension and hyperspecial points . . . 36 1 . 1 1 . Absolute and relative local Dynkin diagram ; the index . . . 37 1 . 1 2. Reduction to the absolutely quasi-simple case ; restriction of scalars . . . 38 39 . .. . . . 1 . 1 3 . The case of simply-connected groups 39 . 1 . 1 4. Example : general linear groups 1 . 1 5 . Example : quasi-split special unitary groups in odd dimension . . 40 1 . 1 6. Example : quasi-split but nonsplit orthogonal groups . . . 42 . . . . . . . . . . . .
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43 2. The building . . . . . . 2. 1 . Definitions . . . . . . 43 2.2. Affine structures, facets, retractions, topology and other canonical structures on &?/ . . 44 .. . . . . . 45 2.3. Metric and simplicial decomposition . . . . . 45 2.4. Dynkin diagram ; special and hyperspecial points . 2.5. Action of (Aut G)(K) on &?/ and Ll ; conjugacy classes of special and 46 hyperspecial points . . .. . .. .. . . . .. ... .. . 47 2.6. Behaviour under field extensions 2. 7. Example : groups of relative rank 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.8. Example : SL3 and GL3 .. . ... . . .. . . . 48 . . .
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AMS (MOS) subject classifications (1 970). Pri mary 20G25 ; Secondary 20G30, 1 5A63 . © 1 9 79, American Mathematical Society
29
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2.9. Example : general linear groups . . . 2. 1 0. Example : special unitary groups . .
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68
0. Preliminaries. 0. Introduction. This is a survey of some aspects of the structure theory of reduc
I.
tive groups over local fields. Since it is mainly intended for "utilizers", the main emphasis will be on statements and examples. The proofs will mostly be omitted, except for short local arguments which may give a better insight in the way the theory operates. When proofs are available in the literature (which is not always the case !), references will be given ; references to [8] are often conditional, as explained in § 1 . 5. We shall not try to give the historical background of the results exposed here. Let us merely recall that the theory was initiated by N . lwahori and H. Matsumoto [15], who were considering split semisimple groups, that quasi-split and classical groups were later on studied by H . Hijikata [13], and that, in the generality given here, most results are due to F. Bruhat and the author [6], [7], [8], [9]. For further information, one may consult the introduction of [8]. 0.2. Notations. The following notations will be used throughout the paper : K denotes a field endowed with a nontrivial discrete valuation w, the value group no with 7r E l1 w(K x ) ( c R) is also called r. l1 represents the ring of integers, p its prime ideal and K ojp the residue field. We always assume K complete and K perfect. We consider an algebraic group G defined over K whose neutral com ponent G o is reductive, and call S a maxim al K-split torus of G, N (resp. Z) the normalizer (resp. the centralizer) of S in G, " W the finite group N(K)/Z(K) (as =
=
REDUCTIVE GROUPS OVER LOCAL FIELDS
31
usual, ?(K) stands for the group o f rational points of ? over K), X * = X * (S) = Hom K (S, Mult) (resp. X* = X* (S) = Hom K (Mult, S)) the group of characters (resp. cocharacters) of S, V the real vector space X* ® R, (/) = r/J(G, S) c X * the set of roots of G relative to S, v W the Weyl group of the root system (/) which we identify with a normal subgroup of v W (equal to v W if G = G0) and Ua, for a E r/J, the unipotent subgroup of Go normalized by S and corresponding to the root a (i .e., the group called U
K-isomorphic to the additive group. Indeed, the choice of a "Chevalley basis" in the Lie algebra of G determines a system of K-isomorphisms Xa : Add -> Ua (an "epinglage") satisfying the commutation relations of Chevalley [10, p. 27]. Since K is a local field, its additive group is filtered and so are the groups Ua(K), "par transport de structure". The terms of those filtrations are conveniently indexed by affine functions on V: for a E (/) and r E r, a r is such a function and we set
+
(I)
If we transform the Chevalley basis by Ad s for an element s E S(K), the system (x a) is replaced by Cx�) = (x a o a(s)) and, setting X�_,_ r = x� (w-1 [r, oo]), we have (2)
Thus, the terms of the filtrations of the groups Ua(K) are unchanged but their indexation has undergone a translation. The same conclusion holds for an arbitrary change of Chevalley basis (one just has to replace s by a rational element of the image of S in the adjoint group). We may express that conclusion in a more invariant way as follows. There exist an affine space A under V, a system r/J.r of affine functions on A and a mapping a ,_. Xa of r/J.r onto a set of subgroups of G(K) with the following property : to every Chevalley basis, there corresponds a point 0 E A such that r/J.r consists of all functions a: x
(3)
,_.
a(x - 0)
+
r
(x E A ; a E (/), T E r)
and that, if Cx a) denotes the "epinglage" associated with the given basis, the group corresponding to the function (3) is given by ( 1) . The group S(K) operates by translations on A in such a way that, for s E S(K), we have Xa
(4)
From (2) it follows that the translation v(s) E V of A induced by s (i .e., defined by x + v(s) for x E A) is given by
s(x) (5 )
=
a(v(s)) = - w(a(s)) for every a E r/J.
More generally, the normalizer N(K) of S(K) in G(K) operates on A by affine trans · formations in such a way that (4) holds for any s E N(K). 1 .2. The apartment A(G, S, K). Our purpose is to generalize the above results to an arbitrary group G in the following form : to G, S, K, we want to associate an
J.
32
TITS
affine space A = A(G, S, K) under V on which N(K) operates. a system !JJ . £ = !JJ . £ ( G , S, K) of affine functions on A and a mapping a ,_. Xa of !JJ. £ onto a set of
subgroups of G(K), such that the relation 1 . 1 (4) holds for s E N(K), that the rector parts v(a) of the functions a E Waf are the elements of !JJ , and that, for a E !JJ , the groups Xa with v(a) = a form a.filtration of Ua(K). We first proceed with the construction of the space A ; the set !JJ . and the Xa ' s will be defined in § § 1 . 6 and 1 .4. The relations (5) show us the way. The group X*(Z) of K-rational characters of Z can be identified with a subgroup of finite index of X * . Let l.l : Z(K) -+ V be the homomorphism defined by ( 1)
x (l.l (z)) = - w (x(z))
for z E Z(K) and
X
E X*(Z) ,
and let Zc denote the kernel of l.l. Then, A = Z(K)/Zc is a free abelian group of rank dim S = dim V. The quotient W = N(K)/Zc is an extension of the finite group v W by A . Therefore, there is an affine space A ( = A(G, S, K)) under V and an extension of l.l to a homomorphism, which we shall also denote by l.l, of N in the group of affine transformations of A. If G is semisimple, the system (A, l.l) is canonical, that is, unique up to unique isomorphism. Otherwise, it is only unique up to isomor phism, but one can, following G. Rousseau [19], "canonify" it as follows : calling � Go the derived group of Go and S 1 the maximal split torus of the center of Go , one takes for A the direct product of A(!2G0, Go n S, K) (which is cano nical) and X AS1) ® R .The affine space A is called the apartment of S (relative to G and K) . The group N(K) operates on A through W . 1 . 3. Remark. Since V = Hom(X*, R) = Hom(X *(Z), R) , the groups Hom( X * , F) and Hom(X*(Z), F) are lattices in V and one has
(I)
Hom(X*, F)
c
l.I (Z(K)) = A
c
Hom(X * (Z), F ) .
I f G is connected and split, both inclusions are equalities, but in general they can be proper. Suppose for instance that G Ru K Mult, where L is a separable extension of K of degree n, and let F1 be the value group of L. The group X*(Z) is generated by the norm homomorphism Nu K , hence has index n in X*. On the other end , A is readily seen to be equal to n Hom(X * (Z), F1). In particular, the first (resp. the second) inclusion ( 1 ) is an equality if and only if the extension L/K is unramified (resp. totally ramified) . A semisimple example is provided by G = SU3 with split ting field L ; exactly the same conclusions as above hold with n = 2 (indeed, in that case Z = Ru K Mult). One can prove that the .first inclusion ( 1 ) is an equality when ever G splits over an unramified extension of K. 1 .4. Filtration of the groups Ua(K). Let a E !JJ and u E Ua(K) - { 1 } . It is known (cf. [3, §5]) that the intersection U_au U_a n N consists of a single element m(u) whose i mage in v W is the reflection ra associated with a, from which follows that r(u) = l.l(m(u)) is an affine reflection whose vector part is 'a· Let a (a, u) denote the affine function on A whose vector part is a and whose vanishing hyperplane is the fixed point set of r(u) and let !JJ ' be the set of all affine functions whose vector part belongs to !JJ . For a E !JJ ' , we set Xa = {u E Ua(K) I u = 1 or a (a, u) � a . The following results are fundamental. 1 .4. 1 . For every a as above, Xa is a group. 1 .4.2. If a, E !JJ ' , the commutator group (Xa, Xp) is contained in the group gen erated by all Xpa+qfl for p, E N * and pa E !JJ ' . =
·
}
{3
q
+ q{3
REDUCTIVE GROUPS OVER LOCAL FIELDS
33
v(a) a v(a) n a, About proofs, references and generala izations.u n n Au) a(a, u) -a is a valuation of the root datum
Clearly, the Xa s with = form a filtration of U0(K). We denote by Xa + the union of all Xa+• for e e R, e > 0 (of course, Xa+ = Xa+• for e sufficiently small). From 1 .4.2, it follows that Xa+ is a normal subgroup of Xa, and we set Xa = = are the quotients of the filtration of Ua(K) Xa/Xa+ · Thus, the Xa s, for in question. It is obvious that for e N(K), one has 1 Xa = Xa•v(n ) . 1 . 5. Let us identify with V via the choice of an "origin" 0, and, for every e (/J and e Ua(K), set ({Ja ( = (e R). The assertions 1 .4. 1 and 1 .4.2 essentially mean that the system of functions (q;a)a E 0 (Z(K) ; ( Ua(K))a E 0), as defined in [8, 6.2]. That fact itself is roughly equivalent with (actually somewhat stronger than) the ex istence of a certain BN-pair in the group generated by all U0(K) {cf. [8, 6.5 and 6.2.3(e)]), and with the existence of the affine building of G over K (cf. §2 below and [8, §7]). Those results have been announced in [6], [7] and [8, 6.2.3(c)]), but complete proofs by the same authors have not yet appeared (though the case of classical groups is completely handled in [8, § and quasi-split groups are es sentially taken care of by [8, 9.2.3]). In the meantime, proofs of closely related results have been published by H. Hijikata [14] and by G. Rousseau [19]. In the sequel, quite a few statements will be followed by references to [8] ; this will usually mean that the quoted section of [8] contains a proof of the statement in question 1 .4. 1 1 .4.2 For the sake of simplicity we have assumed that w is discrete and K perfect. In fact, much of what we shall say until §3.3 remains valid (with suitable reformula tions) without those assumptions, provided that 1 .4. 1 and 1 .4.2 hold, and this has been shown to be always the case except possibly if char K = 2 for some groups G whose semisimple part has factors of exceptional type and relative rank (cf. [8, § 10], [25] and [19]). 1 .6. f/Jaf · For every affine function on whose vector part = belongs to f/J, one has an obvious inclusion X2a 4 Xa (if 2a ¢ (/), we set X2a = { I }) and the quotient Xa/X2a has a natural structure of vector space over K (cf. 3 . 5 . 1 ) whose dimension is finite and will be denoted by In particular, if char K = Xa is a p-group. An affine function such that = e (/J is called an of G (relative to S and K) if '# 0, that is, if Xa is not contained in Xa+ • · U2a(K) ( = Xa+• if ¢ ([J) for any strictly positive constant e. We denote by f/J.f(G, S, K) = (/Jaf the G, i.e., the set of all its affine roots. Note that if 2a ¢ (/), one has e (/Jaf for every e Ua(K) - { I } ; in particular, if(/J is reduced, (/J af = e (/J, u e Ua(K) - { I }}. 1 .7. For every affine function such that = e (/J, we denote by Aa the set oo)), by oAa its boundary a - 1 (0) and by ra the affine reflection whose vector part is the reflection r0 (cf. 1 .4) and whose fixed hyperplane is oAa. The sets Aa (resp. oAa) for E (/Jaf are called the (resp. the of and the are defined as the connected components of the complement in of the union of all walls. The facets of the chambers are also called the of thus, the chambers are the facets of maximum dimension. If G is quasi-simple the facets (and in particular the cham bers) are simplices, if G is semisimple they are polysimplices (i.e., direct products of simplices) and in general they are direct products of a polysimplex and a real affine space. '
'
-
10],
once and are admit ed.
a A a v(a)The affine root system d(a). p, a a v(a) affine root d(a) affine2a a(a,rootu)system of u a, u) I a affine Weyl group. Half-a apartv(a)ment{a(s, chambers, a-1([0, half-apartments walls) A,A chambers a facets A;
�2
a
J . TITS
34
The group w generated by all ra with a E War is called the Weyl group of the affine root system War · (If G is not semisimple, this is a slight abuse of language since W depends not only on War but also on the subspace of V generated by X * (§G o n S), where §G0 denotes the derived group of G0.) If G is semisimple, W is the affine Weyl group of a reduced root system (cf. [5, VI. 2. 1]) whose elements are propor tional to those of W, but which is not necessarily proportional to W, even if W is re duced (cf. the examples in §§ 1 . 1 5, 1 . 1 6). Clearly, War is stable by the group W = ).J(N(K)) (cf. § 1 .2). It follows that the half-apartments, the walls and the chambers are permuted by W, and that W is a normal subgroup of W. 1 .8. Bases, local Dynkin diagram, characteristic dimensions. The Weyl group W is simply transitive on the set of all chambers (i.e., it permutes the chambers transi tively and the stabilizer of a chamber in W is reduced to the identity). Let C be a chamber and let L0, · , L1 be the walls bounding C. For i E {0, · · · , 1}, let a; be the unique affine root such that L1 = oAa, and ta,. ¢ War · The set {a,. I i = 0, · · · , 1} is called the basis of Wa r associated to C. Let a,. be the vector part of a,. and let us introduce in the dual of V a positive de finite scalar product ( , ) invariant by the (ordinary) Weyl group v W. To War. one associates a (local) Dynkin diagram L1 = L1 (Wa r) obtained as follows : The elements a, of a basis are represented by dots v,., called the vertices of the diagram ; if 2a1 E Wa r. the vertex v,. is marked with a cross ; two distinct vertices v,., vi are joined by an empty, a simple, a double, a triple or a fat segment (edge of the diagram) according as the integer }..ii 4(a1, ai)2j(a,., a,)(ai, a1.) equals 0, 1 , 2, 3 or 4 (in the latter case, ai is a positive multiple of - a,.) ; if }..,i 2 or 3 (which implies that (a; , a,.) # (ai, ai) ) or if }..ii = 4 and ai # - a, , the edge joining v,. and vi i s oriented b y a n arrow pointing toward the vertex re presenting the "shortest" of the two roots a; and ai. Since the chambers are permuted simply transitively by W, 1 . 8. 1 . the Dynkin diagram does not depend, up to canonical isomorphism, on the • •
=
=
choice of the chamber C.
It is easily seen that the system (A , War) is determined up to isomorphism by the Dynkin diagram L1 and the dimension of A (i .e., the relative rank of G). The Coxeter diagram underlying the Dynkin diagram-i.e., deduced from it by disregarding the crosses and arrows-is the Coxeter diagram of W, hence the Coxeter diagram of an affine reflection group (cf. [5, V.3.4, and VI.4.3], where our "diagrams" are called "graphes"). Conversely, consider any Coxeter diagram which is the diagram of an affine reflection group, orient all double and triple edges and possibly some fat ones, and mark some vertices (possibly none) with a cross. Then, the diagram thus obtained is the local Dynkin diagram L1 of some group G over some field K if and only if, for
every vertex v marked with a cross, all edges having v as an extremity are double or fat and none of them is oriented away from v .
The necessity of the condition is obvious. As for the sufficiency, the classification of §4 even shows that for any given locally compact local field K, every diagram satisfying the above condition is the local Dynkin diagram of some semisimple
REDUCTIVE GROUPS OVER LOCAL FIELDS
35
group G over K: indeed, it is an easy matter to list all irreducible diagrams in ques tion, and one verifies readily that they all appear in the tables of §4. Note that the above statement, or alternatively the tables of §4, provide the classification of all affine root systems, for a suitable "abstract" definition of such systems, which the interested reader will have no difficulty to formulate (cf. also [8, 1 .4], where the affine root systems are called "echelonnages", and, for the reduced case, [17]). If the vertex v of L1 represents the affine root a, we set d(v) = d(a) + d(2a) ( = d(a) if 2a ¢ f/J.r) , where the function d is defined as in § 1 .6. The integer d(v) of course depends not only on L1 and v but on the group G itself. In the tables of §4, the value of d(v) is indicated for every v whenever it is not equal to 1 . G is split or if the residue field K is algebraically closed, all d(v) are equal to 1 . 1 . 9. Root system attached to a point of A and special points. For x E A , we denote by f/Jx the subset of (/) consisting of the vector parts of all affine roots vanishing in x, and by Wx the group generated by all reflections ra for a E f/J.r and a(x) = 0 (cf. § 1 .7). To x, we also associate as follows a set Ix of vertices of the local Dynkin diagram L1 : there is an element w of the Weyl group W which carries x in the closure of the "fundamental chamber" C and one sets Ix = { v.- l wx ¢ L,-} , with the notations of § 1 . 8 ; that Ix is independent of the choice of w follows from well-known pro perties of Coxeter groups : cf. , e.g., [5, V. 3 . 3 , Proposition 1 ] . The objects (/Jx, Wx, Ix depend only on the facet F containing x and will also be denoted by f/JF, WF• IF· The set f/Jx is a (not necessarily closed) subroot system of (/) whose Weyl group is the vector part of Wx and whose (ordinary) Dynkin diagram is obtained by delet ing from L1 the vertices belonging to Ix and all edges containing such a vertex. The set Ix has a nonempty intersection with every connected component of L1 and, con versely, every set of vertices with that property is the set Ix for some x. The point x is called special for f/J.r if every element of the root system (/) is pro portional to some element of f/Jx, that is, if (/) and f/Jx have the same Weyl group. When it is so, W is the semidirect product of Wx by the group of all translations contained in W; similarly, if G is connected, W is the semidirect product of W, by li ( Z(K) ) = Z(K)/Z, (cf. 1 .2). The fact for a point x to be special can be recognized from the set of vertices lx as follows. A vertex of the Coxeter diagram of an irreducible affine reflection group is called special if by deleting from the diagram that vertex and all adjoining edges, one obtains the Coxeter diagram of the corresponding finite (spherical) reflection group. (Equivalently : such a diagram being the Coxeter diagram underlying the extended Dynkin diagram-"graphe de Dynkin complete" in the terminology of [5]-of a reduced root system, the special vertices are the vertex representing the minimum root and all its transforms by the automorphisms of the diagram.) Clearly, such vertices exist. Now, x E A is special if and only if I, consists of one special vertex out of each connected component of £1. In particular, special points always exist. In the tables of §4, the special vertices are marked with an s or an hs ("hyperspecial points" : see below). 1 . 1 0. Behaviour under field extension and hyperspecial points . Let K1 be a Galois extension of K with Galois group Gal(KifK) = 61, and let S 1 be a maximal K1 split torus of G containing S and defined over K. Such a torus exists for instance in the following cases : if G is quasi-split over K (obvious !) ;
If
36
J . TITS
if Kt is the maximal unramified extension of K [6(c), 3, Corollaire 1 ] ; i f the residue field K i s finite and Ktf K is unramified . (The latter condition is necessary as is shown by the following example due to Serre : suppose that e has even order and no subgroup of index 2, and that G is the norm one group of a division quaternion algebra ; then G splits over Kt but none of its maximal tori does.) Let A t = A (G, St. Kt) be the apartment of St and let (}) 1.r = f}) . r{ G, St . Kt) be the corresponding affine root system . The Galois group 8 operates on At ("canonified" as in § 1 .2) "par transport de structure", and A can be identified with the fixed point set A7. That identification is not quite obvious. To characterize it, we have to describe an operation of N(K) on A7 (cf. 1 . 2) . First observe that A� clearly is an affine space under V. Let now n E N(K), let Nt be the normalizer of St in G and let Vt be the canonical homomorphism of NtCKt) into the group of affine transformations of A t . Since the conjugate nSt i s a maxi mal Kt-split torus o f Z, there exists z E Z(Kt) with t n ' = n z- E NtCKt). Upon multiplying z by a suitable element of (Z n Nt) (Kt), one may choose it so that Vt(n') stabilizes Af. Let now v (z) be the element of V defined by the relation 1 . 2( 1 ) where m must be replaced by the valuation of Kt . Then n = ' n z operates on A� through Vt(n') o v ( z) . That this action is independent of the choices made and indeed defines an operation of N(K) on A � is best seen by using the "build ing" of G over Kt defined in §2 : that building contains A� and is operated upon by G(Kt), hence by N(K), and one verifies that N(K) stabilizes A� and operates on it as described above. Note that, more generally, the results of §2.6 show that if s; is any maximal K1-split torus of G containing S, A can be naturally identified with an affine subspace of A(G, s;, Kt) ; much of what we shall say here extends to that situation . 1 . 1 0. 1 . If KtfK is un ram ified, f}).r consists of all nonconstant restrictions a l A , with a E (j)l af ·
That is no longer true in general when KtfK is ramified . An obvious example is provided by the case where G is split over K. Then, St = S, At = A, and if we identify A with V as in § 1 . 1 , we have f}).r = {a + r l a E (}), 7 E r} and (})ta r = {a + r l a E f/J, T E rt } , where rt denotes the value group of Kt · From I . I 0. I , it follows readily that 1 . 10.2. If KtfK is unram ifie d, every point of A which is special for f}),.r is also special for f}).r . The above example shows that that assertion becomes false without the assump tion on Kt fK. A point x E A is called hyperspecial if there exist Kt , St as above such that Ktf K is unramified, that G splits over Kt and that x is special for rJJ 1 .r. Then, it is easily seen, using I . I 0.2, that the same holds for any Galois unramified splitting field Kt of G and any choice of St (assuming that such a torus exists). More in trinsic characterizations of the hyperspecial points will be given in 3 . 8 . If G is quasi-split and splits o ver a n unramified extension of K, hyperspecial points exist. Indeed, take for Kt the minimum splitting field of G and (obligatorily) St = Z, let a[ , , a, be a basis of the root system f})(G, St) invariant by e and choose at, · · ·, a1 E rJJ 1 .r so that v(a,.) = a,. and that { a t. · · · , a1 } is stable by G (the possibility of such a choice readily follows from the description of A and f}).r given in § 1 . 1). Then, the equations a t = · · · = a1 = 0 define an affine subspace of A t invariant by G (in fact a single point if G is semisimple), and every point invariant by 8 in that subspace belongs to A and is clearly hyperspecial.
do
. . ·
REDUCTIVE GROUPS OVER LOCAL FIELDS
37
Suppose G is quasi-simple. We say that a vertex v of the local Dynkin diagram is A such that lx = {v} are hyper special (a property which depends only on v obviously). In the tables of §4, hyper special vertices are marked with an hs. Let now K1 be the maximal unramified extension of K. The group G is said to be residually quasi-split over K if there is a chamber of A 1 stable by Gal(Ki fK), and hence meeting A . We say that G is residually split if A1 is fixed by Gal(KifK), that is, if G has the same rank over K and over KI > i.e. , if S 1 = S. For an explanation of the terminology and another definition, cf. 3. 5.2. 1 . 1 0. 3 . If the residue field K i sfinite, G i s residually quasi-split. If K is algebraically closed, G is residually split. By a well-known result of R. Steinberg, if K is algebraically closed , G is quasi split. From that, it follows that : 1 . 1 0.4. Every residually split group is quasi-split . If K is finite and, more generally, if G is residually quasi-split, G has a "natural splitting field". Indeed, there is a smallest unramified extension K' of K on which G is residually split, namely the smallest splitting field of S 1 (which does not depend on the choice of that torus), and the group G, being quasi-split over K', has a smallest splitting field K" over K'. The field K" can also be characterized among all splitting fields of G over K as the unique one for which the pair consisting of the degree [K" : K] and the ramification index e(K"/K) is minimal for the lexicographic ordering. 1 . 1 1 . Absolute and relative local Dynkin diagram ; the index. In this section, K1 denotes the maximal unramified extension of K, and A I > SI > (/J1.r have the same meaning as in § 1 . 10. As in the classical, "global" situation (cf. [22] and the references given there), one associates to G, K, S1 (in fact, to G, K alone : cf. §2.4) a local index consisting of the Dynkin diagram J 1 of (/J 1 .r (absolute local Dynkin diagram), the action of e = Gal (Ki fK) on J 1 "par transport de structure", and a 6-invariant set of vertices of J I > called the distinguished vertices. The latter are characterized as follows : to define (/J 1 .r, one uses a chamber C 1 of A1 whose closure contains a chamber of A (such a C 1 exists by 1 . 1 0. 1 ) , and then, the distinguished vertices are those representing the elements of the basis of (/J1.r asso ciated to C1 (§ 1 . 8) whose restriction to A is not constant . Residually quasi-split and residually split groups can be characterized as follows in terms of the index. The group G is residually quasi-split if and only if the orbits of e in the set of all nondistinguished vertices are unions of full connected com ponents of .JI> and G is residually split if and only if G operates trivially on L1I> all vertices of J 1 are distinguished and the smallest splitting field of the connected center of G is totally ramified. The index of G determines its relative local Dynkin diagram L1 = Ll(G, S, K) and the integers d(v) (cf. § 1 . 8) uniquely. We shall indicate an easy algorithm which allows us, in most cases, to deduce the latter from the former. First of all, there is a canonical bijective correspondence v ..... O(v) between the vertices of L1 and the or bits of e in the set _of distinguished vertices of L1 1 . For every vertex v (resp. every pair { v, v'} of vertices) of Ll, let .dl. v (resp . .d1, vv ') denote the subdiagram of L1 1 obtained by removing from it all vertices not belonging to O(v) (resp. O(v) U 0( v')) and all edges containing such vertices, and let L1 1 . v (resp. L11. vv ') be the hyperspecial (with respect to G) if the points x E
38
J.
TITS
subdiagram of Ll 1 . v (resp. Ll 1, vv' ) consisting of its connected components which con tain at least one distinguished vertex. Then, il 1 .., together with the action of 8 � Gal(KifK) on it and the set of distinguished orbits it contains, is the index (in the sense of [3] and [22]) of a semisimple group of relative rank one over K, the integer d(v) is half the total number of absolute roots of that group and v is marked with a cross in il if and only if the relative root system of the group in question has type BC1 (if K is finite-or more generally if all vertices of il 1 . v are distinguished-that means that il1 . v is a disjoint union of diagrams of type A 2) . As for the edge of f/J joining v and v', its type is determined by il1 , vv' • O(v) and O(v'). If no connected component of il1 , vv' meets both O(v) and O(v'), then v and v ' are joined by an "empty edge". Otherwise, 8 permutes transitively the connected components of il1, vv' and the result can be described in terms of any one of them, say il�. vv' . I f the latter has only two vertices v 1 E 0( v) and v � E 0 (v'), then v and v' are joined in il in the same way as v 1 and v� in il�. vv' · Thus, we may assume that il�, vv' has at least three vertices. Suppose first that il�. vv' is not a full connected component of il 1 . Then, there is an "admissible index" (i .e., an index appearing in the tables of [22]) of relative rank 2 whose underlying Dynkin diagram is il�, vv' and whose dis tinguished orbits are O(v) n il�, vv ' and O(v ') n il�. vv' ; indeed, it follows from the assertions 3.5.2 below that to { v, v ' } is canonically associated a quasi-simple group defined over a certain extension of K and having such an index. The relative Dynkin diagram corresponding to that index, which can be computed by simple explicit formulae given in [22, 2.5], provides the nature of the edge joining v and v' in il. The following table gives the result in the case where all vertices of il�. vv ' are dis tinguished (e.g. , in the case where the residue field K is finite) ; in the first row , which represents il�. vv' • the sets O(v) n il�. vv ' and 0( v') n il�. vv' are circled :
il�, vv'
corresponding edge in il
There remains to consider the case where il�. vv' is a full connected component of ilb which means that v, v' are the two vertices of the local Dynkin diagram of a quasi-simple factor of relative rank I of G (cf. § 1 . 1 2). Here we shall restrict our selves to the case where all vertices are distinguished and simply refer the reader to the tables of §4 which give il in all the cases that can occur. 1 . 1 2. Reduction to the absolutely quasi-simple case ; restriction of scalars. We shall now indicate how the local Dynkin diagram-with the attached integers d(v) and the index of an arbitrary group G can be deduced from those of related ab solutely simple groups. First of all, those data are the same for G and for the adjoint group of G0• Thus, we may assume that G is connected and adjoint, hence is a direct product of K simple groups. Then, the Dynkin diagram-with the d(v) attached-and the index of G are the disjoint unions of the Dynkin diagrams and the indices of its simple factors.
REDUCTIVE GROUPS OVER LOCAL FIELDS
39
There remains to consider the case where G is K-simple, which means [3, 6.2 1 ] that G = R u K H, where L is a separable extension o f K, H is an absolutely simple group defined over L and Ru K denotes, as usual, the restriction of scalars. We shall, more generally, assume that G = Ru K H for an arbitrary reductive group H; this allows us to decompose the extension L/K into its unramified and its totally ramified parts and to handle the two cases separately. If L/K is totally ramified, the index, the local Dynkin diagram and the integers d(v) are the same for G, K as for H, L. If L/K is unramified, the index of G, K consists of [L : K] copies of the index of H, L permuted transitively by Gal(KJ!K) whose operation on the whole diagram is "induced up" from the operation of Gai(KJ!L) on one copy, the relative local Dynkin diagram of G, K is the same as that of H, L, and the integers d(r) are [L : K] times as big. 1 . 1 3 . The case of simply connected groups. In § 1 .7, we have seen that the Weyl group W of <Paf is a normal subgroup of W = N(K)/Z(K). When G is semisimple and simply connected, one has W = W. In this and in other instances, nonsimply connected groups behave with respect to the "local theory" in a way similar to non connected groups with respect to the classical theory . 1 . 1 4. Example . Genera/ linear groups. Let D be a finite dimensional central divi sion algebra over K. The unique extension of the valuation w to D will also be denoted by w. Suppose that G = G Ln . D• the algebraic group defined by G(L) = GLn(D Q9 L) for any K-algebra L, and take for S the "group of invertible diagonal matrices with central entries", that is, the split torus whose group of rational points S(K) consists of all diagonal matrices Diag(s b . . · , sn) with s,. E K x . The homomorphisms e,. : Mult --> S defined by e,. (t)
=
Diag( l , . . .
, I, I,
t-I, 1 , . . . , 1 ) 1
with the coefficient t- 1 in the ith place (i = . . · , n) form a basis of X * and hence of V = X * Q9 R. If (a,.) 1 ,. ;,;; n is the dual basis in the dual of V, the relative roots of G are the characters a,.i = ai - a,. (i o;f the group UaJK) consists of the matrices u ,.j(d)
=
1
+
,
((g r ) )
j),
with g r s
=
o� o{ d (d E D),
and N(K) is the group of all invertible monomial matrices n(O' ; dj , . . . , dn)
=
((g ,.j)) with g ij
=
or(j) dj ,
where (}' is a permutation of { 1 , . . · , n} and d,. E n x ( = D has, with the notations of 1 .4,
-
{0}). F or d E n x ' one
(1) where 0' is the transposition o f i and j, di = d, d,. = - d-1 and dk We may identify the apartment A with V in such a way that ).J (n(O' ; db . . · , dn))
(� v,. e,) � v�e,. =
with v� cil
=
=
1 for k
*
i,
j.
v,. + w(d,) .
'To avoid confusion, we ad here to the notations of [8, § 1 0] which, unfortunately, i mpose this somewhat unnatural choice of the basis (e,) (and, consequently, of (a1)) . This remark also applies to § § 1 . 1 5 and 1 . 1 6.
40
J.
a, u), a + I i, j E
From ( 1 ) and the definition of a( Thus
1/Jaf
TITS
it now follows that a(a;j , u;j(d)) = aij + w(d).
{1, · · ·, =f. j, r = { ij r and the filtration of Ua,/K) by the groups Xa with v(a) = aij (cf. § 1 .4) is the image of the natural filtration of D by the isomorphism d ...... uij(d). In particular, for any a E I/Ja1 , the integer d(a) of § 1 .6 is equal to the dimension of the residual algebra of D over K. The description of the walls and half-apartments is readily deduced from The chambers are prisms with simplicial bases, one of them, call it C, being defined by the inequalities a1 < 2 < · · · < n < a1 w(n- 1) , where n-1 denotes a uniformizing element of D. The corresponding basis consists of the affine roots . .-+1 = · · · , and n 1 + w (n-1 ) , and we see that the local Dynkin diagram is a cycle of length (affine diagram of type An_ 1 ) . The special points are all the points of the one-dimensional facets of the chambers, that is, all the points L: where v,- - v1 is an integral multiple of w(n-1) for all they are hyperspecial if and only if D = K. 1 . 15. Let be a separable quadratic extension of K. The valuation of extending w will also be called w, and we denote by n-1 a uniformizing element of by F1 the value group w (U ) = Z w(n-1 ) and by -. the nontrivial K-automorphism of Let n be a strictly · · ·, positive integer and set I = In £2n+ 1 , we consider the hermitian form h : ((x_ m · · · , Xn), (Y- m · · · , Yn)) ,..... l: X� Y ; Xb YO · (1) -
(2)
n}, i
E r},
(2). a a + a.- (i I, n -I) n a v,-e,i; Example. Quasi-split special unitary groups in Lodd dimension. L L, L. {±I, ±n}. + n)±iE e;(t) I i}. t, a_,- -a,-c, dEL a,- c�c + d�1/J a I 2a,E I}. {a,, ±I+ ((g,s)) + ((g{u.-,s)/)c) I cEgL},_,-, 0 c•, -c', g.-;.-c; gI 0,-c, dEL,c g,c<sc + g,s I dE L, + 0}. I, i), } -n, n} L -i i. For cE L, m(u,-j(c)) u _ _ c (2) - c I. m(u,-(c, u_,-( c ·
iEJ
Suppose that G is the algebraic group S U(h) and let the torus S be defined by S(K) = {Diag (d-m · · · , dn) ld,- K and d_;d; = do = for all The homomorphisms e,- : Mult --+ S (i = 1 , · · · , defined by = Diag(d_m · · · , dn) with d_; = d,t- 1 , dj = I for j =f. form a basis of X * . If we denote by (a.-h ,; ,- ,; n the dual basis and if we set and a,-j = + aj , we have = { ij i, j E I, = j =f. ± i} U Ii For such that = 0 and i, j E I + d withj =f. i, we define the following elements of G(K) : with g_j, i = g .-. j = and all other = 0, uij(c) = = 0. = other I all = with d, and = d) , = u,{c Then, UaJK) = Ua.(K) = {u,-(c, d) d + d< = 0} and U2a j (K) = {u.-(0, d) d d< = The group N(K) consists of all matrices of determinant one of the form n(a ; d_m . . . , dn) = ((g,-j)) with g,-j = o/(j) dj , where a is a permutation of U { 0 = { · · · , which fixes 0 and pre serves the partition of I in pairs ( and the d,-'s are elements of such that d:..,- d,- = I for all one has, with the notations of 1 . 1 4, =
= _ ,- j( - c- 1)u,-j(c)u -; , -j( - - 1 )
where a is the permutation (i, -j) (j, - i), d_; = c- 1 , d_j = ( c•) - 1 , dj = - c, d; = c• and all other d, are equal to Similarly, for c , d as above with =f. 0 (and hence d =f. 0), - d - 1 , (d•)- 1 ) u,-(c, d) u_,-( - c(d•)- 1 , (d •) - 1 ) d)) = (3) n(a ; d_,, · · · , d,.) =
REDUCTIVE GROUPS OVER LOCAL FIELDS
41
(i, - d_; (d•)-1, d0 - d•d-1, d; d . , vn E A v�(i) + w(d;). d_m"' • dn)) (� ) � cc,(2),d 0, a (a;j, uij(c))c 0,a;ja(+a;,wu;(c),(c,a(a,d))u), a; + w(d), dE LF' {wd(d)l+ dd•E 0,da+(2a;d•, u,-(0,0},d)) 2a; + w(d). r E r rEF') a;j + r 2a; + r) L c u,{c) d u,.(O, d)). {d/ dEL, d + d• 0}) d(2a; + r) d(a;j + r) 2 LfK d(a) a;+ +d + rd•E f/J.r0,, w(d) r}. {u, { c, d)i c •c a; + r E f/J.r cEL I {w(d) c•c + d+ d• 0}.
where CJ is the transposition i), = = = and all other are equal to 1 . We may identify the apartment with V i n such a way that, for Vt o . . R and setting v_; = v;, one has
d,
(4)
v; e; with
V; e; =
v(n(CJ ;
= V;
From (3), (4) and the definition of it follows that = for =1!for as above and =1= for x with = = Setting = u' = we see that for (resp. (resp. i s an affine root for all i, j. Furthermore, the filtration of Uaij(K) (resp. U2a,(K)) by the subgroups Xa is the image of the natural filtration of (resp. its intersection with the subgroup = by the isomor phism ,_. (resp. ,_. In particular, the corresponding values of the integer of § 1 .6 are = 1 or according as = 1 and is ramified or not. To determine under which condition !we first note that, with the notations of § 1 .4, Xa;+rl2 = By definition, = � !if and only if Xa,+r!Z (/:. Xa;+r/2+< . Uza, for every strictly positive c . That means that there exists such that (5)
=
r = sup
More precisely, an easy computation shows that, with the notations of § 1 .6, the group Xa;+rnl X2a,+r is isomorphic to the residue field of L or is trivial according as whether or not is given by (5) for some thus, we see that, in the first case (i .e., when !t = or 1 according as is unramified or ramified. If we set = sup 1 = a real number which is strictly negative if is ramified and char K = and = otherwise, the right hand side of (5) can be written = and we conclude that
a; + rorE f/J.r), d{w(a;(d)+ I dEr) L, d2 + c;d• + 0}, L/K L/K w(c•c) + o 2w(2,c) + o, 0 I I I f/J.r {a;{a;j + +r E E r E±2Fi, 1r+ o}.r1 } {2a; + i E rEF' } =
i, j
I, j
!- r i I,
u
E
=1-
u
r
I,
Let us show that (6)
Lf
K is ramified, o �
F' . x, y EyL x + x• + I y + y• ( w(x) w y) o. 1 xyo, + 0 7r1), (x +o. y) + (x + y)• + I 0 w(x + y) F F1 F'; Lf L/K r 2F1 F' 2F1 + o + w(7r1) . F' a;(j2F+1 + o) 2a;F1+; E Fl> 0 a1 if
Indeed, assume the contrary and let be such that = = 0 and = = Upon multiplying by a suitable unit of K, we may assume that 1 = (mod but then = and > which contradicts the maximality of In view of (6), one of the following holds : (7) (8)
K is unramified and =
i s ramified,
In both cases, U the affine functions
=
r and
=
and
=
=
therefore, the walls are the vanishing sets o f and the inequalities < < r, With r
42
J.
TITS
< < an < ! w(n-1) define a chamber. The corresponding basis is {a� > a_J.2, , a-n+J.m 2a_n w(n-1)} if o is an even multiple of w(n-1) and {a-n + 1 w(n-J),
a2
· · ·
· · ·
,
+
a-n+l .m · a_J.2, 2ai } otherwise. It follows from (7), (8) that in the first case, 2a1 is an affine root if and only if L/K is unramified, and in the second case (where L/K is necessarily ramified) 2a_n w(n-1) is never an affine root. As a result, we see that, whatever the value of o, the local Dynkin diagram, together with the attached integers d(v) (cf. § 1 . 8) are ·
·
+
X
(9) 3
2 2
(;
2 2
or
(10)
(r-i
if n = 1
)
if n = 1
)
I
according as L/ K is unramified or ramified. A point v = 1:; 7=1 v,.e, E A is special if and only if either v,. E rl for all i or v,. 1w(n-J) E rl for all i. It is hyperspecial if and only if LJK is unramified and v1 E F1 for all i, which means that /" consists of the vertex at the right end of the diagram (9). 1 . 1 6. Example. Quasi-split but nonsplit orthogonal groups . Let L be a separable quadratic extension of K and let n be an integer � 2. In the space Kn EB L EB Kn , viewed as a (2n 2)-dimensional vector space over K, we consider the quadratic form n (x0 E L ; x1 E K for i i= 0) q : (x_m . · xn) >--> .E x_ ,.x,. Nu KXo
-
+
.,
1= 1
+
(where Nu K : L ---> K denotes the norm), and we suppose that G is the orthogonal group O(q). The elements of G(R), for any K-algebra R, are conveniently repre sented by (2n 1) x (2n 1) matrices ((g1i))-nM, J &n where g1i E R if both i and j are not zero, g0i E L® KR if j i= 0, g10 E HomK(L, R) if i i= 0, and g00 E HomK(L, L) ® K R. For S, we take the group of diagonal matrices Diag(d_m · , dn) with d_1d1 = 1 for 1 ;;::;; i ;;::;; n and d00 = id. The characters a1 : Diag(d_"' . ' dn) d 1 for 1 ;;::;; i ;;::;; n form a basis of X * (S) and if we set a_1 = - a1, aij a1 + ai and I = { ± I , · · · ± n} , we have
+
...... _
+
,
=
1J
= {a1i l i, j E l,
j
i=
± i}
U {a1
..··
I i E I},
a root system of type EnHere, we shall simply describe the affine root system 1J.1 and the local Dynkin diagram without giving the details of the calculations, which can be found, in a more general setting (covering also the groups handled in the previous section) in [8, 1 0. 1]. Calling again F1 the value group of L, one has, for a suitable identification of A and V, af
= {a,, +
T I i, j E /, j
i= .
± i, T E r}
u {a, + T
I i E /, T E rJ } ·
I f the extension Lj K i s unramified, the inequalities < a1 < · · · < a n < w(n-) - an-I define a chamber, the corresponding basis of <1>.1 is {ah a_J.2, · · ·
0
,
REDUCTIVE GROUPS OVER LOCAL FIELDS
43
w(n') } and the local Dynkin diagram, together with the at is
d(v),+
a-n+! . m a-n+! . -n tached integers
(1)
The special vertices are the two endpoints on the ramified side of the diagram (the two endpoints of the diagram if = 2) ; both correspond to hyperspecial points of A. If L/K is ramified, the inequalities O !w(n) define a chamber, the corresponding basis is
n < a1 < ... < an < + }
{ ab a- ! . 2• .. a- n + ! . m a_ n · ,
and the local Dynkin diagram is
!w(n)
The special vertices are the two endpoints of the diagram and they do not cor respond to hyperspecial points. Note that in the unramified case, the Weyl group W is an affine reflection group of type Bm whereas in the ramified case, it is of type en . 2. The building. The 86 = 86(G, K) K can be constructed by 2. 1 . "gluing together" the apartments of the various maximal K-split tori of G. More precisely, a definition of 86 is provided by the following statement where by "G(K) set", we mean a set with a left action of G(K) on it. A = A(G, K) § 1 .2. 86 A N(K) ).1 § 1 .2) 86 = u g EG(K) a, Xa § 1 .4 Aa a- 1 ([0, oo ) (N.B. The "canonicity" of the building 86 is the same as that of A : cf. § 1 .2.) The proof roughly goes as follows. We assume that G is semisimple (which is no essential restriction). Modulo 1 .4. 1 and 1 .4.2-as explained in § 1 . 5-the existence of 86 is proved in [8, 7.4]. It is then clear that there is a "universal" G(K)-set f) with the given properties, which is obtained by taking the quotient of the direct product G(K) x by a certain equivalence relation. The canonical mapping of fj in the building of defined in [8, 7.4.2] is obviously surjective, and it is also injective be cause, as is readily verified, the stabilizer of a point of 88 contains the stabilizer of its image in of. Thus, of maps onto any G(K)-set 86 with the required properties and, using [8, 7.3.4], one shows that the stabilizers of the points of cannot be bigger in 86 than they are in of without "eating more of N(K)" than they are allowed to by the prescribed action of N(K) on A . The sets with E G(K) are called the o f the building. The apart ment can be identified with "the" apartment of the maximal split torus gS. That gives a one-to-one correspondence between the apartments of 86 and the maximal K-split tori of G : indeed, is the only apartment stable by gS(K) (the proof of [8, 2.8. 1 1] shows that) and gN(K), which determines gS, is the stabilizer of
Definitions. building
of G over
Let phism, only onebeG(K)-set given as incontainiThen, therehaviexingststheonefoland,lowinupg propert to uniiqeues: isomor n g and therootgroup the group stabilizes oAf and operat esthone half-apartment it through (cf fixes and for every) poigA,affine ntwise. =
A
A
gA gA g
apartments
gA
J.
44
TITS
g A in G(K). "In most cases", gA can also be characterized as the fixed-point set i n {s E gS(K) Iw(x(s)) 0 for all characters X E X*(gS) } , &8 o f the group o f units gso =
=
but that i s n o t always true (for more precise statements, cf. §3.6). In this context, it i s worthwhile to note also that the half-apartment Aa (for a E (]J.r) is nel'er the fixed-point set of the group Xa : indeed , if &8 is metrized in the way described below (§2.3), there is a constant c such that for every point x E Aa at distance d of the wal l oAa, the whole ball with center X and radius cd is pointwise fixed by Xa (cf. [8, 7.4. 33]). If 51 is the maximal split torus of the center of G o and if Gh · · · , Gm are the almost simple factors of Go, the building :!J' is canonically isomorphic with the direct product of the buildings :!J'(Sb K) (which is an affine space) and M(G, , K) (i I , · · · , m) . If G i s K-anisotropic (i.e., if S { I }), :?1 consists of a single point. If G RLI K H, where L i s a separable extension of K and H is a reductive group over L, the build ings f!J'(G, K) and :33 (H, L) are canonically isomorphic. =
=
=
2.2. Affin e structures, facets, retractions, topology and other canonical structures &8. Since the stabilizer N(K) of A in G(K) preserves its affine structure and its partition in facets, each apartment gA of ,:d ( with g E G(K)) is endowed with a
on
natural structure of real affine space and a partition in facets. Those structures agree on intersections. Indeed , 2.2. 1 . If A ' and A" are two apartmen ts, there is an element of G(K) which maps A ' onto A " and fixes the intersection A ' n A" pointwise ; furthermore, A ' n A" is a closed com•ex union offacets in A ' (hence also in A") [8, 7.4. 8]. From that, we deduce a partition of :!8 in facets, among which those which are open in apartments are called chambers. In particular, if Go is quasi-simple (resp. semisimple), :53 is a si mplicial (resp. polysimplicial) complex. Given two facets of :!8 , there is an apartment containing them both [8, 7.4. 1 8] . I n particular, given two points x, y E .'?1, there is a n apartment which contains them and it follows from 2.2. 1 that, for t E [0, I ] c R, the point ( I - t)x + ty, which i s well defined i n any such apartment, is i ndependent of it. The set { ( I - t)x + ty I t E [0, I ] } is called the geodesic segment joining x and y in ::d. Let A' be an apartment and let C c A ' be a chamber. For every apartment con taining C, there is a unique isomorphism of affine spaces of that apartment onto A ' which fixes C pointwise. In view of 2.2. 1 , all those isomorphisms can be glued together in a mapping PN ; c : f!8 --> A' called the retraction of :.:d onto A' with center C. Clearly, geodesic segments are mapped by PA' ; C onto broken lines (connected unions of finitely many geodesic segments). The building fE i s commonly endowed with a topology i nvariant by G(K) which is most naturally defined via the metric considered below (2. 3), but which can also be more canonically defined as the weakest topology such that all p A' ;C are con tinuous. If the residue field K is finite, that topology makes :33 into a locally compact space and coincides with the "CW-topology" (that is, the quotient topology of the natural topology of the disjoint union of all apartments). Otherwise, it is strictly weaker than the latter. In all cases, the topological space :.:d is contractible ; indeed , for every point x E 88, the mappings rp 1 : :.:d --> 3B defined by rp t (y) tx + ( I - t)J' form a homotopy from the identity to the retraction of f!8 onto {x} [8, 7.4.20]. A subset of f!8 is called bounded if its image by some retraction PA';C i s bounded , in which case its image by every such retraction is bounded, as i s easily seen . As =
45
REDUCTIVE GROUPS OVER LOCAL FIELDS
usual , a subset H of G( K ) is called bounded if for every K-regular function / on G, the set w (f( H )) is bounded from below. The action of G(K) on PJJ is bounded and "proper" in the following sense : in the mapping (g, b) >-+ (gb , b) of G(K) x PJJ in dJ x PJJ , bounded subsets of G(K) x PJJ - i .e., subsets of products of bounded sets are mapped onto bounded sets, and the inverse images of bounded subsets of PJJ x fl/ are bounded. If K i s finite, "bounded" becomes synonymous with "relatively compact" ; in particular, the action of G(K) on PJJ is proper in the usual sense. 2 . 3 . Metric and simplicial decomposition. In various questions, buildings play for p-adic reductive groups the same role as the symmetric spaces in the study of noncompact real simple Lie groups (cf. [24, §5] and the references given there). This section shows some aspects of the analogy ; cf. also [18, 5.32] . Note that, unlike those introduced in §2.2, the structures considered here are not canonical, at least when G is not semisimple . Let us choose in V a scalar product invariant under the Weyl group " W. If G is quasi-simple, such a scalar product is unique up to a scalar factor, and there are various "natural" ways of normalizing it (Killing form, prescription of the length of short coroots, etc.). Canonical choices are also possible-componentwise-if G is semisimple, but not in general . From the scalar product in question, one deduces a Euclidean distance on A, hence, through the action of G(K), on any apartment. From 2.2. 1 , it follows that two points x, y of fl/ have the same distance d(x, y) in all apartments containing them , and the properties of the retractions PA' ; c described in §2.2 readily imply that the building fl/ endowed with the distance function d : i!8 x !fa --> R + i s a complete metric space [8, 2 . 5] . The associated topology coincides with that defined in §2.2. Again using the retractions PA ' ; c one shows [8, 3.2. 1 ] that d satisfies the following inequality, where x, y, m E fl/ and d(x, m) = d(y , m)
=
-!- d(x, y) :
d(x,
z)Z
+ d(y,
z)Z
� 2 d(m ,
z)Z
z, y)Z.
+ -!- d(x,
In Riemannian geometry, that inequality characterizes the spaces with nega tive sectional curvatures (hence is valid in noncompact irreducible symmetric spaces !) ; as in the Riemannian case, it can be used here to prove the following fixed-point theorem : 2 . 3 . 1 . A bounded group of isometries of f!J has a fixed point [8, 3 .2.4]. Interesting
applications are provided by Galois groups ("Galois descent" of the building) and by bounded subgroups of G(K) (cf. § 3 . 2) . In some applications (cf. , e . g . , [2]) it is useful t o dispose of a simplicial decom position of &? invariant under G(K). To obtain it, it suffices to choose a si mplicial decomposition of A invariant under N(K) and finer than the partition in facets-it is easily seen that such a decomposition always exi sts-and to carry it over to all apartments by means of the G(K)-action . If G is semisimple, one can more directly use the canonical barycentric subdivision of the partition of in polysimplical facets. If G is quasi-simple, that partition itself meets the requi rements. 2.4. Dynkin diagram ; special and hyperspecial points. Let C be a chamber of &?. Starting from any apartment containing C, we can , following § 1 . 8 , define a local Dynkin diagram J(G, C) which, in view of 2.2. 1 , does not depend, up to unique isomorphisms, on the choice of the apartment. If C' is another chamber, 1 . 8 . 1 , applied to any apartment containing C and C', provides an isomorphism IPc ' c : ,
B
J. TITS
46
L1(G, C) --> L1(G, C') which, again by 2.2. 1 , is independent of the apartment in ques tion. All those isomorphisms are coherent : if C, C', C" are three chambers, one local Dynkin diagram L1(G) = has lfJc"c IPc"C' o lfJc·c · Thus, we can talk about L1(G, K) of G over K, a diagram which is well-defined up to unique isomorphisms. The same is true of the absolute local Dynkin diagram (§ l . l l), which is nothing else but the diagram L1(G, K1 ) of G over the unramified closure K1 of K, and of the local index (§ l . l l ) . The definitions of §§ 1 .9 and 1 . 1 0 can be immediately transposed to arbitrary points x and arbitrary facets F of the building @ : one chooses an apartment con taining x or F, uses the definition under consideration and deduces from 2.2. 1 that the result is independent of the apartment chosen. Thus, to every point x (resp. facet F) of ?4 is canonically associated a set r, (resp. IF) of vertices of L1(G) and a root system (/J" (resp. (/JF), the latter being only defined up to noncanonical i so morphisms. We can also talk about and points of @. The criterion in terms of I" for a point x to be special (last paragraph of § 1 .9) remains of course valid. A necessary condition for the existence of hyperspecial points is that G split over an unramified extension of K; that condition is also sufficient if G is quasi-split. To every vertex v of the diagram L1(G) is attached an integer d( v) : the definition given in § 1 . 8 made reference to an apartment but the result is independent of its choice, always by 2.2. 1 . If the residue field K is finite, isomorphic with Fq , the number d(v) can be interpreted as follows : a facet F of codimension one and "type " v , that is, such that IF is the complement of v in the set of all vertices of L1( G), is contained in the closure of exactly qd < vJ + 1 chambers (cf. §3 . 5). 2.5. (Aut G)(K) (J(J L1; The group (Aut G)(K) of all K-automorphisms of G and, in par ticular, the group G.d(K) of rational points of the adjoint group Gad of Go, act on ?4 and on the local Dynkin diagram L1 = L1(G) "par transport de structure". Through the canonical homomorphism int : G --> Aut G, that gives an action of G(K) on ?4 and on L1. The action of G(K) on ?4 provided by the definition of ?4 as a G(K)-set coincides with this one if G is semisimple but not in general ; however, the induced actions on L1 are always the same. We call E E(G, K) the image of G(K) in Aut L1. If G L1, E = {I} (another illustration of the "philosophy" of § 1 . 1 3) . Suppose G connected. Then, E is also the image of Z(K) in Aut L1, and it can be computed as follows. We denote by (} a simply connected covering of the derived group of G, by S the maximal split torus of (} whose i mage in G is contained in S, by Z the centralizer of S in G, by S 1 the maximal subtorus of S which is central in G and by z. the image of Z(K) in Z(K). Then, S1(K), z. and Zc = {z e Z(K) I w(x(z)) = 0 for all X e X * (Z) (cf. § 1 .2) are normal subgroups of Z(K) and their product S1(K) z. · Zc is the kernel of the action of Z(K) on L1; thus E = Z(K)/ (S 1 (K) · z. Zc) . If G is quasi-split-in particular if K is algebraically closed Z and Z are tori and the computation of E is particularly easy. Note that, in most interesting cases, the subgroup E of Aut L1 is uniquely determined by the underlying "abstract" group. Two facets F and F' of (J(J are in the same orbit of G(K)-for any one of the two
the
=
special hyperspecial A
specialActpoiinotns. of
on and conjugacy classes of special and hyper =
is semisimple and simply connected, it operates trivial y on i.e. , ·
·
}
47
REDUCTIVE GROUPS OVER LOCAL FIELDS
actions of G(K) on BS described above-if and only if IF and /p, (cf. §2.4) are in the same orbit of E. In particular, if G is semisimple and simply connected, the orbits of G (K) in the set of special points of BS are in canonical one-to-one correspondence with the sets of vertices of L1 consisting of one special vertex out of each connected com ponent.
Suppose G semisimple. If G is K-split, the group G.d(K) permutes transitively the special points of BS : that is an immediate consequence of Proposition 2 in [5,VI.2.2]. A case analysis shows that, for any semisimple G, G.d(K) permutes transitively the special points except possibly if the Coxeter d iagram underlying L1 has a connected component of the form s
s �----�=
or
s s - .
Suppose now that G is quasi-simple and that the Coxeter diagram in question i s one of those above. Then, obvious necessary conditions for G.d(K) (and even (Aut G)(K)) to permute transitively the special points are the existence of an auto morphism of L1 permuting its two special vertices, and the equality of the numbers d(v) attached to them. One verifies that if the residue field K is finite, those conditions are also sufficient.
For arbitrary G, if BS has hyperspecial points, the facets consisting of such pointshence the points themselves ifG is semisimple-are permuted transitively by Gad(K). 2.6. Behaviour under field extensions. The buildings behave functorially with respect to Galois extensions.
More precisely, for every Galois extension K1 of K, we can consider the building !J?J(G, K1), on which the Galois group Gal(KifK) acts naturally (in the nonsemi simple case, one has to "canonify" the apartments-and hence .�-as described in § 1 . 2) , a n d there is a unique system of i nj e c t i o n s (KI >Kz Galois extensions of K with K1
c
K2)
with the following properties : the i mage of cK2K , is pointwise fixed by Gal(K2/K1) ; the restriction of cK2K1 to any apartment of !J?J(G, K1) i s an affine mapping i nto an apartment of @(G, K2) ; t K2K1 i s G(K1)-covariant ; i f K1 c K2 c K3, one has has cK 3K, c K3K2 o tK2K , · The last property allows us t o identify coherently every @(G, K1) with i t s image by every cK2K1 · 2.6. 1 . If KtfK is unramified (or e ven tamely ramified: cf [19]), @ is thefixed point set of Gal{K1/K) in @( G, K1) and the apartment A A( G, S, K) is the intersection of@ with the apartment A(G, SI > K1) of any maximal K1- split K-torus S1 of G contain ing S. Still assuming that Kd K is unramified, one deduces from 1 . 1 0.2 that a point x of fJ1 which is special in @( G, K1) is also special in fJ1 ; iffurthermore G is split o ver KI> the point x is hyperspecial. If KtfK i s wildly ramified, the fixed point set of Gal(KifK) in P-l(G, K1) may be strictly bigger than @ : it then looks like the building fJ1 "covered with barbs". Suppose for example that G i s split and i s not a torus, that K = Q 2 and that K1 i s totally ramified over K and different from K (which implies that KdK i s wildly ramified). The apartment A A (G, S, K) of fJ1 is also an apartment of P-l(G, K1 ). =
=
=
J. TITS
48
Let F be a facet of codimension one of A with respect to K1 which is not a facet with respect to K (cf. the example following 1 . 1 0. 1). By the last assertions of §§ 1 . 8 and 2.4, there i s exactly one chamber of K1) not contained i n A and whose closure contains F; it must of course be fixed by Gal(K1 /K) and cannot be con tained in rJ1J since F is not a facet of fJI. For a proof of the above results and a more detailed analysis of the situation, cf. [19]. 2.7. 1 . The building of a semisimple group of relative rank 1 is a contractible simplicial complex of dimension 1, i.e., All its vertices are special points. If K � Fq and if are the integers attached to the two vertices of the Dynkin diagram, each edge of the tree has one vertex of order q d + 1 and one vertex of order q d ' + 1 (cf. § 2.4 Consider for instance the special orthogonal group of a nonsplit quadratic form in 5 variables over Q 2 : here, = 1, 2, and the building looks as suggested by the picture below. In that case, the vertices of order 5 are hyperspecial and the others are not.
e?J(G,
Example. Groups of relative rank
d d'
d, d'
=
).
d(v) a tree.
K) GL3 . Suppose that G SL3. The building rJ1J 2.8. SL 3 is a 2-dimensional simplicial complex whose maximal simplices are equilateral triangles, for the metric introduced in §2.3. The apartments are Euclidean planes triangulated in the familiar way :
Example. and
=
=
e?J(G,
To picture the building itself, one must imagine it "ramifying" along every edge, each edge belonging to q + I triangles if q card K. The link of each vertex ine?J =
REDUCTIVE GROUPS OVER LOCAL FIELDS
49
is the "spherical building" of SL3( K), that is, the "flag complex" of a projective plane over K, a picture of which can be found in [24] or [26] for the special case where K � F2 • All vertices of 81 are hyperspecial. The building 81( GL3, is the direct product of 81 and an affine line. 2.9. We adopt the hypotheses and the notations of § I . I4. In particular, D denotes a finite dimensional central division algebra over and = GL n . D (thus (K) = G L ( D) ) . Then, the building 81 = 81( , K) can be interpreted as the set % of all "additive norms" in Dn, that is, of all functions Dn � R U + such that
Example. Genera/lK)inear groups. G n K G cp: { oo} cp(xcp(xd) + y) cpin(x)f{cp(x), cp(y)} d) (x v;e; v; i .. . , n} (xt > .. . , xn) G(K) (gcp)(x) cp(g-1x). cp �
(x,
=
+ w(
G
E D n), E D n, d E D) . y
More precisely, if we identify the apartment A with V as in § 1 . 1 4, the mapping A � % which maps .E �=I onto the norm >-+
inf { w(x,) -
l
=
I,
extends-of course uniquely-to an isomorphism of G(K)-sets 81 ..... %, where operates on % by = A norm is special-i .e., corresponds to a special point of B-if and only if there is a basis (b;)1;&;;;&;n of the vector space Dn and a real numberf such that
cp(� b;d,)
cp
=
f + inf { w
)l
(d; i K) Example. Special unitary groups. d d d� I}.
=
I, . . .
, n} (d;
E D) ;
is hyperspecial if and only if it is special and D = K. A similar interpretation of 81(S Ln , D • can be found in [8, p. 238]. The space % has been first considered by 0. Goldman and N. Iwahori [12]. 2. 10. L et L, w, n"t > Ft > -c and o have the same meaning as in § 1 . 1 5. In particular, L is a separable quadratic extension of and Let E be a finite dimensional vector space o = sup { w(d) I E L, + over L endowed with a nondegenerate hermitian form h relative to -c , and suppose that = SU(h). Then, the building 81 of G over K, which is also, by the way, the building of U(h), can be interpreted as the set .AI"h of all additive norms ..... R U { satisfying the inequalities
K
=
G
+ oo}
w (h , x) ) � 2 w (h , �
cp: E
-o +
(x(x y)) cp(x)cp(x) cp(y) (x(x, yE),E), E v;e; w(x_,) w(x0) i n} (x_m . .. ,xn) cp E tF1 E
E
and maximal with that property (cf. [8, p. 239] for a more general result). Suppose further that = £2n +I and that h is as in I . I 5( I ) . Then, the identification of 81 and %h can be described more explicitly as follows : with the notations of § 1 . 1 5, the mapping A ..... .AI"h which maps .E r=I onto the norm >-+
inf {w(x,) - v; ,
+
V;,
- oi l �
�
extends uniquely to an isomorphism of G(K)-sets 81 -+ %h · A norm E %h is spe cial-i.e., corresponds to a special point of 81-if and only if there is a basis (b;)1,,,n of with respect to which h has the form 1 . 1 5( 1 ) and a constant /E such that, for X; E L,
J. TITS
50 ( I)
�
(� b; ;) = inf{ ( ,.) -J, wx
x
w (x_,.) +f, w (x0) - a
II
� i � n
}
(one can then choose the basis (b,) so that / = or !w(n- 1 )). The norm � is hyper special if L/K is unramified and if there is a basis (b;) of such that ( 1 ) holds for !=
0
0.
E
3. Stabilizers and centralizers. From now on, G is assumed to be connected.
3 . 1 . Notations ; a EN-pair . For every algebraic extension K1 of K with finite ramification index and every subset Q of the building &?J(G, K1), we denote by G(K1)0 the group of all elements of G(K1) fixing Q pointwise. If Q is reduced to a point x, we also write G(K1)x for G(K1)0. Note that if F is a facet of &?J(G, K1) and if x is a point of F "in general position", one has G(K1)F = G(K1)x. The stabilizers G(K)x of special (resp. hyperspecial) points x E fJ?J are called special (resp. hyper special) subgroups of G(K) . We recall that if G is semisimple and simply connected, the group W N(K)jZ(K) coincides with the Weyl group W of the affine root system Waf · As be fore, we set fJ?J &?J(G, K). G(K)x for every facet F of fJ?J and 3 . 1 . 1 . Suppose that W = W. Then G(K)F every x E F. Furthermore, if C is a chamber of A = A (G S), the pair ( G(K)C, N(K)) is a EN-pair (or Tits system : cf [5], [23]) in G(K) with Weyl group W. In that case, the groups G(K)x for x E fJ?J are called the parahoric subgroups of G(K) (cf. [8]), but we shall avoid using that terminology here in order not to prejudge of its most suitable extension to the nonsimply connected case. An alternative con struction of the building fJ?J starting from the above EN-pair (which can be defined independently of the building, as we shall see) and using the parahoric subgroups defined by means of that EN-pair is given in [8, §2] . Let Q be a nonempty subset of the apartment A whose projection on the building of the semisimple part of G (cf. last paragraph of 2. 1 ) is bounded . For any root a E 1/J, let a(a, Q) denote the smallest affine root whose vector part is a and which is positive on Q. Let 1/J' be the set of all nondivisible roots-i.e., all roots a E !/J such that !a ¢ 1/J-and let rJ;' + (resp . rJ;'-) be the set of all nondivisible roots which are positive (resp. negative) with respect to a basis of 1/J, arbitrarily chosen. Set N(K)0 N(K) n G(K)0 and let Zc and Xa be defined as in § § 1 .2 and 1 .4. Then one has the following group-theoretical description of G(K)0 (cf. [8,6 .4.9, 6.4.48, 7.4.4]) : If X±(Q) denotes the group generated by all xa(a, {}) with a E 1/J'±' the product mapping TI aE
=
=
,
=
=
---+
=
·
x+(Q)
·
REDUCTIVE GROUPS OVER LOCAL FIELDS
51
every bounded is contofaianedpoiinntaxmaxi mal one and every maximal bounded subgroupsubgroup is the stxaofbibellG(K) iozngser G(K)" o f to ainfacetpartiofcular,minimspecial dialmsubgroups ension of areG(K)" ismala maxi m al bounded subgroup o f G(K), maxi bounded subgroups. Gselisysemi simabiplliezanders osifmtplheyvertconnect efd,ththeebuimaxildinmgal bounded subgroups ofG(K)conare preci t h e st i c es o t h ey form jugacy classes, where 11 > · · · , 1, denote the relative ranks of the quasi-simplefactors ofG. Various decomposiV tions. C A A(G, A C; B G(K)O is a special subgroup G(K). D G G(K)C; B Iwahori subgroup alG(K)v u+ Y G V A Y v Z(K). Y+ Y D) Z(K)+Z(K)v-l(Y+), Bruhat tion. {BgB G(K) g EBN(G(K)}K)B BnB ..... v(n)3. 3(nn.1.EEN(K) N(K)) decomposi v(n) w, BnB BwB, qw BwB/B d(v) w0(C) rC,1 · ·· r1w0, v, (rl > ··· , r1) r;; d(v):qw d r(v)d(v;) . v BwB/B d(v;), +g(K)U+(K)I g E G(K)} lwasawa+(Kdecomposi tzioEn.Z(K)) G(K) G(K)OZ(K)U G(K)OzU ) .... . v(z) { G(K)O ( Y. Cartan decomposition. 0zG(K)O ..... v(z) (z E Z(K)+) G(K) G(K)OZ(K)G(K)O G(K) {G(K)OgG(K)Oig E G(K)} Y+. G(K) Y+· That algebra is commutG(K)O ative. Some group schemes. fJI.
It is now clear that if
P-1,
From 3. 1 . 1 , it follows that the above two statements give a complete description of the maximal bounded subgroups in the simply connected case : if fJI ; TI �= 1 (I; + 1)
For an analysis of the nonsimply connected case, cf. [8, 3 . 3 . 5]. 3.3. Let be a chamber of S) . We identify with the vector space in such a way that 0 becomes a special point contained in the closure of in particular, of Set R.f · C (a "vector chamber") and = if K is finite or, more generally, if is re sidually quasi-split, and if is simply connected, is an of (cf. §3.7). Let be the group generated by all Ua for which ale-and hence is positive and let be the "intersection of and W ", that is, the group of all translations of contained in W; thus, is the image of by the homomor phism of § 1 .2. Set = n [J (closure of and a subsemi group of One has and the mapping is a bijection of the set I onto W. If and = we also write as usual. If K Fq, the cardinality of (used for instance in [1]) is given by the following formula is a reduced in terms of the integers of § 1 . 8 : set w = where and let be the vertex of L1 repre word in the Coxeter group W and senting then q d with I; �=l In particular, we have another inter d v q r < v > where pretation of q < > denotes the fundamental reflection cor responding to the vertex of L1. More generally, for any K, the quotient has a natural structure of "per fect variety" over K, in the sense of Serre [Publ. Math. I.H.E.S. 7 ( 1 960), 1 .4], and, as such, it is isomorphic to a K-vector space of dimension I; �=l with the above notations. 3 . 3 . 2. One has and the map ping is a bijection of onto 3.3.3. One has = and the map ping is a bijection of onto In particular, we see that if K is finite, the convolution algebra of all functions ..... C with compact support which are hi-invariant under (Heeke al gebra) has a canonical basis indexed by For the proofs and some generalizations of the above results, cf. [8, §4]. 3 .4. The results of this section and the next are special cases of results which will be established in [9] . =
=
=
=
=
=
=
=
=
=
=
5
J.
2
TITS
It is well known that the maxi mal bounded subgroups of SL,.(K) are the group
SL,.(o) and its conjugates under GL,.(K). It is natural to ask whether, more gener
ally, the maximal bounded subgroups of G(K) can always be interpreted as the groups of units of some naturally defined a-structures on A positive answer is provided by the statement below. In this section, we denote by • • the derived group of and by pr•• the canonical projecti on •• , (cf. the last paragraph of 2. 1 ) for any pr ••
G. G 3. 4 .1 G f!4(G, K1) ..... f!4(G K1) K • 1 3.4.1. IfthQereisisaanonempt yaffinesubsetgroupof anscheme<§ apartment0 overof f!4 uniwhoseque upprojecti oqnue isomor (.Q) is bounded, smoot h t o uni phitshm,e generi with tchfiber efoll<§owingopropert ies: f <§ 0 i s G; for everyis equalunrami.fi ed 1Gal)0 (cofis extensiwhere on KQ1 oifs Kidentwitifih eridngwio.thf iinttsegers tihmeagegroupin t o G(K canoni c al the building<§!!4(G, K1) (cf. . If G isleyspligroupt, theschemes group schemes <§"cfiber associG.ated to the special points x of f!4 are3.3.the44..32.Cheval wi t h generi LetasKabove ebed Galthe ocanoni is extecnsial oimn ageof KofwiQthinrin!!4g1 of inf!4te(G,gersK1) let 1 be anandunrami.fi f!4 be l e t 0 1 Then <§01let<§is thbee group scheme overscheme over deducedwitfrom <§0cfiber by change ofsuppose base. Con versely, a smoot h group h generi G and that,!!41 <§ becomes a group scheme <§0 with 0 byas ichange o f base from t o 1 n 3.4.1; hence then can(Qbe1) isidentstableifiedbywith a subset andof if it isK)poi(cnf.twise fixedwhoseby inverseQ' image by in f!4 we denote by Q, one hasQ,<§ <§0• G(K1)0 ..... G(K1)0', K1 3.4.1, <§0 -> 0 <§0 0 0 is surjective. 3.3.45 ..4Reduct . The reducti o n homomor p hi s m <§ ( G( Q 0 ion Q A(G, <§0, 0 o,
o, K
oK1,
3. 1), 2.6). Clearly, 0 depends only on the closed convex hull of pr•• (O) . The following two statements are easy consequences of the definitions.
<§o(oK)
Q
c
oK 1 ,
(2.6).
=
oK 1
o
1
oK1,
o
pr••
Gal(KdK),
c
f!4(G •• , 2.6. 1 ) pr •• If is any nonempty subset of the closure of the inclusion homomorphisms for as in define a morphism of group schemes <§0, which we denote by Po·o· We represent by G and Po·o the algebraic group defined over K and the K-homomorphism obtained from and Po · o by re duction mod !J. 0 o) K) -> G (K) mod !J. Let be as in 3 .4. Our next purpose is to investigate the group G . We assume, without loss of generality, that c S). Then, the well defined split torus scheme whose generic fiber is S is a closed subscheme of and its reduction mod !J , called S, is a maximal K-split torus of G . The character group of S is canonically isomorphic with the character group X * of S and will be identified with it ; similarly, we identify the cocharacter group of S with X* . The neutral component G0 of G possesses a unique Levi subgroup containing S, which we denote by GB'd ; it is defined over K. We suppose convex. Let F be a facet meeting and of maximal dimension with that property. Since F satisfies all the conditions imposed on the reductive group Grpd is defined and it also contains S. One shows that S Glfd -> Grpd ; if F c 0, that is nothing else but the restriction of PoF to G&•d. In the sequel, we shall be mainly concerned with the group Grpd. The notion of is usually defined for split groups (cf. [1 1 , XX, 2.8], [20, § 1 ]). In view of the next statement, we extend it as follows to
Gal(Ki fK),
=
=
Q
Q
Q,ty map of onto itselfextends uniquely to t h e i d enti an isomorphism coroot associated with a root a
REDUCTIVE GROUPS OVER LOCA L FIELDS
53
arbitrary reductive groups : if 2a is not a root, we simply take the coroot associated with a in the split subgroup of maximal rank defined in [3, §7] ; if 2a is a root, we define the coroot associated with a as being twice the coroot associated with 2a (X* being written additively). 3 . 5 . 1 . The root system of G'fd with respect to S is the system f/JF (cf 1 .9) ; in partic ular, its Dynkin diagram is obtained from the local Dynkin diagram Ll(G, K) by deleting the vertices belonging to IF (cf. 1 . 9) and all edges containing such vertices. The coroot associated with a root a E f/JF is the same for Grpd as for G. If Da denotes the unipotent subgroup of G"'fd corresponding to a, the group Da(K) is nothing else but the group Xa of § 1 .4, where a is the affine root vanishing on F and whose vector part is a. Applying that to the unramified closure of K, one gets the following immediate con sequence. 3 . 5.2. The index of Gr;.d over K, in the sense of [3) and [22], is obtained from the local index of G by deleting from L1 1 all vertices belonging to the orbits O(v) with v E IF (the notations are those of § 1 . 1 1 ) and all edges containing such vertices. In particular, �f G is residually quasi-split (resp. residually split), Grpd is quasi-split (resp. split). When F is a chamber, then G is residually quasi-split (resp. residually split) if and only ifG'fd is a torus (resp. a split torus). If G is simply connected, the group G0 is connected. In general, the group of com ponents of G0 is easily computed when one knows the group 31 = E(G, K1 ) (cf. §2.5), where K1 is the maximal unramified extension of K. Here, we shall give the result only in the case of a facet. 3 . 5 . 3 . The group of components of GF is canonically isomorphic with the intersec tion of the stabilizers of the orbits O(v) with v E IF in the group 31 • A component is defined over K if and only if the corresponding element of 31 is centralized by Gal(KdK). lf K is finite, every component of GF which is defined over K has a K rationa/ point (by Lang's theorem). The groups Gr;d give an insight into the geometry of the building through the following statement : 3 . 5.4. The link of F in is canonically isomorphic with the spherical building of Gr;d over K, i.e. the "building of K-parabolic subgroups" ofG"'fd (cf [23, 5.2]). The groups Gr;d also provide an alternative definition of the integers d(v) of § 1 . 8 . Suppose F is of codimension one and let v be the complement of IF in the set of all vertices of Ll. Then, Gr;d has semisimple K-rank 1 and d(v) is the dimension of its maximal unipotent subgroups, or, equivalently, the dimension of the variety G�/ P F• where PF is a minimal K-parabolic subgroup of G�, the neutral component of GF. This, together with 3 . 5.4, implies the interpretation of d(v) given in 2.4. If G is residually split, G�fPF is a projective line, hence d(v) = 1 ; in particular, we re cover the last statement of § 1 . 8 . While 3.5.2 gives a n easy algorithm t o determine the type o f Gr;d, 3.5. 1 , applied to the unramified closure of K, actually provides the absolute isomorphism class of that group. Here is an immediate application of that. Suppose that G is quasi simple, simply connected and residually split and that F is a special point. Then, Gr;d is a simply connected quasi-simple group except if the local Dynkin diagram is the following one : (I ) *
B
J. TITS
54
and if IF is the vertex marked with a * · Indeed, it is readily verified that in all other cases, (/JF contains all nonmultipliable relative roots of G, and the assertion follows from 2.23 and 4.3]. In the exceptional case above, Grpd is a special orthogonal group, hence not simply connected. Using the fact that in a simply connected group the derived group of the centralizer of a torus is also simply connected, one easily deduces from the preceding result the following more general one. Let us say that a special vertex of the absolute local Dynkin diagram L11 is good if it is not the vertex * of a connected component of type ( 1 ) of that diagram. Then ifG is semisimple and simply connected and if U vEI F O(v) contains a good special vertex out of each connected component of L1 1 > the derived group of G'}d is simply connected. 3.6. Fixed points of groups of units of tori. Let M be a subgroup of the group of units Sc = { s E S(K) I w (x(s)) = 0 for all X E X * } of S. We wish to find under which condition the apartment A = A(G, S) is the full fixed point set �M of M in �. From the properties of the building recalled in §2.2, one deduces that A = � M if and only if, for every facet F of A of codimension one, the only chambers con taining F in their closure and fixed by M are the two chambers of A with those pro perties. By 3 . 5 .4, that means that the image M of M in S(K) has only two fixed points in the spherical building of Grpd over K. If a is any one of the two nondivi sible roots in C/Jp, that condition amounts to a(M) =f. { 1 } . Thus, we conclude that : 3.6. 1 . A necessary and sufficient condition for A to be the full fixed point set of M in � is that a(M) ct. I + p for every relative root a E C/J. In particular, if K has at least four elements (resp. if K � F2) A is always (resp. never) the full fixed point set of Sc in B. The preceding discussion also gives information on the fixed point set of the group of units S1.c of a nonsplit torus S1 which becomes maximal split over an unramified Galois extension K1 of K: one applies 3.6. 1 to the action of S1 .c on � (G, K1 ) and one goes down to � by Galois descent, using 2.6. 1 . In that way, one gets the following result for instance : If S 1 is an anisotropic torus which becomes maximal split over an unramified Galois extension of K, then S 1 (K) has a unique fixed point in the building �. By contrast, it is easily shown that if S1 is a maximal torus of G = SL2 whose splitting field is ramified, then S1(K) necessarily fixes a chamber of � and possibily more than one 2 ; for a similar torus S 1 in PGL2, S1(K) may have a single fixed point in � and may have more than one. 3.7. lwahori subgroups ; volume of maximal compact subgroups. In this section, we suppose G residually quasi-split ; remember that that is no assumption if the residue field K is finite ( 1 . 10.3) . To every chamber C o f the building �. w e associate a s follows a subgroup Iw( C) of G(K), called the lwahori subgroup corresponding to if G� denotes the neutral component of the algebraic group Gc (cf. 3.4), Iw( C) is the inverse i mage in �c(o) = G(K)C of the group G�(K) under the reduction homomorphism �c(o) ---> Gc(K). Clearly, all lwahori subgroups of G(K) are conjugate. From 3 . 5.2, it follows that G� is a solvable group, hence is the semidirect product of a torus t by a uni-
[4,
C:
2
This answers a question of G. Lusztig.
REDUCTIVE GROUPS OVER LOCAL FIELDS
55
potent group D. By general results on smooth group schemes, it follows that Iw(C) is the semidirect product of f(K) by a pronilpotent group Iwu(C) ; if K has finite characteristic p, I wuC C) is a pro-p-group, and if K is finite, f(K) is of course a finite group, of order prime to p. If x is a point of the closure of C, the i mage of G� by the homomorphism [ixc : Gc --> Gx is a Borel subgroup B of Gx, and the kernel of Pxc is a connected uni potent group. It follows that Pxc maps G �(K) onto B(K) and consequently, by 3.4.4, that Iw(C) is the inverse image of B(K) in G(K) x = �xCo) under the reduc tion homomorphism. Thus, the Iwahori subgroups of G(K) can also be defined as the inverse images in the stabilizers G(K) x , for x E :JB, of the K-Borel subgroups of the reductions Gx . Now, suppose that K is finite. Then, G(K) is a unimodular locally compact group of which the Iwahori subgroups are compact open subgroups. Therefore, there is a unique Haar measure fl. for which the Iwahori subgroups have volume I . From the above, it follows that, for any x E :JB , the volume of G(K) x with respect to fl. i s the index [ Gx(K) : B(K)] where B is any K-Borel subgroup o f Gx. If x is "in general position" in the facet F of :JB containing it, one has Gx = GF and the assertions 3.5.2 and 3 . 5 . 3 provide an effective way of computing that volume knowing the local index of G (together with the correspondence v >-> O(v) of 1 . 1 1 ) , the set lx of vertices of L1 and the group S(G, K1) (cf. 2.5), where K1 is the unramified closure of K. 3 . 8 . Hyperspecial points and subgroups. From 3 . 5 . 1 and 3 . 5 . 3, one easily deduces the following characterization of the hyperspecial points of :JB defined in § 1 . 1 0 :
81
=
3 . 8 . 1 . A poin t x of :JB is hyperspecial if and only if the neutral component of the group Gx is reductive, in which case Gx itself is connected and hence reductive.
One can also show that the schemes �x correspond ing to the hyperspecial points x are the only smooth group schemes over o with generic fiber G and reductive reduction. Thus, the hyperspecial subgroups of G(K) can be characterized as the groups of units of such group schemes. 3 . 8 . 2 . Suppose that K is .finite and that G(K) possesses hyperspecial subgroups
(a
condition satisfied, for instance, if G is quasi-split and has an unramified splitting field : cf 1 . 1 0) ; then, the hyperspecial subgroups of G(K) are among all compact subgroups of G(K), those whose volume is maximum . The proof, using § § 3 . 5 and 3 . 7 , i s not difficult. 3 . 9 . The global case. Let L be a global field . For every finite extension L' of L and every place v of L', we denote by ou (resp. 00) the ring of integers of L' (resp. of the completion L�). Let H be a reductive linear group defined over L. We suppose H embedded in the general linear group GL n and, for every L' and v as above, we set H(ov) H(L�) n GL n (o0). Another way of viewing that group consists in con sidering the ocgroup scheme structure .Yt'0L "on H" defined by the standard lattice oz in Ln-in more precise terms, .Yt'0L is the schematic closure of H in the standard general linear group scheme 'fl.!En, oL- ; then H(o 0 ) .Yt'0L( o 0) . For any ring R con taining oL, we denote by .Yt'R the group scheme over R deduced from .Yt'0L by change =
=
of base.
3 . 9 . 1 . A t almost allfinite places v of L, .Yt'o is the group scheme .Yt'x associated with , a hyperspecial point x of the building :JB(H, Lv) ; hence H(o0) is a hyperspecial sub group of H(L0).
J. TITS
56
Indeed, let L' be a Galois extension of L over which the group H splits, and let
Jll'�L ' be a Chevalley group scheme over o u with generic fiber Jll'u . Since the group schemes Jll' " L ' and Jll'�L ' have the same generic fiber, they "coincide" at almost all
places of L'. Since almost all places of L' are unramified over their restrictions to L, the assertion now follows from 3.4.2, 3.4.3 and 2 . 6. 1 . 3 . 1 0. Examp e . Genera/ linear groups. Suppose that G GLn. The Iwahori sub groups of GLn(K) are the subgroups conjugated to
l
=
B = { ((g1)) l g11 E o x , g1i E o for i
<
j and g1i E p for i � j } . n Let (b.-h,g;; ; ,;n b e the canonical basis o f K . F o r 1 � r � n, let A , b e the lattice i n Kn generated b y { b J n: l i � r } U {b,.j i > r } and l e t P, b e the stabilizer o f A, in GLn(K). Thus, P, is the group of all matrices whose determinant is a unit and which have the following form
(
)
r n - r r o 11:- 1 0 n - r no -0- '
I
where the notation means that the upper left corner i s an r x r matrix with coef ficients i n o, the upper right corner an r x (n - r ) matrix with coefficients i n 11:- 1 0, etc. The group B i s the centrali zer i n GLn(K) o f the chamber described in § 1 . 1 4. The subgroups P, are special and every special subgroup i s conjugate to any one of them. The P /s are the stabilizers of the points of (JJJ contained in a one-di mensional facet of with the notati ons of §2.9, the points fixed by P, correspond to the norms of the form
C
C;
(xl >
···,
xn) ,..... inf ({ w (x,.) + w (11:) -
el i �
r } U {w (x,.)
-e i >
l
r} )
for some constant c E R. If v i s any such point, the scheme �. i s the Chevalley scheme "on" GLn defined by the lattice A,. One can describe the scheme �c. whose group of units is B, by embedding G Ln i n GLn 2 by means of the sum of n times the standard representation, and considering in Kn 2 the lattice A 1 EE> • • • ® A n. Note that B i s the stabilizer of any point of The corresponding statement for G = PGLn is not true. For instance, the i m age i n PGL n(K) of the group generated by B and by the l inear transfo.-mation
C.
(1 )
is the stabilizer of the "center of gravity" y of the chamber of (JJJ (PGLm K) project ing That group is also a maximal bounded subgroup of PGLn(K). The scheme �Y can be described by means of a lattice in the Lie algebra of G P G Ln on which G acts by the adjoint representation. If F denotes the cyclic group of order n generated by the reduction mod p of the i mage of ( l ) i n PGLm the group Gy i s the semidirect product of F by a connected grou p ; in particular, its group of com ponents is cyclic of order n. 3 . 1 1 . Example. Quasi-split special unitary group in odd dimension. We take over all hypotheses and notati ons from 1 . 1 5 and denote by oL the ring of i ntegers of L. o (we recall that o is de Let ). E L be such that ). + ).< + l 0 and that w (il) fined as sup { w (d) I d E L, d + d• + I 0}). We suppose the uniformizing element 11: 1 chosen in such a way that (i\11:1) + (i\11:1)• 0 : if L/K is unramified the pos-
C.
=
=
=
=
=
REDUCTIVE GROUPS OVER LOCAL FIELDS
57
sibility of such a choice is obvious and if LfK is ramified it follows from 1 . 1 5(6). let (b;)-n & i & n be the canonical basis of LZn+I . For 0 � r � n, we consider the basis (bfrl ) _n & r & n defined by hfrl = b ,./71:1 for i < - r, b,(r) = b,. for - r � i � 0 and b,(r) = ).b,. for i > 0, and we denote by A, the oclattice generated by that basis. Note that if o = w ( ).) = 0, which is always the case except if L/K is ramified and char K = 2, A, is also generated by the basis {b ,./11:1 I i < - r} U {b, l i � - r } . The stabilizer P , o f A, in is also the stabilizer o f the point v , o f V = A c &a (with the conventions of 1 . 1 5) determined by if i � r, a ,. ( v , ) = -t o = H o + w (11: 1 )) if i > r.
G(K)
The points v , are the vertices of the chamber defined by the inequalities -to < a1 < < a n < to + !w(7r 1 ) ; they correspond, by v ,_. I., to the vertices of the dia grams (9) and ( 1 0) of 1 . 1 5 in the natural order, from left to right. The scheme '§., is the ocstructure on defined by the lattice A,. We shall now briefly investigate the algebraic group G., obtained from '§., by reduction mod p. We choose r once and for all and use primed letters to designate the coordinates with respect to the basis (b,('l ) . With those coordinates, the her mitian form h is given by r h((x,.) , (Yi) ) - Xo Yo + L..J ( A x,. Y-i + AX_,. Yi) i= l 'T {!--, (x,.' Gl(E) is a monomorphism ; in other words G., can be viewed as a subgroup of Gl(E). From thi s point on, we must treat separately the unramified and the ramified case. First case. L/K is unram(fied. Then, E is also a vector space over the residue field L of L. By reduction mod 'lr j , the antihermitian form 11: 1 hj).< becomes the antiher mitian form h 1 : ((x;), (y;)) ,_. 1; 7=r+ l (x/y'__ ,. - x'!,. y;) in E, with obvious nota tion. Let E0 be the kernel of that form, defined by the equations x; = 0 for I i i > r, and let A,.o be the inverse image of E0 in A. We now consider the restriction of h to Ar.o x Ar. o which, by reduction mod p , becomes the hermitian form · · ·
G
' ' ' < ' " , < '< ' , '< ' ).< '
in E0• Finally, G., can be described as the stabilizer of the pair (hi> h2) in the group RL!K (SLL(E)) , that is, the special linear group of the L -vector space E "considered as an algebraic group over K" by restriction of scalars. Le� E1 be the subspace of E defined by the equations x;. = 0 for - r � i � r, and let lz 1 denote the restriction of t�e antihermitian form h 1 to E1 x E1 • Then, G., clearly contains the group SU(lz1) x SU(h2), which is nothing else but its Levi subgroup G��d (cf. 3 . 5). Observe that, in conformity with 3 . 5 . 1 , the diagram obtained from the diagram (9) of 1 . 1 5 by deleting its (r + I)st vertex and the adjoining edges is a diagram of type BC, x Cn- n which is indeed the type of the relative root system of S U(h 1 ) x SU(h2). Second case. L/ K is ramified. Then, the scalar multiplication by 11:1 in A" reduced mod 11:, provides an endomorphism ).! : E --> E, obviously centralized by G.,, and
58
J. TITS
whose kernel is equal to its image ))( £) . The quotient E = Ej )) ( E) is canonically isomorphic with the quotient A ,jn1A, and will be identified with it. From the fact that )) is centralized by G. , it follows that the projection E -> E induces a homo morphism of K-algebraic groups G. -> GL(E ) whose kernel is unipotent. Here, we shall only describe the image G., of that homomorphism, leaving as an exercise the determination of the full structure of G.,. By reduction mod nb the antihermitian form n 1 h/A� becomes the alternating form lz1 : ((.x;), {j;;)} >--+ .E 7=r+ l (x; y'_,. - x'_,. y;.) in E. Let E0 be the kernel of that form, defined by the equations .x; = 0 for I i i > r, and let A,,0 be the inverse image of E0 in A ,. We now consider the function q : A ,,0 -> K defined by q(x) = h(x, x) . By reduction, it becomes the quadratic form q : E0 -> K given by T
-
q-c<X- ;' ) -r &,i'i£ r )
= X- o· 2
T
- .t...J "'
i= l
X _ ; X; . _,
_,
Finally, the group G. is the group of all elements of SL(E ) stabilizing lz1 and induc ing on E0 an element of the (reduced) group SO( q). Let E1 be the subspace of E defined by the equations .x; = 0 for - r < i < r and let h1 denote the restriction of h1 to E1 x E1 • Then Sp(h 1 ) x SO( q) is a Levi subgroup of G.,, which is the iso morphic image in that group of the Levi subgroup G�;d of G.,. As in the unramified case, we can test the statement 3 . 5 . 1 , this time by using the diagram ( 1 0) of 1 . 1 5 x Cn -r· which provides, for G�;d , a root system of type Note that, also in the unramified case, we could have, instead of the restriction of h to A, , 0 x A ,,0, considered its "contraction" q : A ,,0 -> K defined by q(x) = h(x, x), thus making the treatment of the two cases still more similar. On the other hand, we have introduced A in order to reduce the case distinction to a minimum ; in the unramified case, as well as if char K =P 2, we could have replaced A by 1 every where, thus simplifying the equations somewhat. 3. 1 2. Example. Quasi-split but nonsplit special orthogonal group. Now, we take over the hypotheses and notations of § 1 . 1 6 except that we take for the special orthogonal group SO(q). We shall not, as in §3 . 1 1 , treat that example in any systematic way. Our only aim here is to give an example of a vertex v of the building such that G. is not connected. We suppose that L/K is unramified. Let A be the lattice o n EB noL EB o n in Kn EB L EB Kn where oL is the ring of integers of L. The stabilizer P of A in G(K) is also the stabilizer of the point v E V c A defined by a,. ( v) tw(n) for 1 � i � n, which is a vertex of the chamber described in § 1 . 1 6 . I n the diagram ( 1 ) o f § 1 . 1 6 , I. is the vertex a t the extreme left. A s i n §3. 1 1 , one can describe G. as a subgroup of GL(E), where E is the K-vector space A/nA. By reduction, the form q , restricted to A , becomes the quadratic form ij_ : (x;)-n'i£i'i£n ....... 1:: 7=1 x_;.X; in .£, with obvious notations (.X; belongs to the residue field of L if i = 0 and to K otherwise). Let be the two-dimensional kernel of q defined by the equations x,. 0 for i =P 0, and let q be the quadratic form in E = i}f.£0 image of q . Clearly, G. preserves the form q . Therefore, the projection E -> E induces a K-homomorphism G. -> O(ij). One verifies that that homomorphism is surjective (in particular, G. is not connected) and that it maps G�·d isomorphically onto SO(q). If 1: denotes the nontrivial K-automorphism of L, the linear transformation (x;)-n'i£i'i£n >--+ (xm X-n+ b . . · , X_t . x6, Xt > . . . , Xn-b x_n), which belongs to P, provides by reduction an element of G.(K) which is mapped into O(ij) but not into SO(ij).
B,
G
=
=
.£0
REDUCTIVE GROUPS OVER LOCAL FIELDS
59
4. Classification. 4. 1 . Introduction. To finish with, we give the classification of simple groups in the case where the residue field K is finite, which will be assumed from now on. We recall that, in the characteristic zero case, that classification has been given first by M. Kneser [16] . The tables 4.2 and 4.3, together with the comments in §4.5, provide a list of all central isogeny classes of absolutely quasi-simple groups over K. For each type of group, they give the following information, where K1 denotes the unramified closure of K: a name of the shape a X where the symbol X represents the absolute local Dynkin diagram L1 1 ( 1 . 1 1 ) with the notations of [8, 1 .4.6]-except that our C-BC cor responds to the C-BCIII of [8]-and where a is the order of the automorphism group of L1 1 induced by Gal(K1 /K) ; for residually split groups, a 1 and the superscript a is omitted from the notation ; note that the index on the right of X is the relative rank over Kh hence equal to the number of vertices of L1 1 minus one ; primes, double primes, etc. are used to distinguish types of groups which would otherwise have the same name ; the symbol representing the affine root system (or echelonnage) in the notations of [8] ; in the residually split case, that symbol coincides with the name of the type and is not given separately ; note that the right part of the symbol gives the type of the relative root system (/J and, in particular, that the index on the right of it is the relative rank over K, hence equal to the number of vertices of the relative local Dynkin diagram L1 minus one ; the local index (§ 1 . 1 1), the relative local Dynkin diagram L1 (§ 1 . 8) and the integers d(v) attached to its vertices (§ 1 8) ; the action of Gal(K1 /K) on L1 1 through a cyclic group of order a (see above)-is essentially characterized by its orbits in the set of vertices of Llb orbits which are exhibited as follows : the elements of the orbit O(v) corresponding to a vertex v of L1 ( 1 . 1 1 ) are placed close together on the same vertical line as v (in the few cases, such as ZDm ZD�m• etc., where two vertices of L1 are on the same vertical, the correspondence v ...... O(v) should be clear from the way the diagrams are drawn) ; since K is finite is residually quasi-split ( 1 . 1 0.3), hence all vertices of L1 1 are distinguished except for the unique anisotropic type dA d-l (§ 1 . 1 1 ), and there is no need for a special notation like the circling of orbits, as in [22] ; hyperspecial vertices (§ 1 . 1 0) are marked with an hs and the other special vertices (§ 1 .9) with an s ; the index o f the form, i n the "usual" sense o f [3] and [22] ; for simplicity, we do not represent that index by a picture but rather by the corresponding symbol in the notation of [22] ; we recall that that symbol carries, among other, the following in formation : the absolute type of the group, the absolute rank, the relative rank (already provided by the symbol representing r/J.1) and the order of the auto morphism group of the ordinary Dynkin diagram induced by the Galois group of the separable closure of K. In the case of the inner forms of Am the diagrams are, for technical reasons, re placed by explanations in words. =
.
G
60
J. TITS
4.2. Residually split groups. Local Dynkin diagram
Name
Index [22]
A cycle of length n + 1 all vertices of which are hyperspecial
I
Z I
• - --+--< :
B-Cn (n G; 3)
1= 1 ::; ;:.::::::r --11 - u - - I
hs
C-Bn (n G; 2)
C-BCn (n G; 2)
C-BC1
I s
; I
I s
Z I
1--
- -
1---
cn<,Dn
- t-----1:::+::::1 s
- -t-----1:�::::1 s
� s s
2A Cn1 l n Z ,
2A C 1J 2. 1
REDUCTIVE GROUPS OVER LOCAL FIELDS Na m e Dn
(n ;:::;; 4 )
� �
Index [22]
Local Dynkin di agram
h•
- - - - ---
hs
61
h· hs
lD Cl l
n, n
hs hs
£6
1 £0
6, 6
hs
£7
I
I
hs
I
hs
I
hs
£�. 7
EH. s
F�. 4
hs
2£2 6, 4
s
I
hs
I
s
t:::::)::4
I
z I
G2 , 2
62
J. TITS
4 . 3. Nonresidua/ly split groups. Name
(m
Affine root system (notations of [8, 1 .4.6])
Local index and relative local Dynkin diagram .
Index [22]
The absolute local Dynkin diagram .d, is a cycle of length md on which Gai(K, /K) acts through a cyclic group of order d generated by a rotation of the cycle. The relative diagram is a cycle of length m all vertices of which are special but not hyperspecial and carry the number d.
� 3, d � 2)
.d , is as above, with m
0
2
d
L1 is
(d � 2)
=
d
s
1 A �:!_1, 1
s
.d , is as above, with m 1 , or, i f d 2, consists of a fat segment whose vertices are permuted by Gai(K,/ K ) and .d 0 =
=
=
(d � 2)
__.-+-----! - - - - - - � � - - - - --�
(m
� 2)
ZA ;m- 1
(m
� 3)
1 A f:J-l, m-l
2
2
2
2
:
:
:
:
1 A �'!!1 , o
I1l m 2A 2'm-l,
1==>=::}----i - - - - - - t-----t=<;::::!
hs
C-BC�- 1
c:
3 X i· -::: s
2
-T
. _ _ _ __ _
2
21 - - - - - - 2
c:
>
hs
__,
"7
:)
3 s
2A�!?- 1, m-1
X
ZA 3,OJ1
63
REDUCTIVE GROUPS OVER LOCAL FIELDS
Name
2A � m
(m
Affine root system (notations of [8, 1 .4.6])
c:
C-BC�
� 2)
C-B,.-1
21 - - - ---12
3
2 I
:::
1--- - - 1
I z
2
1-- - - - 1
I
I
I
B ,.,,._l
I I
;
s
�---- -
C-BC!�1
2
(n � 3)
I
C-BC�'!... 1 i
:
c:
3
X I
s
0
1l n-1 2A �n-1,
1------1
I
�
s
2 I
:
______
I
,
21 · - - - - - 2
<;9 x .......,.
s
2A� �. m
2A�,1l1
>I (
I ndex [22}
hs
hs
s
I
C-B Cf V
- --
X 1-4--1
s
� 3)
:------=> -- -
�
(n � 3)
2 C3
Z I
s
2B, .
(m
2
3
C-B Cf V
2 C2m-l
:
X I
2A ;
2B-C,.
Local index and relative local Dynkin diagram.
s
(
I
I
s
J..
( I
Ci;l-1, m-1
2 ......2.
s
z> q, 1
64
J. TITS
Name
z c2m
(m
Affine root system (notations of [8, 1 .4.6])
..::::::- -----1
em
� 2)
z cz
Local index and relative local Dynkin diagram.
------
I ) s
2 I
I - -- - - 1
(m � 3) 2 C-B3
c:
3
X I
s
:
2
: I
(m
� 2)
2 C-Bz
I s
c2,
s
:--- - - I
I
=
I
2
2
_...._
2
1- - - - -
2
2 I
2
zn�mzJ m-
1
s
2DJ,2l1
s
�----I ;
2
3 2 x ....-
s
C-BCl1 1
{
c:::
C-BCl
C-BCM 1
I
2
�
s
2 C-Bzm
: I
Z: I
2
s
C-BC�- 1
2
2
I
I
<
AI
2 C-Bzm-i
1
I ndex (22]
2
I -- - - - 1
<
2
� s s
_...._
2l m 2D£m+1,
_...._
2
I :
2 I
s
zn 3,
REDUCTIVE GROUPS OVER
Name
2D n
(n
� 4)
2D �
(n
� 4)
2D�m
Affine root system (notations of [8, 1 .4.6])
Bn- 1
Local index and relative local Dynkin diagram.
>-----. - -- -2
�
zm��-1
� ::
-<:=:- -- - --� - - --
....
2
� m��-2
s
s
B-Cm
I ndex [22]
� --::::::=--2 2 - - --
� 3)
(m
1-----
:: I
I
C-Bn-2
65
LOCAL FIELDS
1D�m, 2l m
2
�
2
_ _ _ _ __
-�
s
2D �m+l
(m
B-BCm
� 3)
c: 3
X I
:--- ---- -
:
2D£2m+l, > m
2
f-----L-( :
2
c I
2
2D �
�
B-BC23
3 D4
Gz
I I
hs
2m,2)2 2
<s
3
X
I
S
2
I� 3
r-:r-1
'This " echelonnage" is missing in the table of [8, p. 29) .
3 D 4, 2
.
66
J. TITS
Affine root
system Name
4D m 2
(m G:
[8, 1 .4.6])
---
3)
s
C-BCfl'
4D m+ l 2
C-BC!,._1
4fl s
<=:----- �
C-BC!J!_1
4D 4
(m G:
Local index and relative local Dynkin diagram .
(notations o f
3)
..:.. ...,...
2
2
---
- --
�
s
� �s s c: X
3
s
: �
2
2fl4,(2)1
:- - - --+-� � 2 2 2 ....., 4 - - -s - - - - --
--
r n
5, 1
x ---.c
s
2£ 6
3£6
2£7
F4 G�
Fl
l D�� +l, m-1
...,...
C2 3 4
C-BCf
2 D��. m- 1
2 _, 4 .,.
2
--
Index [22]
s
� 2
.....
hs
2
<:£G 3
I ) I
1£6,1 62
3I s
� 2
2£6,2 4
2
2
s
£�. 4
67
REDUCTIVE GROUPS OVER LOCAL FIELDS
Interpretation.
4.4. We shall now repeat the classification in classical terms. The following enumeration gives a representative of every central isogeny class of absolutely quasi-simple groups over K and, in each case, the name of the cor responding type, with the notations of the first column of the Tables 4.2 and 4. 3. SLm of a d 2-dimensional central division K-algebra d The type is Am d -1 · � of a hermitian form in variables (r � 3) with Witt index over a quadratic extension L of K. If L/ K is ramified, we assume # 4 because the case = 4 is more adequately represented by an ordinary special orthogonal group in 6 variables or a quaternionic special orthogonal group in 3 variables according as = or I . If the form is split, the type is 2 A�_ 1 if L/K is unramified and = + 1) or = 2n) otherwise. If h is not split, one has = + 2 and the type is according as L/K is un or ramified or ramified. of a quadratic form in variables � 6) with Witt index over K. If is even, we denote by L the center of the even Clifford algebra of which is isomorphic to K EB K (form q of discriminant one or Arf invariant zero) or is a quadratic extension of K. If L/ K is unramified (in particular if L = K EB K), we assume # 6 because the case = 6 is more adequately re presented by a special unitary group in 4 variables. If = 2r' (resp. 2r ' + I ), the type is (resp. If = + L is a quadratic extension of K and the type is 2 r ' l or according as L/K i s unramified or ramified. Finally, if r = + 3 (resp. ' + 4), the type is (resp. 2 � ' +2 The in � 4 variables is of type of a hermitian form in variables � re lative to the standard involution. The Witt index is always maximal and the type is zc,. of a a-quadratic form in variables � 3) relative to an involution a of the quaternion algebra whose space of sym metric elements is 3-dimensional (cf. [21], [23] ; if char K 1= a-quadratic amounts to a-hermitian and the group is also the group of an form relative to the standard involution). Let be the Witt index of the form and L the center of its "even" Clifford algebra Cl(q) (cf. [21]). If LjK is unramified (in particular if L � K EB K), we assume ¥- 3 because the case r 3 is more ade quately represented by a special unitary group in 4 variables; if furthermore = 2 ' , we also assume r # 4 because the case = 2 ' = 4 is more adequately re presented, through the triality principle, by an ordinary special orthogonal group in 8 variables with Witt index One always has � � + 3. If = If = + 1 (resp. + 2), one has L � K EB K and the group is of type or according L is a quadratic extension of K and the type is 2D; (resp. as L/K is unramified or ramified. If + 3, one has L � K EB K and the type is Let L denote the splitting field, which is a cyclic extension of degree 3 or a Galois extension of degree 6 with Galois group @53 . If L/K is unra mified (hence cyclic of degree 3), the type is 3D, ; otherwise, it is The type has the same name G2, F4, E6, E7 or E8 as the absolute type of the group.
Speci2).al linear group (mdSpeci SU(h) h r r'al unitary group r r r'n (r 2 2n B-Cn (rh CB C 2A;_1 2B-C,,+I r 2r' Special ortr'hogonal groupr SO(q) q r (r q r rr 2r' D +2rD,, C-B,, B,,) . r 2B,,2r'I 2, D ). + sympl e cti c group 2n Cn. r (r 2) Special unitary group quaternion r (r Special orthogonal group SO(q) quaternionic 2, specir'al unitary antihermitian r r r r r 2. 2D;. 2r'r 4D,)r2r' 2r'2C-B,_ 2r'r 2r', 1 r 2r' 4D,. Quasi-split triality D4. G�. Split exceptional groups. =
=
68
J. TITS
Quasi-split groups of type E6• The type is 2£6 or Fl according as the quadratic splitting field L is unramified or ramified. Nonquasi-split groups of type E6 and E7• They are the forms of E6 and E7 con structed by means of a central division algebra of dimension 9 and 4 respectively ; their types are 3E6 and 2E7• 4.5. Invariants. All types of groups listed in the Tables 4.2, 4.3 exist over an arbi trary field with finite residue field. The central isogeny class corresponding to a given name is always unique except in the following cases. The isogeny classes of type dA md-l for d � 5 are classified by the pairs of op posite central division algebras of dimension d2 over their number is therefore !q:J(d), where qJ is the Euler function. The isogeny classes of the types B-Cm 2B-Cm C-Bm 2C-Bm C-BCn and Fl are classified by the ramified quadratic extensions of namely the extension always called L in §4.4. The groups of type G2 are classified by the Galois extensions L of which are either cyclic of degree 3 or noncyclic of degree 6. 4.6. The classification kit. The following experimental facts provide a handy way of reconstructing the classification. First note that, except for dA d-!> each type in the Tables 4.2 and 4.3 is completely characterized by the local Dynkin diagram and the integers d(v) attached. Now, consider a connected Coxeter diagram of affine type and rank (number of vertices) at least three, attach an integer to all vertices, mark some of them (possibly none) with a cross and orient each double or triple link with an arrow. Then a necessary and sufficient condition for the existence of a semisimple group G having the resulting diagram as its relative local Dynkin diagram with the given integers as d(v) is that all subdiagrams formed by the pairs of vertices belong to one of the following types, representing the ordinary Dynkin diagrams of quasi-split groups of relative rank two :
K
K;
K,
d
X
d
d'
K
d d �
X
d
�
3
1-------t
The group G can furthermore be chosen to be absolutely quasi-simple if and only if the integers d(v) are relatively prime or if the underlying Coxeter diagram is a cycle. As for the types of relative rank one, whose underlying Coxeter diagram is 1-1 , they can be obtained as "limit cases" of types of higher ranks, but we shall not elaborate on that. REFERENCES 1. A. Borel, Admissible representations of a semi-simple group over a localfield with vectors fixed
under an Iwahori subgroup, Invent. Math. 35 (1 976), 233-259.
2. A. Borel et J.-P. Serre, Cohomologie d'immeubles et de groupes S-arithmetiques, Topology 15 (1 976), 2 1 1 -232.
REDUCTIVE GROUPS OVER LOCAL FIELDS
69
3. A. Borel et J. Tits, Groupes reducti/s, Publ. Math. Inst. Hautes Etudes Scientifiques 27 (1 965), 55-1 50. 4. -- , Complements a /'article "Groupes reducti/s", PubI. Math. Inst. Hautes Etudes Scien tifiques 41 (1 972), 253-276. 5. N. Bourbaki, Groupes et algebres de Lie, Chap. IV, V, VI, Hermann, Paris, 1 968. 6. F. Bruhat et J. Tits, (a) BN-paires de type affine et donnees radicielles, C. R. Acad. Sci. Paris 263 (1 966), 598-601 ; (b) Groupes simples residuellement deployes sur un corps local, ibid. , 766-768 ; (c) Groupes algebriques simples sur un corps local, ibid., 822-825 ; (d) Groupes algebriques simples sur un corps local : cohomologie galoisienne, decompositions d'lwasawa et de Carlan, ibid . , 867-869. 7. -- , Groupes algebriques simples sur un corps local, Proc. Conf. on Local Fields (Drie bergen, 1 966), Springer, New York, 1 967, pp. 23-36. 8. -- , Groupes reductifs sur un corps local. I, Donnees radicielles valuees, Publ. Math. lnst. Hautes Etudes Scientifiques 41 (1 972), 5-251 . 9. �. Groupes reductifs sur un corps local. II, Schemas en groupes (en preparation). 10. C. Chevalley, Sur certains groupes simples, T6hoku Math. J. 7 (1 955), 1 4-66. 11. M. Demazure et A. Grothendieck, Schemas en groupes. III, Sem. Geom. Alg. 1 962/64, Lec ture Notes in Math. , vol. 1 53 , Springer, New York, 1 970. 12. 0. Goldman et N. Iwahori, The space ofp-adic norms, Acta Math. 109 (1 963), 1 37-177. 13. H. Hijikata, On the arithmetic of P-adic Steinberg groups, Mimeographed notes, Yale Univ., 1 964. 14. -- , On the structure ofsemi-simple algebraic groups over valuation fields. I, Japan. J. Math. 1 (1 975), 225-300. 15. N. lwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Heeke ring of P-adic Chevalley groups, Publ. Math. Inst. Hautes Etudes Scientifiques 25 ( 1 965), 5-48 . 16. M. Kneser, Galois-Kohomologie halbeinfacher algebraischer Gruppen uber P-adischen Kor pern, Math. Z. 88 (1 965) , 40-47 ; 89 (1 965), 250-272. 17. I. G. MacDonald, Affine root systems and Dedekind's 7J function, Invent. Math. 15 (1 972), 91-143. 18. G. Prasad, Lattices in semi-simple groups over local fields, Advances in Math. (to appear). 19. G. Rousseau, Immeubles des groupes reductifs sur les corps locaux, These, Paris XI, ( 1 977). 20. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1 , pp. 3-27. 21. J. Tits, Formes quadratiques, groupes orthogonaux et algebres de Clifford, Invent. Math. 5 ( 1 968), 1 9-4 1 . 22. -- , Classification of algebraic semisimple groups, Proc. Summer Inst. o n Algebraic Groups and Discontinuous Groups (Boulder, 1 965), Proc. Sympos. Pure Math. , vol. 9, Amer. Math. Soc., Providence, R. 1., 1 966, pp. 33-62. 23. -- , Buildings of spherical type and finite BN-pairs, Lecture Notes in Math., vol. 386, Springer, New York, 1 974. 24. -- , On buildings and their applications, Proc. Intemat. Congr. Math. (Vancouver, 1 974), vol. 1 , Canad. Math. Congr., 1 975, pp. 209-220. 25. -- , Resume de cours, Annuaire du College de France pour 1 975-1 976, pp. 49-56. 26. -- , Classification of buildings of spherical type and Moufang polygons : a survey. Coli. Intern. Teorie Combinatorie, (Rome, 1 973), Atti dei Convegni Lincei 17 (1 976), 229-246. CoLLEGE OF FRANCE, PARIS
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), pa rt l , pp. 7 1 -86
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
NOLAN R. WALLACH The purpose of this article is to give an introduction to the representation theory of real, reductive Lie groups (see § I for the precise class of groups to be studied). Our main goal is the Langlands, Harish-Chandra classification of irreducible re presentations of such groups. §§7 and 8 are devoted to a study of intertwining operators for GL(2, R) and GL(2, C) and to a detailed exposition of how the classification specializes to GL(2, R) and to GL(2, C). The representation theory of reductive Lie groups is a vast subject. A complete survey of the subject would be a very ambitious task. It would also involve a size able monograph rather than an article of modest size. It is for this reason that we limit ourselves to a single path through the subject leading to the classification. Many important aspects of the theory are hardly discussed (e.g., the detailed theory of the discrete series [5], [6], [19], intertwining operators [13], [14], the Plancherel theorem [8], [9], [10]) . However, we feel that the material covered will give the reader an idea of the flavor of the subject. Much of the material that is not covered in this article will appear in other articles in this volume. There are several surveys of the subject of representations of semisimple Lie groups that are available (notably [20]). The reader interested in a deeper pursuit of the subject should con sult the monographs of G. Warner [22] . 1. The class of groups to be studied. 1 . 1 . Let ill be an algebraic group over C defined over R. We denote by ili (R) the
real points of ill and by ili0 the identity component of ill . 1 . 2. The type of group we will be studying will be a Lie group, having the following properties : ( I ) There is an algebraic group ill defined over R and a Lie group homomorphism r : G ---> ili(R) so that r(G) is open in ili(R) and r : G ---> r(G) is a finite covering. (2) rn o is reductive. (3) Ad( r ( G)) c Ad(@ 0 ) . Here Ad denotes the adjoint representation of ilia on the Lie algebra of ilia. 1 . 3 . Our basic examples are GL(n, R) and GL(n, C) (looked upon as the real points of an algebraic group defined over R). One of the main reasons for in troducing a broader class of groups than linear groups i s to include the "meta plectic" group, a two-fold covering of Sp(n, R).
G,
A MS (MOS) subject classifications ( 1 970) . Primary 22E45 .
71
Q
1 9 7 9 , American Mathematical Societ;·
NOLAN R. WALLACH
72
1 .4. Let G be as in 1 .2. Let X( G) denote the group of all continuous homomor 1 , x E X( G)} . phisms of G into Rx R - (0). Set aG {g E G l lx(g) l 1 . 5. I f H is a Lie group, w e will denote b y f) the Lie algebra of H. Ad will denote the adjoint representation of H on f) . fJ c will denote the complexification of f). 1 .6. Let G satisfy the conditions in 1 .2. Then it is not hard to show that there is a nondegenerate, Ad( G)-invariant, symmetric bilinear form, B, on g and an involu tive automorphism, 0, of G such that : ( 1 ) B is 0-invariant and (X, Y) - B(X, 0 is positive definite. (2) There is a maximal compact subgroup, K, of G so that Ka c {g E G I O(g) g} is an open subgroup. For such G we fix B, 0, K. Let c denote the center of g. Set °C {X E c I OX X}. Then exp ec) K n exp (c). Seq = {X E cl X = - X} . Let = exp (3). Then is called the split component of G. 1 .7. It is not hard to show that the map x 0G --+ G, (z, g) f-> zg is a Lie group isomorphism. 1 . 8 . We now look at our examples. If G = GL(n, F), F R or C, set B(X, Re (tr XY). Set OX = - X * (here X * denotes the conjugate transpose of the matrix X). 0G {g E G l ldet gl 1 }, Z = {all a > 0}. K O(n) if F = R and K U(n) if F = C. =
=
=
Y)
=
=
=
=
=
Z
=
O
Z
Z
=
=
=
=
Y)
=
2. Admissible representations. 2. 1 . We fix G as in 1 .2, B and K as in 1 .6. 2.2. Let H be a separable Hilbert space. Let GL(H) denote the group of all invertible, bounded operators on H. A representation (tr, H) of G on H is a homo morphism n : G --+ GL(H) so that The map G x H --+ H, g, v --+ n g)v is continuous. (tr, H) is said to be unitary if n(g) is unitary for g E G. 2.3. Let II · · · II denote the operator norm on End( H) . The principle of uniform boundedness implies that if w c G is a compact subset then ll n (g) II ;;:;:; C(w) < oo for g E w . This i mplies that if ( , ) is the Hilbert space structure on H and we set (here denotes Haar measure with total mass 1 on f K (tr(k)v, K) then ( , ) gives the same topology on H as ( , ) and n i K is unitary relative to ( , ). We therefore assume that n i K is unitary. 2.4. Fix a Haar measure on G (G is unimodular). lff E C�(G) (C"" with com pact support) we can define n(f) : H --+ H by and l i n { ! ) II ;;:;:; C(w) ll / lh . if = f G f(g) (tr(g)v, w) ( 1 ) ( tr{f )v, supp f c w (see 2.3, 11 / 11 1 the D norm of f). Iff E C""(K) set nK (f) equal to (n ! K) (f). 2.5. Let C(K) denote the set of equivalence classes of irreducible, unitary re presentations of K. If r E fix (..-T' V7) E T · Then dim V7 < oo. Set a7(k) (dim V7)tr ..-7( -1 ) , E K. Then 7r K (a7) is a projection operator on H. Set trK(a7)H H7 . Then H (;B 7H7 Hilbert space direct sum. if dim H7 < oo for 2 . 6. A representation (tr, H) of G is said to be
(
( I)
(v, w)
=
w)
tr(k)w) dk dk dg
dg
k k C(K)
r
EC(K).
=
=
admissible
=
2.7. A representation (:>r, H) is said to be irreducible if the only closed n(G) invariant subspaces of H are (0) and H.
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
2.8. THEOREM (HARISH-CHANDRA [2]). (1) If ( :n:, H) (:n:, H) (2) lf (7r:, H) E EBrE F Hr � 7r:(g)v
73
is an irreducible unitary re presentation of Gis tanhenadmissibilse admi s si b l e . then ifv the map g representisarealtionanalof Gytiandc. ifF is a finite subset G. c
�(K)
2.9. Let (7r:, H) be an admissible representation of Set H0 equal to the algebraic sum of the Hr > r E �(K). If v E H, set H(v) = .E k E K C7r:(k)v. Then H0 = { v E H I dim H(v) < oo }. 2. 1 0. If v E H0 then Theorem 2.8 implies : ( 1 ) If X E g , n(X)v = ( j )7r:(exp tX)vlt= O exists. (2) If k E K, X E g then 7r:(k)7r:(X)v = 7r:(Ad(k)X)7r:(k)v. 2. 1 1 . 2. 1 0 (2) implies that if X E g , v E H0 . Then 7r:(k)7r:(X)v E 7r:(g) · H(v). Thus dim H(7r:(X)v) < oo. This implies 7r:(X)v E H0 for v E H0 , X E g . Just as in the case dim H < oo we find ( 1 ) If X, Y E g then 7r:[X, Y] = :n: (X)7r:( Y) - 7r:( Y)7r:(X) = [7r:(X), 7r:( Y)] on H0• 2. 1 2. 2. 1 0, 2. 1 1 lead naturally to the notion of a ( g , K)-module. A (g , K)-module is a complex vector space V such that ( 1 ) V is a g-module. That is, there is a linear map g ® V � V, X ® v � Xv such that [X, Y] · v = X · Yv - Y · Xv, X, Y E g . (2) V is a K-module. That is there is a map K x V � V linear in the V-variable such that 1 · v = v and k 1 · (k2 v) = (k 1 kz) · v, kt. kz E K. (3) If v E V then dim .E k E K Ck · v < oo and if W c V is a finite dimensional K-invariant subspace of V then : (a) K acts completely reducibly on W. (b) The map K x W � W, k, w � k · w is continuous (hence real analytic). (4) If X E t and v E V then
d dt
Xv
=
exp X · v l e= O ·
(dfdt) t
(5) If k E K, X E g , v E V then k · X · v = (Ad(k)X) · (k · v) . 2. 1 3 . I f V, W are K-modules (resp. g-modules) then w e denote b y Hom K ( W, V) (resp. Hom 9( W, V)) the space of linear maps A : W � V such that Ak · w = k · A w, k E K, w E W (resp. AXw = XAw, X E g , w E W). If V, W are (g , K)-modules we set Hom 9, K( W, V) = Hom 9( W, V) n Hom K ( W, V). 2. 14. A ( g , K)-module V is said to be admissible if for any r E �(K), dim Hom K ( Vr > V) < oo . We set V(r) = .E A EHomK
G then
is anbadmi ble represent is irreducible ifand only if is an irreduci le ssiK)-moudl e. ation of i G intertwining operator (g ,
2. 1 8. If (7r:,-, H,-), = 1 , 2, are representations of then an from (7r:17 H1) to (7r:z, Hz) is a continuous linear map A : H1 � Hz so that A o 7r:1(g) =
NOLAN R. WALLACH
74
are said to be equivalent if there is a bijective intertwining operator A. If are unitary representatives of then are said to be from to unitarily equivalent if there is a bijective unitary intertwining operator from to
7Cz(g) 1e1 1e17C,z7C. z 7Ct. 7Cz G 7Ct. 7Cz 1e1 7Cz. Ifequi(7C;valent , H,) ifareandunionltary,y ifadmissiandble re present a t i o ns 7C are uni t a ri l y of G t h en 7Ct. z are isomorphic (that is, there is a bijective element of unitary ((Xv,kv, kww)) ( v,( v,w),Xw),k K, Let witbeh 7Cananadmiadmissisbsileble, K)-module. necess_ a ry and suffi c i e nt condi t i o n for unitary re presentation of G is that be unitary. G C(G) a
2. 1 9. THEOREM (HARISH-CHANDRA [4]).
V,. 1
H omg, K ( V"I' v"2)).
2.20. A ( g , K)-module, V, is said to be product ( , ) on V so that : (1) E (2) X e g.
V,.2
if there is a positive definite inner
=
=
2.21 . THEOREM (HARISH-CHANDRA [2), [3]).
A
V
V
=
V
(g ,
V"
2.22. The results of this section imply that the determination of the set of equivalence classes of irreducible unitary representations of is the same as finding the set C(g , K) of isomorphism classes of irreducible, unitary, admissible ( g , K)-modules. This can be done in two steps. The first is to determine the set ll(g , K) of isomorphism classes of irreducible, admissible (g , K)-modules. The second is to determine which elements of D(g , have a unitary representative. The first step has been carried out in Langlands' classification (an alternate classification has been given by Vogan which should have important and far-reaching conse quences). The second step is far from complete. There should be some discussion of the literature on this problem in the lectures of Knapp and Zuckerman [14].
K)
3. Infinitesimal characters. 3 . 1 . Let g be a Lie algebra over a field. Let T(g) be the tensor algebra over g . Let /(g) be the right and left ideal in T(g) generated by the elements X ® Y - Y ® X - [X, Y]. Set U(g) = T(g)/l(g). Letj : g --+ U(g) be defined by j(X) = X + /(g). 3 .2. ( U(g),j) or U( g) is called the universal enveloping algebra of g . We note that j [X, Y] = j(X)j( Y) - j( Y)j(X) = [j(X), j( Y)]. U(g) is called the universal envelop ing algebra of g because of the following universal mapping property : If cp : g --+ an associative algebra with unit such that cp[X, Y] = [cp(X), cp( Y)], then there exists a unique algebra homomorphism cp of U(g) into so that (a) cp(l ) = (b) cp(j(X)) cp(X). 3 . 3 . In particular, every g-module is naturally a U(g)-module.
E, E I,
E
=
form a basis of Let Xt.
3.4. THEOREM (cF. [11]). Xp · · · X�· U(g).
· · ·
,
Xn
be a basis of Then the monomials g.
3.5. Theorem 3.4 is usually referred to as the Poincare-Birkhoff-Witt (P-B-W) theorem. In particular it implies that j : g --+ U(g) is injective. We therefore suppress
75
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
thej and look upon g as a subalgebra of U(g) under the commutator of the associa tive multiplication. 3.6. Let be as in 1 .2. Let g be its Lie algebra (as per our conventions). Let ge .2" be the center of U(ge). be the complexification of g . Let .2"(ge) then Ad(g) : B e -4 B e extends to an automorphism of 3.7. We note that if g then Ad(g)z z. U(ge). I .2 (3) implies that if z .2" and 3 . 8 . Recall that a Cartan subalgebra {J of g e is a subalgebra of g e such that ( I ) {J is abelian. (2) If e {J then ad(h) diagonalized on B e · (3) {J is maximal subject to ( I ), (2). 3.9. Cartan subalgebras always exist. (E.g., if GL( , R) then g Mn(R) MnC ). We can take {J to be the diagonal and Be with matrices in Be · ) 3 . I O. Let {J be a Cartan subalgebra of Be · If {) * set (gc)a gc , Set L1 {) * I :1(gc) a :1Then L1 is called the root system of ( ge. {)). We have � aE igc)a {J Be · 3 . I l . Let B also denote the complex linear extension of B to Be (see 1 . 6). Then B l � xlJ is nondegenerate. Let ( , ) denote the dual bilinear form to B llJx � on {) * . ) ) a where If a e L1 set sah a E {J is defined by B(Ha, ) h e {) . Let W(L1) be the group generated by the a , L1, in GL({J). W(L1) is called the Weyl group of L1. We let act on {) * by · A A o 3. I 2. If W(L1) then it is well known (cf. [11]) that there is @0 so that Ad(g) l � · 3 . 1 3. A system of positive roots for L1 is a subset, of L1 satisfying ( 1 ) L1 LJ+ U disjoint union. and (2) If L1 then a R. Let {JR We note that if L1 then � R a . Then BI �R x �R is {)R i ( ) :1Then {)� :1- 0 . L1 is finite. positive definite. Let {) � For {) � set L1 1 ( ) > 3 . I4. Let LJ+ be a system of positive roots for L1. Set n+ = � aE&+(ge)a. Set Then o is called a Borel subalgebra of Be · Set n {J o .E aE &+(gc) -a . 3 . I 5. By P-B-W, U(g c) = U({)) EB (n- U(ge) + U(g e)n+). It is easily checked that .2" c U({J) Ef1 n- U(g c) and .2" c U({J) EB U(ge)n+. Using these observations we see that if .2" -4 U({J) is the projection into the U({J)-factor in the above direct sum decomposition then .2" -4 U({J) is an algebra homomorphism. · I, 1 , tJ. (h) 3 . 1 6. Let 1-': U({J) -4 U( {J) be defined by tJ.{ l ) Set r f1. o p.
G
=
eG e
geG
=
h C G n [X, Y] XY - YX {X e i [A X] {a e a + 0, a e 0}. a(H)X, H e{)}. h - 2(a(h)/(a, a H H s a e h a(h), 1 W(L1) s s se g e• s + LJ , + {-al a e LJ } a, {3 e LJ+ a, a{3 +e {3 e (a, {3)+ e{3 e LJ+. H {he a h 0, a e L1}. h e , LJ+(h) {a e a h 0}. + n+ . p: p: h - o(h) tJ.(h1h2) tJ.(hi)fJ.(hz). Theh, s map is injective t o with imaisgienequal {he s h e }. the choice ofsm suchSetthat then there is A so that cie s aofhomomorphi If dependent e extension to of the linear map h A(h)Set foroff) Ainteo C. Here) A:A (p(z) , Cie s thThen and only if A' sA for somes e =
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
3. I 7. THEOREM (HARISH-CHANDRA, CF. [1]). ( I ) T: .2" -4 U({J) U(q)W C<�l = U({J) I · W(L1) (2) r LJ+. r Tsc · � · x(I) I E {) * (3) X : .2" -4 U({)) U({J) -4 x(z) A(r(z) ) , z .2". =
=
=
=
-4
(4)
{) * , X A (z W(L1) .
=
) z .2".
XA
=
XA ' if
=
76
NOLAN R. WALLACH
3. 1 8. The hard part of Theorem 3. 1 7 is ( 1 ) . (3) follows from the fact that U(f)) is integral over U(f))WCJl . (4) is just a restatement of
(I).
3. 1 9. Let V be an admissible K)-module. Then V EB rE O" CKl V(r). If fl' tff (K). Since dim V(r) < oo we see that fl' l vcrl can be put then V ) c: V(r), in upper triangular form. 3.20. Set for a homomorphism X : fl' -->
= zE z · (r r E (g, C, = {v E (z (z}) v = E d (g, = = { E(g, z = == x(z)v, z E (g, W(L1). Then there exist elements ub · · , uw of so that Let= be the order of iens a (g,is admi K)-module, finihteerly generat e d as a U(g)-module, and if = for some t h s si b l e . Furt more, if Vis nonzero, then has a nonzero irreducible quotient. G G. G P G. G a0 h E a0 h g h E a0 =1h{X=Eh).gl B(X, =i\Eat, g g•a0= {XEgl i\(h0h) · X = ).i\(EA.h)X, h EAa+0}.= {i\ AE Al={i\il.(hEatl g• h E ) 0 0 Mn0g0=NE0GL:.lEA+g•. gPM0g0-1 P. G. P A0 = a0 G.G P0 = G kP0k-1 G = KP0 k P0 G = G P oM0 = M0(PP 0, A0 A0 (P0, A0G.) N M0 M = P (P, A) = MN. M = oM A A A) MP. A A 0 P0 . unique P = MN, (P, (P,A) A) K M K. (P, A) = MN = oMAN. Hu) V;r;
Vl
-
X
d
0 for
Z
f!',
depending only on v } .
The discussion in 3. 1 9 implies that V is the K)-module direct sum of the V;r;'s. 3.2 1 . If v is irreducible and v;r; of. 0, then v;r; v. Let w v VI . v fl'}. Since V;r; V it is clear that W of. 0. But W is clearly a K)-sub K) module of V. Hence W V. X is called the infinitesimal character of the module V. 3.22. THEOREM (CHEVALLEY, HARISH-CHANDRA, CF. [22]). ·
U(q)
3.23. THEOREM (HARISH-CHANDRA, CF. [22]). If V V Vx X• V
U(f))
W
L; u,.U(q)WCJl .
V
4. The principal series. 4. 1 . Let be as in 1 .2. Let c: be a subgroup of Then P is said to be a parabolic subgroup of if (1) P is its own normalizer in (2) Pc contains a Borel subalgebra of g c (see 3 . 1 4) . 4.2. Let be a maximal abelian subalgebra of contained in flt) 0}. If then ad diagonalizes on (indeed if then #- (0), Let, for ad Set ). of. 0} . Fix so that of. 0, Set > 0}. Set in Then Set equal to the centralizer of exp is a parabolic subgroup of If is a parabolic subgroup of then there is so that c: 4.3. The Iwasawa decomposition of says that if is as above. Thus if P is a parabolic subgroup of then :=J for some E K. Hence KP. 4.4. Fix ) as in 4.2. Then is known as a minimal parabolic pair. chosen as in 1 .6. is a split component of n K (see 1 . 5). 4. 5. Let be a parabolic subgroup of Let be the unipotent radical of Let n OP. Then P · with a split component of chosen as in 1 .6. is said to be a parabolic pair or p-pair. (P, is said to be standard if c: and P :=J Each p-pair is conjugate in G to a standard p-pair. 4.6. If is a p-pair, then M satisfies the conditions of 1 .2. Suppose that is standard. Then M n K plays the role of for G. The notion of admissible will be relative to n 4.7. Let be a standard p-pair. P Let (a, be a finitely
77
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
apeman) a PP e PP a
generated admissible representation of M. Let op denote the modular function of MN. Then = 2 = 2 CH) if = exp H and 2pp(H) = tr(ad Hln) , H e n.
4.8. We form a representation (11:(P, a), HP,u) of G as follows : ( 1 ) Set HP,u equal to the space of all measurable functions /: K --+ Hu such that K and 11! 11 2 = I K 1 1/( ) 11 2 < 00 . K n M, = e K. (2) If f e HP,u extend f to G by f( ) = op(P)- 1 1 2 a( ) 1/( ) for Set (11:(g) j) (x) = f(g- 1 x) ( = 11:(P, a)).
k kdk, pEP, k f(km) a(m)-lj(k), m E k E kp p 11: 11:(P, is an admissible representation of V(P, Vu V(P, V,.., V(P, V(P, V(P, Vu V(P, Vu -
4.9. THEOREM. (
to
a), HP,u) = I(P, a)
G.
4. 10. Theorem 4.9 can be derived from Theorem 3 .23. 4. 1 1 . Set a) equal to the corresponding admissible (g, K)-module (i.e., denote the (m, K n M)-module corresponding a) = 11: = 11:(P, a)). Let a.
4. 12. If f e a), f is real analytic by Theorem 2.8(2). We define j(j) = /(1 ), fe a). Then it is easy to see ( 1 ) j: a) --+ is a surjective (m, K n M)-module homomorphism. 2 ) n · a). Kerj ( :::> This implies that is a quotient of the (m, K n M)-module V(P, a)/n V(P, a). 4. 1 3 . The following result can be proved using Theorems 3.22, 3 .23, the sub quotient theorem (Harish-Chandra [3], Lepowsky [16], Rader [18]) combined with a technique of Casselman using the asymptotic expansion of matrix entries of admissible representations (cf. Milicic [17]). For a proof that does not use asymp totic expansions, see [21]. 4. 14. THEOREM. P = MN. (1) # (2)
Let V be an admissible K)-module. Let p-pair,V nV. Then If Vis .finitely generated then VjnVis a.finitely generated (g,
(P, A)
be a standard K)-module.
(m, M n
4. 1 5. The particular case of 4. 14, (P, A) = (P0 , A 0) has an interesting consequence, M0 = M0 n K. Hence a finitely generated, admissible (m, K n M) module is finite dimensional. 4. 1 6. Let be a nonzero admissible, finitely generated (g, K)-module. Then 1 � dim VJn 0 V < oo by the observation of 4. 1 5. Let W be a nonzero irreducible quotient of f no as an (m, K n Mo)-module. Then no acts by v e (n3)c. We can therefore look upon W as a representation (a, Hu) of M0 • Set � = op�12a. Let p. : W be the composition of natural maps. Define, for v e C(v)(k) = p.(k-1 v) . Then clearly, C : HPo. e = H and C(k · v) = · C(v). a
V--+
V V V V --+ Set
4. 1 7. LEMMA.
11: = 11:p0, 0•
Then
V,
k
C((Xv)) = 11:(X) C(v) jor v
·
E V,
X e g.
PROOF. We note C(k X · v) = C(Ad(k)X k · v) . Hence 11:(k)C(X · v) C(Ad(k)X · kv) . Thus it is enough to show that C(X · v) ( 1 ) = (11:(X)C(v)) ( 1 ) , for X e g, v e ( 1 ) is not hard in light of (2) g = f E9 n0 EEl n0 , and the discussion in 4. 1 2.
(I)
·
·
V.
4. 1 8 . 'THEOREM (THE SUBREPRESENTATION THEOREM OF CASSELMAN) . If
=
V is an
78
NOLAN R. WALLACH
irreduciq)for ble admian appropri ssible atK)-modul Vis isaomor e irreducieblethenrepresent tionpofhic with a K)-submodule of (g ,
(g ,
V(P0,
M0 •
PROOF. Let C be as in 4. 1 6. Then C
4. 1 9. CoROLLARY. V"
¥-
0 and hence C is injective.
Let V be anisiranreduci ble, badmi K)-module.ationThenof V is isomorphic with where irreduci le, admissibslesible represent V V11 anThenirreducible, admissible K)-module. Let be a standardhasp-paainonzero rLet V beirreduci ble M)-KmodulM)-modul e quoti ent. then W V11 W is an i r reduci b le e quot i e nt o f and Vis isomorphic with a K)-submodule of (g ,
(n, H)
G.
4.20. Let (P, A) be a standard p-pair, P = MN. Let be an admissible, irredu cible ( g , K)-module. Then V/n V is a finitely generated, nonzero, admissible (m, K n M)-module. Hence V/n V has an irreducible, nonzero quotient. Corollary 4. 1 9 implies that this quotient is of the form for (0', H11) an irreducible admissible representation of M. Arguing as in 4. 1 6 and Lemma 4. 1 7 we have : 4.2 1 . THEOREM.
(P, A)
(g ,
P = MN.
(1) V/n V (2) If
(g ,
( m, K n
(m,
n
V(P, o-p1 12 0').
V/n V
=
5. Exponents. 5. 1 . We retain the notation of §4. 5.2. Let (P, A), P = MN be a standard p-pair. Then M n P0 = * P0 is a minimal parabolic subgroup of M. Then *P0 = 0M0A0*N0, *M0 = 0M0A0• (*P0, A0) will be used for M in the same way as (P0, A0) is used for G. We note that N0 = * N0N. 5.3. If V is an admissible, finitely generated ( g , K)-module then it is easy to see that ( 1) V/n0 V = ( V/nV)/*n0( V/nV). 5.4. Let be as in 5 . 3 . Let (P, A) be a standard p-pair. Set E(P, = { v - pplv a weight of a on V/n V} . 5.3(1) implies that E(P, V) = {,u l n l,u E E(P0, V)} . 5.5. I f (P, A ) is a p-pair let A ) b e the set o f weights o f a o n n. Then Z(P, A) is called the set of roots of (P, A). An element A e (P, A) is said to be simple if · · · A, ;;::; 2, with A; E (P, A), = 1, · · · , A cannot be written in the form A1 o Set Z(P, A) equal to the set of simple roots of Z(P, A). 5.6. Set o zo (Po , A o) = {al> · · · , az} . Let Hl> · · · , HI E a o be defined by a,-(Hj) = O;j and H,. .l 3 (3 the Lie algebra of the split component of G chosen as in 1 .6). Set D0 = L, RH,.. Then a0 = 3 EB '1>0, orthogonal direct sum. 5.7. Let �.. e Dt be defined by (�,., aj) = Oij · If F c { 1 , · · · , 1} set D6(F) = { v E D � l v = - L..-rtF x,-{3 ,. 'L. iEFY ;a;, x,. > 0, Y.- ;;::; 0 }.
V
Z(P,
r.
+
Vj)
+ + r Z Z,
5.8. LEMMA (LANGLANDS [15]) . D6
Z
is the disjoint union of the z; i + If and then
i
D6 (F).
5.9. If v e D t(F) set V0 = - 'L. irtFX,-{3; if v = - 'L. i rtFx.-{3,L. .-EFY;a,- . If a, {3 E D � we say a 2::: � i f a - {3 = L,z;a;, z,. E R , ;;=:; 0, = I , · · · , I. 5 . 1 0. LEMMA (LANGLANDS [15]) .
A,
).!
E D6
A 2::: ).!
A0 2::: ).!0 •
5. 1 1 . Let
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
79
"# 0}. Set nF Set aF = {H e aola; H = 0, e F}. Set F = {A e exp aF. Then MFNF ,E ,;Ez F(noh · Set MF equal to the centralizer in G of F PF and (PF, AF) is a standard p-pair. It is well known (cf. [22]) that every standard p-pair is of the form F A F
( ) i � �(PA 0, A0)IA.I .F (P , ). 5. 1{1,3. LEMMA. LetanVirbereduci an ibrreduci ble,ssiadmi ssible atK)-module. Then thsoerethexiat st ... , /}, and l e , admi b l e represent i o n * E(u, P and Vi s i s omorphi c wi t h a K)-submodul e of ) t h en 0 V(PF, Note. L(V, P ) cff ( . E(V, P0) L(V, P0) {1, *n0(VfnV) (P, A) V (VfnV)f Vf n 0 W VfnV E(V/nV, *ePE(W, 0) *P0). e E(W, *P0) +e , i + . ... 5.14. cffiVG) 1, . e E(V, P0) aninircffreduci e K)-modul easuchstandard that tp-hepequiair, valence clandass ofan5.15. Viirreduci s LEMMA. in cffbl1e(G)admiLetbutssiVisbbelnote represent iG).atibolThen t h ere are (P, A), n e of thsoen that I 1, and if (1) ifV ee cffiM), then anda) Visx(a)l isomorphic with a subrepresentation of V(P, 5.13.5.16. 6.1. v, we G, §5.Z. w ( ) x(z)I, x Z. v, we 2 e £ (G/Z) C2(G) V,, G. C2(G)6. 3 .If Vcff is a representative of an element of cffiG) then there is a 1-dimensiona/(1) K)-module W so that the class of We cff2(G). =
=
(g,
F c:
Re (vlv0) e D�(F)
ve
=
(a, Hu) of MF,
if
(g,
a).
In particular, vq E t MF) PROOF. Let o = { R e v lv0 I v e be a minimal }. Let p. e element. Then p. e D�(F) for a unique F c: . . . , /} . Let be the correspond = we see that there is r; e ing standard p-pair. Since so that Re r;lv0 = p.. This implies that there is a quotient of s o that r; I f r; ' then r;' J . = r; J • . Thus i f p. = - ,E ;,uX;{j; .E .-EF Y;a ; X; > 0, e F, Y; � 0 then Re r;'lv0 = - ,E ;,uX;{j; .E iEFz,-a; with z; R. If we show that z; � 0 we will have completed the proof of the lemma. Sup pose only Z;1 , . . , z;, < 0. Then Re r;'lv0 ;;;; - .E ;ruX;{j; + .E iE F-< i� o . . . , ;, JZ;a;. " " ' Set p. Re r;'lvo• p. = - .E irt=FX;{j; + ,E ;EF- 0 for i = . · , I. =
(g,
a u
(2)
A
a(
=
(u, Hu) M a n Do
H
i x(exp H) =
q) .
The proof of this lemma is essentially the same as the proof of Lemma
6. The discrete series and the Langlands classification. We maintain the notation of Let (n, H) be an irreducible unitary re presentation of G. If H then l (n(g)v, )l = f!> v . w(g) satisfies ( 1 ) f!> v . w(gz) = f!>v . w(g) for g E Z E This is because n z = a unitary character of ( n, H) is said to be square-integrable if lf> v . w for all H. 6.2. Let denote the set of isomorphism classes of the (n, H) an irre ducible square-integrable representation of THEOREM (HARISH-CHANDRA [6], CASSELMAN, CF.
(2)
(g ,
c:
d(G).
V®
MILICIC [17]).
NOLAN R. WALLACH
80
6.4. Theorem 6.3 is proved using the relationship between the elements of P0) and the leading exponents of in the sense of Harish-Chandra (cf. Warner [22, Chapter This relationship is due to Harish-Chandra and Cassel man (see also Milicic [17]).
E(V,
V 9]). If V ise aWrepresentati !-dimensional K)-modul so that V ®veWofisanunieltaery.ment of then there is a 6.6 LetatioFn of so that(seeif E(5.Va,7). Let then be anlJ i oiorreduci b l e admi s si b le represent D�(FurtFh).ermore Then ifF' has a uni1}queandnonzero,satiisrfireduci beleabove , submodul e denot ed es t h propert i e s for then with is isomorphic with if and only ifF' F and is equivalent 6.5. CoROLLARY.
tff1 (G)
(g,
5. 1 5 .
This result follows from the fact that V(P, o') is unitary if a is unitary and
6.7. THEOREM (LANGLANDS [15]).
E Re J(Pp, a).
MF'
V(Pp, a)
a' .
{ 1 , . . · , /}
·
( a, Ha)
5.6,
v E
MF
a' J(PF', a ' )
{1,.. ,
c
J(Pp, a)
c
*(Pp) o)
=
a
6. 8 . The first part of Theorem 6.7 is proved by using the relationship between intertwining operators and certain limit formulae. The second statement is proved using the limit formulae alluded to above and an argument similar to the proof of Lemma 5. 1 3 . Theorem 6 . 7 combined with Harish-Chandra's classification o f the elements of tff2(G) (see [6]), Lemma 5. 1 3 and Lemma 5. 1 5 give the complete classification of irreducible (g, K)-modules.
6.9.
7. The principal series for GL(2, R) and GL(2, 7. 1 . Let G = GL(2, F), = R or We take (}, B, etc. as in 1 .8 . We take P0 to be the group of upper triangular matrices in G and P0 to be the group of lower triangular matrices in G. M0 is then the group of all diagonal matrices in G. 7.2. We denote by [hh 2] the diagonal matrix with diagonal entries h1o h2• We also use the notation
C). K,
F C. h n(x) [b �J A 0) =
7 . 3 . We note that !:; (P0,
= x1 - Xz .
n(x) =
and
[! ?].
= {A} and that !:: (P0, 0) = { - .A} with .A([x1, x2])
A F. kh k2 dn
7.4. Up to normalization of Haar measure on N0 (resp. N0), (x) = dx ( resp. dn(x) = dx) where dx denotes Lebesgue measure on 7.5. If F = (resp. R) we define for k1o k2 E Z (resp. E {0, 1 }), z�o z2 E k = lhl i· 1 -k 1 l z l zz - :l, Then �k" kz, ' " •z is the most general ir .;k J , f<:l, ZJ, reducible admissible representation of M0 • We extend .;k� > kz, • �> •z to P0 (resp. P0) by making it take the value I on n(x) ( resp. n(x)) .
C, �([hhhz]C) h:1h�2 h f thenkh k2, are as in 7. 5 and if z (A(kh zh JNo f(kn) dn A converges morphism ofabsolutely andinto uniformly for k K and defines a K)-module homo kz ,
V(Po , a)
Re( 1 - z2)
zh z2
7.6. LEMMA. If V(Po, .;k" kz, '" zz)
Zz)f)(k) =
=
E
V(Po , a) (a = .;k" kz, • � > •zl·
(f)
(g,
<
0, if
E
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
8I
PROOF. (A(f))(k) = JNh5o
P
G
K,
(I) Writing n as n(x) a direct computation gives
a(n(x))
{II)
=
[ ( I + x2)-1 1 2,
(I
+ x2) 1 1 2] .
Set d = dimRF. Then (II) implies that the integrand in (I) is given by {III) which is integrable on F if Re(z 1 - z2) < 0. The fact that Af E V (P0, a) is proved by a simple change of variables. The uniform convergence of the integral implies that A is a (g, K)-module homomorphism. 7.7. We now focus our attention on the case F = R . In this case K = 0(2) . We use the notation e; for elements of { 0, I } . 7.8. Set 1J = diag(I , - I) E K and set
k(()) =
[
c<_:� s () sin
- Sill
{)
COS
()] {)
'
() E R .
S0 (2) = { k(O) I O E R } , K = S0 (2) U 7JS0 (2). 7Jk(0)1J-t = k( - 0) . 7.9. We now describe cff (K). Let W b e an irreducible finite dimensional K module. Then there is w0 E W so that k(()) · w0 = ei10w0, () E R and I E fixed. k((})1JWo = e-it wo1Jw0. Thus if I f= 0, w is completely determined by I. Since we can replace w0 by 1JWo we can take I > 0. Let W1, I = I , 2 , · · · , denote the K-module we just described. If I = 0, set W0 = { w E Wlk(())w = w, () E R} . Then 1J · W0 = W0. = W0, dim W = I and 1JW = w, w E W or 1JW Hence since 1}2 = I we see = w, w E W. We use the notation W8 for the trivial representation of K and Wb for the case 1J --> - I . 7. I O. We leave it to the reader to check that Adi K on g c is a direct sum of two copies of W8 and one copy of W2 • 7 . I l . We note that W1 ® wk = wt+k EB wl t-k l for k > o . = W1, c = 0 or (2) W1 ® ® W?J' = Wfl" with s" = c + s ' mod 2 . 7. 1 2. Let rt denote the equivalence class of WI, I > 0. Let r6 denote the equi valence class of W5, c = 0, 1 . 7 . I 3 . Using the formulas of 7 . 5 we see that if V is an admissible (g , K)-module then g · V(r1) c V(rl+2) + V(rlt-z1 ) + V(r,) if I > 0. (2) 9 V(r6) c V(r6) V(r2), s = o, I . 7. 14. Since (P0, A0) i s a minimal p-pair we can take o M0 = M0 n K = = I , 2 } . Let V = V(.;,�o •z. z�o zz • P0) . .;,�o ez, zpz l oMo = .;,�o •z with {[7] 1 > 7]2] 11}; = ±
Z,
W
-
I. /,
(I) W5 (3) W5 (I)
.
I, i
+
82
NOLAN R. WALLACH
2 7JzD 7Ji17J�2 •
.;,�> , ([7] 1 ,
= Frobenius reciprocity implies that dim (r) is the num ber of times .;,�> ,2 occurs in r l oMo· 7. 1 5. We leave it to the reader to check that if 'l:J Tl• I = 1, 2, . . . , Tb r5. c = 0, 1 , then
E
= .;o,o· = .;1.1 · = .;o.o E8 h 1 if I is even, I > 0. = if I is odd. E8 7 . 1 6 . Let v = Po) . Then 7. 1 5 implies dim
( 1 ) T8l oM0 (2) TBi oM0 1:1 l oM0 1:t l oM0
((3)4)
and (I) If c1 (II) If c1
V
.;1.0 V(.;.;,J0, E. 21, ZJ, Z2'
E
(r) ;;£ 1 for r
V
E tff(K)
VV V 1 LEMMA. Set, for in C, in a whereSet is the gammaThenfunctifioVn (cfV(P0, then as a K-module V is isomorphic with Vor /f/ E then set Let us denote by either +
+
c2 cz
is even then is odd then z
7. 1 7.
k (z) =
= (r5 ) E8 EB 1>o V(rzt) . = EB go V(rzt+l ) · Z
k
.,. I 12F(z/ 2)F ((z + 1 ) /2)/F(( l
+
z
+ k) j2) F (( l + z - k)/ 2)
r [23]). 0' = .;,�> •z. •�> •z · 0') = (Po . 0'). If Re (zl - Zz) < 0 A = A(c[, cz, Z[ , Zz) . Af = ak(zz - z1 )f r8 · V(rk)
r6
ro
PROOF. 7. 1 6 implies that A is a scalar on each V(r) . Using the formula at the beginning of the proof of Lemma 7.6 and an easy computation, it follows that, on V(rk), A is given by the following scalar : (1)
s:oo
((1
+
ix)/( 1
+
In ( 1 ) use the change of variables Then the integral becomes
(2)
S
tr/2
x2) 1 1 2) -k (1
e it
+
x2) H+zi-zz) /2 dx .
= ( 1 + ix) /(1 + x2)1 1 2,
2 0 cos(k ) (cos( )) - Cl +znz l
t t
dt.
-
;;£
.,.;2 t
;;£
.,.;2.
3 24C. 263, G C)Sl U U ) (SUl2 U2 SI S SU(2)1 V) d SU(2) I, tff(K) d mE Z, tff ( K). n E Z n mn n
The lemma now follows from Whittaker and Watson [23, p. Exercise 9] and the duplication formula for the gamma function [23, p. 0]. 7 . 1 8 . We now concentrate our attention on the case F = For the remainder of this section will denote GL(2, and K will denote ( ) . the circle group) given x S (2) --> (2 7. 1 9 . As is well known, the map by z, k ...... zk is a two-fold covering of ( ) . If (1:, V) is an irreducible unitary representation of K and if z is in then 1:(zl) U(2) is = zkf for some k depending only on 1:. Hence the restriction of 1: to irreducible. are determined up to equi The irreducible, unitary representations of valence by their dimension. If (1:, is the + dimensional representation of then it is well known that 1:(cl) = ca c = ± 1 . Hence k + is even. 7.20. We have completely described is the set of all rm . n with and ;?; 0 and + even. Here rm . n denotes the class of all irreducible (1: , W) with dim W = + 1 and 1: (zl) = zmf for z E St.
REPRESENTATIONS OF REDUCTIVE LIE GROUPS
83
7.2 1 . Let �k1, k2 = �k � o kz , z � o z2 l oMo· It is an easy consequence of the classification of representations of SU(2) that rm . n 1 °Mo = EB � < n+m-2jl / 2 , n
j=O
(m-n+ 2 j ) / 2 •
7.22. Using 7.21 and Frobenius reciprocity we find that if V = V( 0 �k� > �· • � > •2)
P,
then
furthermore each V(r) is irreducible as a representation of K. 0 �
7.23. LEMMA.
�
Set,for r n, r, n integers andfor zin C bn, r(z) 2 z n IT (z - n 2(} - z n 2j). If V(P0, and z then A(kb k , z l > b z z = n( + ) - 1
V=
�k �o k2, z�o •2) 2
if Re(z1
1 )) / (
+
j= l
-
2
)
<
+
-
0
z2) J v crk1 Hz , l k l + k2 1 + 2 il =
l k l - k2 1 + 2 i . i ( 2
-
l
) /.
7.24. This lemma is proved using an argument similar to the proof of Lemma 7. 1 7 . The proof is more complicated and even less enlightening. It is therefore
omitted.
7.25. Lemmas 7. 1 7 and 7.23 imply that the A(kb k2 zb originally defined for Re(z 1 - z2) < 0 have a meromorphic continuation to x The reader is ad vised to consult the lectures of Knapp and Zuckerman for more information on these intertwining operators.
, Cz2) C.
8. The representations of GL(2, R) and GL( , 8. 1 . We retain the notation of § 7. We first look at the case of G = GL(2, R).
= V(P0 , �'�> ,2, z�o •2) Z - {0} . + 1 = 0 mod
8 .2. LEMMA. (1) k = -
2 C). is reducible and only if
V z 1 2.then le b If (2) If z1 z is-reduci en ble. is finitIfe dizm1 ensional andk k -in1,deedkk �� k 1 andk k i2,s itrhreduci 2, then contasaiintss as i t s uni q ue nont r i v i a l submodul e and unique nontrivial quotient module. k (1) � �2 2m, m � 2. l 2. g (g, z2 E
.>1 + ->2 V(P0 , .;,� o ,2, z�o zJ
(a)
-
2
=
if
0, e 1 + e2 = mod dim J = + V/J mod 0, e1 + e2 =
1,
(b) - z2 = + V(.;,2, ,p2, .)/J (.;,2, , �o z2, z )
J(P0, .;, � o ,2, z�o zJ = J V J(.;,2, ,p2, z )
PROOF. Let W c V be a nonzero submodule. Then W = EB W(r) If W(riJ # 0 with lim,_oo j, = oo then W contains V(ri+2k) for all
0. Hence dim V/ W < oo . Indeed, supposej,. and j i+
Hence > j,. + Then j,+1 - j ,. = ( = 0, V/ are nonzero. But V/ W ( V/ W) V/ W)( W) ( ) ) i 7J.+2m) (7 f+2m-2) ( 7J.+2) , . . . , ( (7J. EB >;;�i( V/ W)(h+2u) c EB::z,;=-f ( V/ W)(7J.+2u). Hence = 0. Now 7. 1 3 implies K)-submodule of V/ W. The finite EB ::z,;=-f ( V/ W)(7J;+2u) is a finite dimensional dimensional representations of G restricted to K are of one of the following forms : (a) Wg EB EB 1,o;} W21 or ·
NOLAN R. WALLACH
84
(b) EBo���j W21+ 1 · This gives a contradiction proving (1). (1 ) easily implies that if V is reducible then V has a nonzero finite dimensional submodule or a nonzero finite dimensional quotient module. (2) If W is a finite dimensional (g, K)-module, then W/no W is isomorphic as an Mo-module with o}t;�. J o t:!, Z Jo ZZ ' with zl - Zz = - k - 1 , k G; 0, k E z and cS I + cSz k mod 2. This is just the theorem of the lowest weight for Combining (1), (2) and the fact that the dual (g, K)-module to V(P0, �•�o ez. •�o zz) is V(P0 , �ez. •�o •z . •) gives the lemma. =
G. Set Suppose that Then the class of
V = V(P0, .;•�o ez. •� o zz). = k mod 2. J =
Setez and e 1 is in CiG). 8.4. E §§4, 5, 6 e 1 e2 e e2CiG):k 8. 5. foll8.o6wi. ng condiistipure ons areimagisatnisary,fied: then (3) A(el > is positive semidefinite. 1 8.3. LEMMA. Z, k G; 0
+
z1 - z2 = - k - 1, k e
J(�•�o •z. •�o •z• P0) .
VfJ
( VfJ, P0) = (A, ..:t)- 1 (z2 - z1)A by 8.2 and 4.21 . Using the results of we have a complete classification of irreducible, admissible, (g, K)-modules (see also Jacquet-Langlands [12]). (I) The finite dimensional (g, K)-modules J(.;•�o •z •H2' P0) with z1 - Zz = - k - 1 , k G; 0, k e Z and + = k mod 2. (II) The V(�.1, ,2, ,t . •2, P0) with z1 - z2 ¢ Z - {0} or if z1 - z2 = k e then #- + 1 mod 2. Furthermore V(.;.p2, z � o zz • P0) = V(.; ,2, •�o •2, ,1 , Po) . 1 + (III) For z1 - z2 = - k - 1 and k G; 0, k e Z and 1 + = k mod 2, D• t. ez. k, z t+•2 = V(�•� o <2 . •1· 22' Po) /J(.; • t . •2· • t . •2' Po) . We now give a classification of the irreducible, unitary (g, K)-modules. We first need a lemma. PROOF.
LEMMA. lf Re (z1 - z 2)
(1) z1
+
<
0
e e2
J(P0, .; , t . ,2 . • �o zz)
z2
Z
is unitary if and only if the
E R, .s2, zl> z2)
(2) z1 - z2
< O then ez, zl> Zz) V(P0, �•�o •2. •�o •z> J(P0, �•�o •2. •�o zz) (use Lemma 8.2 and Lemma 7. 1 7). The conjugate dual admissible (g, K)-module to V(P0, �• t . •2, •�o •2) is V(P0, �•�o ez. -•2. _2 1) (this is a computation). Thus J(P0, �•�o ez. •�o •2) is equivalent with its conjugate dual if and only if z1 = - z2 and
=
PROOF. We observe that if Re(z - zz)
A(el >
z2 = - t1 . This implies the necessity of conditions (1) and (2). To complete the proof we observe that composed with the sesquilinear pairing of the V's gives a sesquilinear pairing of the corresponding J with itself. 8.7. Lemmas and combined with Lemma 7. 1 7 immediately give a complete classification of the elements of If( G). We give the list : ( 1) V(P0 , �•� o •2· • t . •z) with e R. (2) D,�o ez. k, . with e G; 0, e 1 + = mod 2 and e R . V(.; •�o •2· •t. zz) with z + z2 e iR, 1 + ez = 0 (3) mod 2 and - z2 e R, - 1 < - z2 < 0.
A(el > .s2, zl > z2) 8.6 8.4 The diuniscrettarye priserinecis.pal series. k Z, k zl >ez2 2 ik The zi The compl e ment a ry seri e s. e 1 (4)8.8. The uniz1tary characters of G.z1 C).
We now look at the case when G We note that :
=
GL(2,
We use the notation of 7.20.
REPRESENTATIONS OF REDUCTIVE LIE GROUPS rn. m
®
r n', m ' =
EB
O�j;£ m in (m, m ')
r n+n', m+m '-2j
and Adl x = ro . 2
85 +
ro .2 ·
8.9. Using the formulae in 8.8 the following result is provided in precisely the same way as Lemma 8.2. 8 . 1 0. LEMMA . Let Re(z1 - z2) � 0. Then V(P0, �k1. k2, 2�. 22) is reducible if and only if
(I) z 1 - Zz = - k - 2, k � 0, k E Z, (2) lk1 - kz l � k.
If V is reducible then the corresponding J is finite dimensional. 8. 1 1 . Lemma 8 . 1 0 combined with the results of give the classification of the irreducible, admissible (g, K)-modules : (I) CiG) = 0 . (II) The finite dimensional representations : J(P0, �k1 . k2, 21, 22) with z1 - z2 = - k - 2, k � 0, k E Z and lk 1 - k2 1 � k. (Ill) The V(�kj, k2, 2 J . Z2, P0) with Re(z 1 - z2) < 0 and z 1 - z2 i= - k - 2, k E Z , k � 0 or z 1 - z2 = - k - 2 and lk1 - k2 1 > k. (IV) The V(P0, �k1 , k2, 2p2) with Re(z 1 - Zz ) = 0 and V(P0, �k1 . k2, 2� > z) V(P0, �k2, k 1. 22, 21 ) if Re z1 - z2 = 0. 8. 12. Using the analogue of Lemma for GL(2, (the proof is exactly the same) and Lemma 7.23 it is not hard to give the following classification of the irreducible, admissible, unitary (g, K)-modules :
§6
4.6
G.
0
C)
(I) The unitary characters of (II) The unitary principal series : V(P0, �k�> kz, 2�> 22), zj E iR, j = 1 , 2. (III) The complementary series : V(P0, �kj, k2, 2�. z) , z 1 + z2 E iR, z 1 - z2
>
z 1 - z2
>
- 2 and k1 - k2
=
0.
E
R,
REFERENCES 1. J. Dixmier, Algi!bres enveloppantes, Gauthier Villars, Paris, 1 974. 2. Barish-Chandra, Representations of semi-simple Lie groups. I, Trans . Amer. Math. Soc. 75 ( 1 953), 1 8 5-243. 3. , Representations of semi-simple Lie groups. I I , Trans. Amer. Math. Soc. 76 ( 1 954) , 26-65. , The Plancherel formula for complex semi-simple Lie groups, Trans. Amer. Math. 4. Soc. 76 ( 1 9 54), 485-528 . 5. --, Discrete series for semi-simple Lie groups. I, Acta Math. 113 (1 965), 241 -3 1 8 . 6 . -- , Discrete series for semi-simple Lie groups. I I , Acta Math. 1 1 6 (1 966), 1 -1 1 . 7. , Differential equations and semi-simple Lie groups, (1 960) unpublished . 8. Harmonic analysis on real reductive Lie groups. I, J. Functional Analysis 19 ( 1 975), 1 04-204. 9. - - , Harmonic analysis on real reductive Lie groups. II, Invent. Math. 36 ( 1 976), 1-55. 10. Harmonic analysis in real reductive Lie groups. III, Ann. of Math. (2) 104 (1 976), 1 1 7-20 1 . 1 1 . N. Jacobson, Lie algebras, Interscience, New York, 1 962. 12. N. Jacquet and R . Langlands, A utomorphic forms on GL(2) , Lecture Notes in Math . , vol . 1 1 4, Springer-Verlag, Berlin, 1 970. 13. A. Knapp and E. Stein, Intertwining operators for semi-simple groups, Ann . of Math. (2) 93 ( 1 97 1 ) , 489-578. 14. A. Knapp and G . Zuckerman, Classification theorems for representations of semisimple Lie groups, Lecture Notes in Math . , vol . 5 87, Springer-Verlag, Berlin, 1 977, pp. 1 38-1 59. --
--
--
-- ,
--
---- ,
86
NOLAN R. WALLACH
15. R. P. Langlands, On the classification of irreducible representations of real reductive groups, Mimeographed notes, Institute for Advanced Study, 1 973. 16. J. Lepowsky, Algebraic results on representations of semisimple Lie groups, Trans. Amer. Math. Soc. 176 (1 973), 1 -44. 17. D. Milicic, Asymptotic behavior of the matrix coefficients of the discrete series (to appear). 18. C. Rader, Thesis, University of Washington, 1 970. 19. W. Schmid, Some properties of square integrable representations of semisimple Lie groups, Ann. of Math. (2) 102 (1 975), 53 5-564. 20. V. Varadarajan, The theory of characters and the discrete series for semisimple Lie groups, Proc. Sympos. Pure Math. , vol. 26, Amer. Math. Soc. , Providence, R. 1 . , 1 973, pp. 45-100. 21. N. Wallach, Lectures on the representations of reductive Lie groups, Proc. Can ad. Congr. 1 977 (to appear). 22. G. Warner, Harmonic analysis on semi-simple Lie groups. I, II, Springer- Verlag, Berlin, 1 972. 23. E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, London, 1 902 (reprinted 1 965). BRANDEIS UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 87-9 1
REPRESENTATIONS OF
GL2(R)
AND
GL2(C)
A. W. KNAPP 1. SL ) . We shall give lists of the irreducible finite-dimensional representa tions, the irreducible unitary representations, and the nonunitary principal series. Then we discuss reducibility questions, asymptotic expansions, and the Langlands classification. Let g = (� �) be a typical element of G = !Fno � 0, an integer.
2(R
Irreducible finite-dimensional representations. nSL2(R). {f R n n (bx d) (x ((ax + c)/ ( bx d)). J n Unitary representations. n I Zyn-2 dx dy l f (z) { JJ (bz d)-n 1( ����). i)-n !F
Space = {g)f ) =
polynomial on +
of degree }, +
Finite-dimensional representations of G are fully reducible. The irreducible unitary representations were classified by Bargmann [1]. We give realizations in function spaces on the line or upper half plane. Realizations on the circle or disc are possible also. (1) Discrete series .@i; and .@;;, � 2. Space for .@i; = ! analytic for Im z .@i;(g)f(z) =
>
01 ll !ll z =
lm z>O
< co
}•
+
The space for .@i; is not 0 because (z + is in it. The representation .@;; is obtained by using complex conjugates. All these representations are irreducible, unitary, and square-integrable. The square-integrability (of a matrix coefficient) will be shown below. Principal series f!JJ+ .iv and qJ-.iv, v E Space for f!JJ ± .iv = if + , + + f!JJ ± .iv(g)f(x) = + if - . + = sgn + + +
(2)
V(R),
R. l bx (bxdl-1d)l-i'fbx((ax dl -c)/(bx 1-i•f((ax d)) c)/(bx d))
These representations are all unitary, and all but qJ-.o are irreducible. Equivalences f!JJ + .iv � qJ+.-iv and f!JJ- .iv � qJ-.-;v are implemented by analytic continuations of intertwining operators that we give below. f!JJ ± .iv is really the induced representation 1 ) with G acting by right translation and with the functions Ind�AN(a eiv. restricted to N = U �). Here MAN is the upper triangular group, a is trivial or signum on M = ± /}, and the character of A is
® ® {
(oet e-0 t)
-->
eiv t .
AMS (MOS) subject classifications ( 1 970) . Primary 22-02 ; Secondary 22E45. © 1979, American Mathematical Society
87
88
A . W . KNAPP
(3) Complementary series
Space for
res
rc•, 0
<
s
<
1.
J-oeoeJoe-oe f(xjx)J(jj dx_ r( bx + )
= {f: R -+ cj IIJII 2 =
rc•(g)f(x) = i bx
+
d l-1 -·
·
ax
-
+
yj 1
dy
s
<
oo}.
c . d
These are irreducible unitary. They arise from certain nonunitary principal series (see below) by redefining the inner product. Others. There is the trivial representation, and there are two "limits of dis crete series," !'Jt and �1· The group action with �t and !'J1 is like that in discrete series, but the norm is different. We have the relation f'JJ- . o � !'J t Ef) !'J 1 .
(4)
Nonunitary principal series.
f1JJ ± .C, ' e
Space = £2 (R, (l + x2)R• Cdx)
f1JJ ± .C(g)f(x) = i bx
+
= sgn(bx
C.
d i - 1 -Cf( (ax + +
d)ibx +
d
c)f(bx
1
+
-H
d))
J( (ax
+
if + ,
c)f(bx
+
d) )
if - .
Reducibility . We can see some reducibility in f1JJ ± .C on a formal level by specializ ing the parameter ' and by passing from z in the upper half-plane to x on the real axis. We obtain the following continuous inclusions : :F,. !;'; !;';
�;i ® !'J;;
!;'; !;';
[IJJ+ . -
f'JJ- .- [IJJ+ .n-1
if n even, if n odd, n � 0 ;
n
if even,
f'JJ- . .. -1
if n odd, n �
There is no other reducibility. The quotient by an vice versa. Asymptotics. Let be the rotation
k8
( -cossin ()() sincos ()())
:F
I.
i s the sum of two !'J 's, and
·
The maximal compact subgroup K = is abelian, and its irreducible represen tations are one-dimensional, -+ eimB with m an integer. We have G
=
k8
{k8}
KA +K with A + = {a1
= (g' e�' ), t � o}
and Haar measure is of the form dg = c sinh if g = Let U(g) be an admissible representation of G, and let q> 1 and q> 2 transform under K accord ing to k 8 -+ eim1B and -+ eimz8• Then
k8
2t dk8 dk8, dt
2t dt, t LP
Thus to test whether a matrix coefficient is in some � 0. test (U(a1) q> to q>2) and use the measure sinh EXAMPLE. !'J;i(k8)(z + i)-" = einB(z + i)-". Then
k8a1k8, .
class on G, it is enough to
REPRESENTATIONS OF
GLz
(R) AND
GLz
(C)
89
(�;;(at)(z + o -n , (z + i) -n) =
Jlm z>OJ ent[x + i(y + I)]-n[e2tx - i(e2ty
+
1)] -nyn- 2 dx dy.
By residues the right side is =
Cn
s�e-nt(y
+ 1 + e- 2 t) 1-2nyn-2 dy,
and this in turn, after the change of variables y Then
s I · . - 1 2 dg G
=
cc�
n>
=
y ' (l + e- 21), is
=
c�(cosh t)-n.
s�(cosh t)-2n sinh 2t dt,
which is finite for 1 . Thus this matrix coefficient is square-integrable on G. A theorem in functional analysis due to Godement [3] implies that all matrix coef ficients are square-integrable on G. In the example, we could see the matrix coefficient was square-integrable by computation. There is a general technique, due to Harish-Chandra, for getting at the behavior of matrix coefficients by means of differential equations. Let
(b ?), e = (8 b),
h=
_
f
=
(? 8)
be a basis for the Lie algebra of G. The D !h 2 + ef + fe is a member of the universal enveloping algebra. For SL2{R), D generates the center of the universal enveloping algebra. (For larger groups, it must be replaced in this discussion by the whole center of the universal enveloping algebra.) It acts as a scalar on each representation in our lists, hence on each matrix coefficient. Take a matrix coefficient whose two K-dependences are according to known characters of K, and regard the matrix coefficient as an unknown function. Then the equation [)(coefficient) = c(coefficient) leads to a second order ordinary differential equa tion on A + , with t as independent variable. The classical substitution is s = cosh t, and the resulting differential equation has three singularities, all regular ; we are interested in the behavior at oo . (If the "known characters" of are trivial, this is Legendre's equation.) This substitution does not generalize well, and Harish Chandra's treatment of this equation amounts to making the substitution z = e-Z t instead. The resulting differential equation has four singularities, all regular, and we expand about z = 0, using standard regular-singular-point theory. The result is that
Casimir operator
s
=
K
=
00
coefficient(a1) = e- < l +C l t i; cne-Znt + e- < 1-C > t i; dne- 2nt n=O n=O except when � is an integer, in which case there may be factors of t that arise from factors log z in the solution. If one of the leading terms vanishes, the whole cor responding infinite sum vanishes. The eigenvalue of D determines � ' and in particular the matrix coefficients of fl/±.C lead to the expansion with � present. From this expansion, we can read off LP-integrability conditions, since we are to integrate for t ;;;; 0 the pth power against sinh 2t dt, which is comparable with e21dt. We see that fl/± ,iv has coefficients in
A . W. KNAPP
90
screte series .!?&� n L2(G). LDi2(G). tempered.
£ 2+•(G) £2 +•
e> e >n Langlands classification.
with � 2 have one for every 0, but not in sum absent, in order to have coefficients in Representations with coefficients in for every 0 are said to be The tempered representations are f!JJ± .iv and .@� with � 1 . Notice how in general the two leading terms give some information about where imbeddings occur as subrepresentations in the nounitary principal series ; Wallach dealt with this point in his lectures. For general Langlands parametrizes the irreducible admissible representations by triples 'It , v), where = MAN is a standard parabolic, 1t is (the class of) an irreducible tempered representation of M, and v is a complex-valued linear functional on the Lie algebra of A with real part in the open positive Weyl chamber. The Langlands representation lp('lt, v) is the unique irreducible quotient of lnd�(7t ® ® I). In our case, = (� !) is minimal para bolic, or P = P There are two (one-dimensional) representations of M = { ± 1} , and the functional v enters a s the complex number ' with Re ' 0 ; the character = e xp , The Langlands list then includes the unique of A is = exp(v log irreducible quotient of f31J±.C for each ' with Re ' 0. P = Here v is irrelevant, and M = We simply get the irreducible tempered representations of The Langlands classification itself does not address the question of what these are, though one of the theorems implies for our that they are subrepresentations of discrete series or unitary principal series. The Langlands classification theorem describes the uni que irreducible quotient more precisely than we have done. Kunze and Stein [4] showed in 1 960 that the operator
(P,G,
P
e" P G. Case minimal. at at) ( t). > Case G. G. G. Intertwining operators. J f(y) dy J yy)Jfl(y) f --+
--+
oo
- oo
oo
- oo
lx
lx
-
G
for g>+.C,
- yJl c
sgn(x -
>
c
dy
for g>-. c ,
intertwines g>C with g>-c. Note that the integral is convergent only if Re ' 0. Later [5] they found a formula in the induced picture, namely f --+ f R f n w- l dn, where N = U �) and w = U b) . This is the composition of two operators, f--+ f R n and a relatively trivial translation operator by w- 1 . Define A(P : : 1t : v )f(x) = fR f(nx) Under the Langlands conditions on v, this in tegral is convergent if/ is K-finite. The theorem is that
P/(fig) d
lp('�t, v)
=
Ind�(7t ®
e"
dti.
® 1 )/ker A(P :
P:
1t :
v) � Image A(P :
( g)>
P:
1t :
v).
2. Other groups. GL2 (R). To pass from SL2 (R) to the group SL�(R) of matrices of determinant ± I , we first induce the representations of SL2 (R). The g>'s and ff's split into two equivalent pieces, and the !?&'s yield irreducibles on SLt(R) that restrict back to EB .!?&- on SL2 (R). This construction gives us the representations of SLt(R). Then to pass to GL2(R), we paste on a character of the group R+(b �) . SL2(C). This group has finite-dimensional representations given by two integer parameters ; the representations can be realized in spaces of polynomials in z and z on The group has no discrete series. We have
.@+
C.
REPRESENTATIONS OF GLz (R) AND GLz (C) M =
{(ei0O 0e-;o)}
91
and
so that the nonunitary principal series is parametrized by an integer n (for M) and a complex number 1: (for A) ; by restriction of functions to N U �), we can realize these representations in spaces of functions on C. See [2] for more detail. The unitary principal series is all irreducible and provides the only tempered irre ducibles, and parameters and ( - n, - lead to equivalent representations. The Langlands classification points to the Langlands quotients of the nonunitary principal series with Re 1: 0 and to the irreducible tempered representations. GL2(C). To an irreducible representation of SL2(C), we paste on a character of c x (6 �) that agrees with the representation on (-6 _£) . In this way we obtain all irreducible representations of GL2(C). =
(n, iv) >
iv)
REFERENCES 1. V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math . (2) 48 ( 1 947), 568-640. 2. I. M. Gelfand, M. I. Graev and N. Ya. Vilenkin , Generalized functions, vol. 5, Academic Press, New York, 1 966. 3. R. Godement, Sur les relations d'orthogona/ite de V. Bargmann, C. R. Acad. Sci . Paris Ser. A-B 225 ( 1 947), 5 2 1 -523, 657-659. 4. R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the 2 x 2 real unimodular group, Amer. J. Math. 82 ( 1 960), 1 -62. 5. -- , Uniformly bounded representations. III, Amer. J. Math. 89 ( 1 967), 3 8 5 -442. 6. S. Lang, SL,(R), Addison-Wesley, Reading, Mass., 1 975. 7. R. P. Langlands, On the classification of irreducible representations of real algebraic groups , mimeographed notes, Institute for Advanced Study, Princeton, N. J . , 1 973. 8. P. J. Sally, Analytic continuation of the irreducible unitary representations of the universal covering group of SL(2, R) , Mem. Amer. Math . Soc. 69 ( 1 967). CORNELL UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 93-105
NORMALIZING FACTORS, TEMPERED REPRESENTATIONS, AND L-GROUPS
A. W. KNAPP* AND GREGG ZUCKERMAN* Several lecturers have alluded to the intertwining operators associated with principal series representations, particularly for SL(2, R) . These operators and their normalizations play a role in the trace formula, in reducibility questions for principal series, and in the construction of an inner product that exhibits certain representations as unitary. We shall review the development of these operators and their normalizations in the context of a linear real reductive group G (as defined in Wallach's lectures) whose identity component has compact center. We shall be especially interested in the application to reducibility questions for unitary principal series and other continuous series of representations coming from parabolic subgroups, since the answers to these questions lead to a classification of irreducible tempered representations and thereby complement the Langlands classification [12]. The answers concerning reducibility are reviewed in §2 in terms of three easy-to-calculate finite groups, denoted W, W ' , and In lectures during the authors mentioned how, in some special cases, the groups W, W ' , R, initially defined in terms of roots, could be defined in terms of co-roots. Building on this presentation, Langlands [14] was able to redefine these groups in general in terJ:ll S of the L-group. We present his definitions, along with an example, in In we summarize earlier work [8], [9] that leads from the R-group to the classification of irreducible tempered representations.
R.
1975-76,
§3. §4
1. Intertwining operators and normalizing factors. In the group G, fix a maximal compact subgroup K and Cartan involution (). To each parabolic subgroup P, we associate the Langlands decomposition P MAN with MA {}-stable and with M a linear real reductive group whose identity component has compact center. To the pair (�. v) , where � is an irreducible unitary representation of M and v is a com plex-valued linear functional on the Lie algebra a of A, we associate the repre sentation Up(�, v, with in G, given by =
(1.1)
x), x
We adopt the convention that G acts on the left in the induced representation. A member of the representation space satisfies exp ( - ( v + pp) log �(m) - 1
f
f(xman)
a)
=
AMS (MOS) subject classifications (1 970) . Primary 22E45 . *Supported by the National Science Foundation.
f(x),
© 1 9 7 9 , American Mathematical Society
93
94
A. W. KNAPP AND GREGG ZUCKERMAN
where pp is the usual half-sum of restricted roots associated with P. If P = P0 is minimal parabolic, these representations comprise the In formula ( 1 . 1), we have assumed that � is an irreducible unitary representation of M, but we shall allow also that � is a nonunitary principal series representation of M, provided we are not working with formulas involving adjoints of operators. For SL(2, R) , one can restrict the functions in the representation space for the nonunitary principal series to the lower triangular group R = ON = {(! m and realize the representations in spaces of functions on R . The representations be come
nonunitary principal series.
g; ± , C(g)f(x) = =
I
- bx + d j-H
1(
-c )
ax - bx + d
sgn ( - bx + d) j - bx +
d j - HJ( !:_�x / ) -
d
if + , if - ,
if g = (��). Kunze and Stein [1 0] showed that the operator f
(1 .2a)
--> -->
(1 .2b)
Joo Joo
-
-
oo
oo
f(x - y) dy j y jl C
for &+.c ,
-
y) dy ( sgn y) f(x I Y jl C
for &- .c,
intertwines &C with &-C when it is convergent, namely for Re � found a formula in the induced picture, namely f -->
L
/
(g wfi) dn,
where w =
( � 6) .
>
0. Later [11] they
_
This is the composition of two operators, f --> J,v f(gfi) dn and a relatively trivial translation operator by w. The first operator intertwines the representation induced from P = MAN with the one induced from = MA N and the same data on MA . In the general case, let P 1 = MAN1 and P2 = MAN2 be two parabolics with the same MA , and define
P
(1 .3) Formally ( 1 .4)
Up/�, J.i, g)A(P2 : P1 : � : J.i) = A(P2 : P 1 : � : J.i) Up1 ( � , J.i , g) .
In general, the integral ( 1 . 3) will not converge but will be defined by analytic continuation. To accomplish this analytic continuation, we need the representa tions to occur in a single space, as J.i varies. This space is obtained by restricting the functions to K. Say � operates in the space H' . We consider functions f: K --> H' satisfying f(km) = � (m)-I f(k) for k E K, m E M n K. Under the action by g , we replacef(k) by f(g- 1 k) and restrict back to K. Thus we are led to define ( 1 . 5)
Up(�,
J.i,
g) f(k) =
exp ( - (pp + J.i) H(g-Ik) ) � (,u ( g- I k) ) - 1 f(�> (g- Ik)) ,
where x decomposes within = KMAN a s n. The analytic continuation of ( 1 3) is accomplished in three stages. In the first stage the essence of the argument can be seen by continuing (1 .2a) if f is smooth
G
.
K(x),u(x)eH<xJ
NORMALIZING FACTORS AND £-GROUPS
- y) f y
95
and compactly supported. Since has compact support, we can integrate over a 0 in a finite Taylor series with remaind finite interval. Expand f(x about er. Each of the main terms can then be integrated explicitly and continues mero morphically to the whole plane. The error term is integrable for Re ' > - - 1 if is the order of the Taylor series. Hence the integral continues meromorphically to the whole plane. In Schiffmann [15] and in [6], it is shown that this style of argument yields a meromorphic continuation of if and are minimal parabolic and is one-dimensional. In the second stage, for minimal parabclics with dim > 1 , Schiffmann shows how to decompose into a product of subgroups and to write as an iterated integral ; the analytic continuation is essentially then reduced to the case dim The third and final stage of the analytic continuation was obtained independent ly by Wallach [17] and in [7]. We can use Casselman's subrepresentation theorem to imbed as an (m, K M) module in a nonunitary principal series representa tion of It follows from the double induction theorem that is a subrepre sentation of a nonunitary principal series and is a restriction of an intertwin ing operator for the nonunitary principal series. Then has an analytic continua tion for K-finite j, and holds if is replaced by a member of K or of the Lie algebra g of G. Moreover, with Haar measures normalized suitably, we obtain =
n
n
(1.3) P1 P2 N1 n Nz. A = 1. n (1. 3) (1. 3) (1.1) (1.4) g
(1. 3)
M�.
(1.6)
AA
for � unitary, if the adjoint is defined K-space by K-space, and
A(P2 : P1 : �: v) R(w-1) A(wP2w-1 : wP1 w-1 : w�: wv) R(w), v. v =
where R(w) denotes right translation by an element w in K representing a member of the Weyl group W(a). These operators tend to have poles at many interesting values of We introduce scalar normalizing factors-in part to eliminate some of these poles, in part to make the operators unitary for imaginary, and in part to make the operators behave nicely under composition. The normalizing factors are not unique, and different choices are useful for different purposes. Again the construction is in several stages. We impose the condition that have a real infinitesimal character. Matters are based on the following lemma [6, p. 544], proved using Weierstrass canonical products.
�
LEMMA
Ifr;(z)onisthaemeromor p, hicfunction in the plane such that (i) r;r;((z)z)1.1.i�s real real axi s 0 on the imaginary axis, r; ( z) r; all z, phic function r(z) in the plane such that r;(z) tr(-z)r(z) hen thereandexir(z)st(s-z)for ais rea/for meromor real z. P MAN A(P:P:�: v) A = 1). A(P:P:�:v) Up(�, v). Up(�, v) v (1.7) A(P: P: �: v)A(P: �: v) = r;(P: P: �: v)I (ii) (iii)
=
=
The first stage of the construction deals with a minimal parabolic and G of real rank one (dim In this case is a self intertwining operator for It is a result due to Bruhat [3] that, for nonzero imaginary, the unitary representation is irreducible. It follows readily that =
P:
96
A. W. KNAPP AND GREGG ZUCKERMAN
with v zpp meromorphic and scalar valued for z in C. One checks that (i) 'l}(P : P : ' : v) is real for v real. (ii) 7J(P : P : , : v) � O for v imaginary. (iii) 7J(P : P : t; : v) 7J(P : P : t; : - v) for all v. (iv) 7J(P : P : t; : v) depends only on the class of t;, and if q; is an automorphism of G leaving K and P stable, 7J(P : P : t;'�' : v) 7J(P : P : f; : v). Then we can apply the lemma to obtain r(P : P : ' : v) and to define normalized operators by =
=
=
d(P : P : f; : v)
=
r(P : P : , : v) - 1 A(P : P : , : v).
The normalized operators satisfy d(P : P : , : v) d(P : P : t; : v)
=
and d(P : P : , : v)*
=
I
d(P : P : t; : - v).
These two relations together imply d is unitary for v imaginary. The second stage is to handle a minimal parabolic for general G. Use Schiff mann's decomposition of a general intertwining operator into operators that are essentially rank one operators, and use the product of the normalizing factors as normalizing factor for the given operator. Then one proves the relation ( 1 .8)
The third stage is to handle a general parabolic P MAN with dim A 1 . We again use the trick of imbedding ' as a subrepresentation of a non unitary principal series representation of M. If we combine this trick with formula ( 1 .8) for minimal parabolics, we are led to the conclusion that ( 1 . 7) holds for our P with dim A 1 and that 7J has the same properties as before. Again we apply the lemma to obtain a normalizing factor r, and we set d r- 1 A . The final stage is for a general parabolic with G general and is handled in the same way as with a minimal parabolic and G general. If the r's are chosen com patibly, the result is as follows [7, p. 2460]. =
=
=
=
1 .2. The normalized intertwining operators satisfy d(P3 : P1 : , : v) d(P3 : Pz : f; : v) d(Pz : P 1 : t; : v) . d(P 1 : P2 : t; : - v), K-space by K-space. d(P2 : P 1 : , : v)* d(P2 : P 1 : t; : v) is unitary for v imaginary. lf w in K represents a member of the Weyl group W(a), then d(P2 : P1 : , : v) R(w- 1 ) d(wP2w- 1 : wP 1 w-1 : wt; : wv)R(w).
THEOREM
(i) (ii) (iii) (iv)
=
=
=
For w in K representing a member of W(a), let dp(w, ,, v)
=
R(w)d(w- 1 Pw : P : , : v).
From (i) and (iv) we obtain the cocycle relation (1 .9)
From (ii) we find that ( 1 . 10)
and hence that dp(w, ,, v) is unitary for v imaginary. The intertwining relation is
NORMALIZING FACTORS AND L-GROUPS
Up(w.;,
(1 . 1 1)
wv,
g)..#p(w,
.;, v)
= ..#p(w, .; , v) Up(.;,
97
v,g)
for in K or in g. For application to adelic situations, a more specific normalization is needed. At almost all places the representation has a K-fixed vector, and the infinite tensor product of local normalized intertwining operators will be defined only if the local operator fixes the K-fixed vector. This condition determines the normalizing fac tor. For example, with PI and P2 minimal parabolic and .; = 1 , we are led to normalize ( 1 . 3) by taking
g
r(Pz : PI : .; : v) =
_
J
N 1 n N2
exp ( - ( pp + ) H (n)) dn ,
v
which can be computed in terms of Harish-Chandra's c-functions. Schiffmann pursued this idea further at an early stage in his work leading to [15]. Langlands was led to conjecture [13, p. 282] in general that a valid normalization is obtained by using the quotient of two £-functions : (1 . 1 2) Here rp is the homomorphism of the Weil group of R into the £-group L(MA) of and pp2 1 p 1 is the contragredient of the representation of L(MA) on the Lie algebra of Lfji n LNz. Arthur [1] proved that (1 . 1 2) is a valid normalization if the Haar measures are normalized suitably.
MA corresponding to .;
® e",
2. The R-group, a first formulation. Fix a discrete series representation .; of M, and consider the corresponding continuous series representation Up(.;, ) with v imaginary. In [7], it is stated how the problem of determining the algebra 'if(.;, v) of operators that commute with Up(.;, v) can be reduced to algebraic problems involving certain finite groups. Fix v. Let W be the subgroup of elements s in W(a) that fix v and the class of .;. If w is a representative in K of an s in W, then one can define .;(w) in such a way that .; extends to a representation of the smallest group containing M and w ; the definition of .;(w) is unique up to a scalar factor equal to a root of unity. Then .;(w) ..#p(w, .; , ) is independent of the representative w , and we can write
v
(2. 1 )
v
p( , .;, )
.;(s) d s v
for it. One sees from (1 . 1 1 ) that the unitary operator (2. 1 ) intertwines Up(.; , ) with itself. The essence of the next theorem is due to Harish-Chandra [4]. THEOREM 2. 1 .
(2. 1 ) ,
The operators for in span the commuting algebra s
W,
v
'if(.;, v).
Despite formula (1 .9), it does not follow that the map (2.2)
s e W --> .;(s)dp(s, .;, v)
.;(s)
is a homomorphism into unitary operators ; the factors need to be chosen com patibly, and there may a priori be an obstruction to making such a choice. How ever, the map (2. 1 ) is at least a homomorphism into the projective unitary group. Let W' be its kernel. THEOREM 2.2.
The group W' is the Weyl group ofa root system contained in the Ll'
A. W. KNAPP AND GREGG ZUCKERMAN
98
=
R
R
roots of (g,of lconsequent ly the posiW'tive iroot s a semi directTheproduct ifthe imap s defi(2.ned2) assetconsitohfestsubgroup e avi n g st a ble s i n i m age o f equals stheoforderlinearlofy independent operators; consequently the dimension of <'(J(�, R a) ; W
W
Ll' .
v)
R.
It turns out that the elements of R are characterized in W as those elements for which the normalizing factor of (2. 1 ) is holomorphic at v . Thus W' and are inti mately connected with the functions 'f)( Pz : PI : e : v) defined by A(P 1 : P2 : � : v)A(Pz : P 1 : � : v) = 'f) (Pz : P1 : � : v)l.
In turn, these functions can be connected by analytic methods with the Plancherel measures of subgroups of these measures have been determined by Barish Chandra. Consequently we can describe Ll' very concretely, as follows. Adjoin to a a compact Cartan subalgebra of the Lie algebra m of .,({, and denote a typical root by (3. Let }. be the Barish-Chandra parameter for the discrete series � Let a be an a-root and suppose 2a is not an a-root. If a has even multiplicity and 2a is not an a-root, define
G;
[J. �.a (v)
=
If a has odd multiplicity, define =
[J.�.a1 2 (v) whereh, a(v) = tan (n(v, =
e(ra )
f.i.E,a (v)
a
=
rr (}. + v , {3 ) .
{Ji a=a
(
IJ
,81 a=ca, c>O
) II a 1 2) or cot (n(v ,
- ( - l ) Z ll a l 2] or
)
(}. + V , f3 ) f�.a(V) ,
a)
I I a 1 2) according as
+ ( - 1 ) 2llai 2J.
Here a is a real root and there is a corresponding homomorphism of and ra denotes the image of - I. Now we can characterize Ll' as
G,
=
Ll'
Is
{ roots a of a a is in W and [J. � . a(v)
=
SL(2,
R) into
0},
where ea denotes the reflection corresponding to a. As before, R
=
E
>
0 for every a
>
0 in Ll'} .
{p p is afiniotrse di�(s)rectcansumbeofsel2-element groups e ct e d so thgatebrathe <'(Jmappi ngis commut (2.2) is aa For s i n t h e operat homomor p hi s m i n t o uni t a ry operat o rs; consequent l y t h e al ( �, tive. There exists a set of positive orthogonal real roots { such that t h e onl y root s i n t h e s p an o f are t h e each r i n i s o f t h e form s ; and each occurs in the decomposition of some r in G tempered G) §2, M Wi
a
Then we have the following result [5], [8]. THEOREM 2.3. (a) R (b) R
Z2 •
v)
(c) (i)
a1.1 . . . saJ.n
(iii)
ai
:Yt
±
ai
;
(ii)
:Yf' =
ah
R R.
..., a
q
}
3. The R-group, a second formulation. An admissible representation of
is if its K-finite matrix coefficients are in £2+•( for every e > 0. It is known that every irreducible tempered representation is a summand in some representa tion of the type considered in and a unitary induced from discrete series on character on A . The R group can be used to decompose these induced representa-
99
NORMALIZING FACTORS AND L-GROUPS
tions and give a classification of the irreducible tempered representations. We re turn to this point in §4. For now, we want to consider W, W', and R from a different point of view. We begin with some motivation in the case that G is split and our parabolic is minimal. Then M is finite abelian and is generated by the elements ra that are the images of ( A �) in the SL(2, R)'s that correspond to each root. Here � is a character of M, and we assume lJ 0. Let the root system be Ll. It is a simple matter to see that Ll' {a E �(ra) 1 } . The elements ra transform under Weyl group elements differently from what one might expect. The correct rule is -
-
=
=
.LI I
=
1 s f. ras{3
=
rs pa = rarf32<j3, a)/l a l2 •
.LI'
As a result, need not be closed under addition within the set of all roots. It is in the co-root system that there is closure under addition. The co-root of a is aV 2a/(a, a), and LJ ' V is closed under addition within LJV. We introduce a kind of dual group G-, not the L-group just yet. G- is a connected real group with a compact Cartan subgroup and root system LJV, arranged so that the roots of LJ ' V are compact and the others are noncompact; it is to have a centerless complexification G0, a Lie algebra g-, and a Cartan decomposition g- t- EB Elements of W' lead to elements in the Weyl group of t-, with representatives in G-. Elements of W lead to elements in the Weyl group of g'C that leave the compact roots stable, hence normalize f"C and r- EB hence normalize r- and and therefore g-. It follows that R injects onto the quotient Normc c; (g)/G-. This style of argument can be pushed to yield a proof that R = .E Z2 in this case. Langlands [14] built on these ideas and gave a formulation in general of W, W' , and R in terms of the L-group. We use the notation of Borel [2] approximately and will follow the L-group constructions given by Langlands [12] when we need them. First let us see what the above example has to do with the L-group. Of course, LGO is just Gc;, up to coverings. The character � is specified by giving �(ra), and ra behaves like the (character associated to the) co-root av. To know aV on � for each aV is to know � as an element of the dual torus L T0 • There is a corresponding homo morphism of the Weil group into LG given by 1 for E ex and � x a E L T; here a is the trivial outer automorphism of LGO, trivial since G is split. The elements of the Weyl group that centralize the image { 1, � x a of are those of W; this statement motivates Theorem 3 . 1 below. The roots whose root vectors are centralized by the image of are those in Ll' ; this statement motivates Theorem 3.3 below. The observation above that R is isomorphic to Normc';;(g)/G could have been stated in terms of and then motivates Theorem 3.4 below. We pass to the general case. Fix a minimal parabolic P0 M0A0N0 in G, and let T be a Cartan subgroup of G containing A0. Let LGO ><1 { 1 , a be the L-group of G, with its maximal torus L TO and Borel subgroup LBO. The "standard relevant parabolics" LP, in the sense of [2], correspond to the standard parabolics P 2 Po of G. Let L (MA) be the Levi component of LP. According to [12], the classes of L-indistin guishable irreducible admissible representations of G correspond to conjugacy classes (suitably defined) of certain kinds of homomorphisms of the Weil group into LG. A representation is tempered if and only if the image of is bounded. It comes from the series of §2 for P if, after conjugation, has image in the Levi component L(MA) of Lp but not in any smaller such Levi component. We may and =
ip-.
=
i p-,
p-
rp(z)
rp
K-
z
=
rp
=
}
rp rp rp
rp(z") } rp
=
A. W . KNAPP AND GREGG ZUCKERMAN
100
shall assume that q>(ex) £;;; L TO and q>(-r) is in the normalizer of L TO. We shall give an example of these notions at the end of this section, illustrating the three theorems below. Fix a summand in a representation Up(f;, v). The corresponding q> goes with a whole L-class of representations in G. Since the image of q> is in L(MA), the £-class consists ofthose representations obtained by inducing from a single L-class for MA. The latter class corresponds with the map q>MA obtained by regarding q> as having image in L(MA), and it is characterized by the data v , the central character of f;, and the infinitesimal character of f;. But the definitions of W, W', and R depend only on v, the central character of f;. and the infinitesimal character of f;. Thus the question arises how to define W, W', and R in terms of q> directly. [Digression. A simple consequence of the discussion above and a lemma due to D. Shelstad is that the cardinalities of R and the L-classes corresponding to q> and q> MA are related by the formula
I ®
I L-class for q> I = R I I L-class for q>MA I · In fact, the representations in the L-class for q> are all the irreducible constituents of all representations Up(f;', v) with f;' v in the L-class for q>MA. The R-group for all these Up(f;', v) is the same as f;' varies. Moreover, Shelstad ' s lemma says that Up(f;Io v) and Up(£;2, v) are disjoint if £; 1 v and £;2 v are L-equivalent but not equivalent. The formula follows.] Let S Cent(Image q>), the centralizer being taken in LGO, and let so be the identity component of S and � its Lie algebra. The Weyl group W(a ) imbeds in the Weyl group of (LGO, L TO). Namely any ele ment of W(a ) has a representative that normalizes the Cartan subalgebra of m and preserves positive roots of m ; in this way an element of W(a) leads to a unique member of the Weyl group of (LGO, L TO). With this identification in mind, we have the following result. =
®
®
THEOREM 3 . 1 (LANGLANDS). W = (Norm (L TO) n S) f(L TO n S), being
taken
in LGO.
the normalizer
We denote the lattices that are given as part of the L-group data by L and LV : L = Hom (Tc. ex) � Hom ( e x ' LTO), LV = Hom (L T0 , ex) � Hom cex , Tc).
These are in duality as follows : If A is in L = Hom(Cx , LTO) and Av is in LV = Hom(L TO, ex), then Av A is a power of z in ex ' and (A, AV) denotes this power. The homomorphism q> on ex ' with values in L T0 , can be written symbolically as 0
q>(z) = zPz• = zr•(zz)•,
(3.1) where p. and v are in L
®
e and p. - v is in L. Formula (3. 1 ) means that .ii.V (q>(z)) = z (zz)<•, M
for all Av in LV. The condition that the image of q> lies in no proper parabolic within P means that q>(-r) aV = - av exactly for the roots of L (MA) and that ( p., aV) =F 0 for all roots of L (MA). (See Lemma 3.3 and the paragraph before Lemma 3 . 1 in [12].)
NORMALIZING FACTORS AND L-GROUPS
10 1
+
1 and - 1 eigenspaces
Let Lt be the Lie algebra of L Tand decompose Lt into the for as L t L + EB Lt - .
rp(..-) = t algebra of XLt+ = I; X
LEMMA 3.2 (LANGLANDS) .
5,
The Lie algebra Lt+, which lies in is a Carlan sub XI" = rp(Cx). = = = =
5.
c�l X means that X is the sum of PROOF. If is in 5, then the condition that and a sum Y of root vectors. Moreover, Y must centralize a member of If Y av , then y yq> C z ) I; Yav I; z
(J.i,
aV)
=
0. Thus
CXav· V)=(v, aV) =O
We are to prove that L t + is maximal abelian in 5. It is enough to see that no Xav 0 centralizes L t + , i.e., that no such av vanishes on with (f.L, av) ()), aV) However, (f.L, av) 0 implies av is not a root of L(MA), as we noted above, and then av # - av. Thus aV does not vanish on
= = = Lt+. L t+ . rp(..-) The group W' is canonically isomorphic with the Weyl group of SupposeSO) injectis ssemiintosimSjplSe0,andandsitmheplquoti y-connect ed.isomor Then e nt i s tphihecgroup F = is a split group and is a minimal parabolito tch. e group. Moreover, F SU(2, 1 1 2 2 ) 1 1 2( 2 g = -2-1 12 2-1 12 THEOREM 3.3 ( LANGLANDS) . (5,
Lt +).
THEOREM 3.4 ( LANGLANDS) . R
(L TO n S)/
EXAMPLE. Let G be
(L T0 n
Gc
=
P
{ 1 } if G
1), which we conjugate for convenience by 0 1 0
0
so as to be able to take
T=
B
0
'
(et+iO e-2iO et-iO)·
The Borel subgroup we use is the upper triangular group. The group G is quasi split but not split, and the L-group LG is LG
=
LGO
>
Z2
=
PGL(3, C)
>
{ 1 , a} ,
where a is a particular realization of the nontrivial outer automorphism of PGL(3, C). Specifically LTO is the diagonal group, L T is L TO >
X1 = ( I
0 0 0 0 0 0 0 0
)
and
X2 = ( I ), 0 0 0 0 0 0 0 0
A. W. KNAPP AND GREGG ZUCKERMAN
102
and we require X{ w0 (xtr)-1 w0 1 , where
=
X2 , X�
=
X1 .
w0 2
Explicitly the action of
=
1.
(00 -10 01 ), 0 rp +1 1 l
O"
on LGO is
x"
=
0
-
and 0" = On the Lie algebra Lt, we have diag(c 1 , c2 , c3)u ( c3, - c2 , - c1) . We assume that the homomorphism maps ex into L TO and maps 'r into X 0" , where w is in Norm(L TO). Up to conjugation of rp, there are two possibilities for w as a member of Norm(L TO)fL TO, namely the cosets of l and w0 • In either case, rp(-r) yields an involution of Lt, with and - eigenspaces Lt Lt+ EB Lt - . If then Lt+ and we are led to discrete series. We shall be interested in the contrary case Say w xV with xv in L TO. The action of on diagonal matrices L TO is in this case the same as the action of O" . We write members of = Hom(Tc, ex) by abbreviating
[w] [w0],
[w] 0 [1]. L
=
=
=
=
))
z-I w-1, a[z] + b[w]. L. {a., ow}
J.(diag(z,
as
A
{[z], [w]}
Here is a basis of Hom(L TO, e x). If
(3.2)
w
=
=
w
=
rp(-r)
za wb
=
denote the dual basis of £V
Let
then
;.v
The action of namely
rp(-r)
=
co.
+
dow·
on L TO yields an action on LV and then one on L, by duality,
(a[z] + b[w])�'<�! - b[z] - a[w]. rprp((z)t) rp(z)�'<<(3.1), ), (3.1) ,u�'<<). L c 1 [z] + c 2 [ w ] 119'<
for z in e x . Since t and then implies )) ® e occurs only in the L part.) Write
Let obtain
be as in =
,U
=
))
=
=
=
-rz-r- 1 is in the Weil group, we (Here the action of on
rp(-r)
with CJ , Cz in e ;
=
-
The condition that ,u )) be in says that of we c2 is in From p. see that the character of T that is to provide data for a principal series representa tion is in this case e H ..... el2 = e. If H
=
diag(t
+ i (), -2i() i () 0,
If Re(c1 - c2)
, t-
>
27 [12],
) , we are led to the character
we are led to nontempered representations. We shall spe-
103
NORMALIZING FACTORS AND £-GROUPS
cialize to the principal series with c 1 - c2 = which corresponds to ein8 on the + part and trivial on the A part, by taking p = These principal series representations are reducible (splitting into two parts) if and only if is even and #- 0. To see what qJ(z) is, we realize as
M
LT0
0, n[z]/2 n[w]/2.
L TO
= diag(a 1 >a2 , a3) mod center.
n
Apply ( 3. ) to the element qJ(z) with c = , d = 0, and then with c = Then
2
(ada2 )(z)
= o,qJ(z)
=
(a3ja2 )(z)
qJ(z)
=
(z"/l z l" "/1 1 0
z " ) -2 0
(z
0 0
=
z , oz>(zt) <•, o.>
0,
d = I.
= z <2p, oz> (zz) <-p, oz> = z"f l z l n
and Hence
1
z"/ l I"· z
z"/1 ) 0 0
1 /3
z ln
mod center,
in the sense that if the cube roots are extracted compatibly, their ambiguity dis appears when we pass to PGL(3, C). To compute the image of 9' completely, there is one more step-to determine qJ(r) = xV x 0'. We must have qJ(z-)2 = qJ( ) and we conclude (xV)(xV) " qJ( - ) We can take -
1
.
xV
=
1
1
,
if n is even,
= diag(i,
1
,
- i)
n
if is odd.
Now we can compute the groups S and S0 that are the subjects of the theorems of this section. There are three distinct cases for the image of qJ, corresponding to = 0, odd, and nonzero even. The pattern of the computation is to compute S1 = Cent qJ(Cx), which is connected, being the centralizer of a torus, and then to compute the centralizer of qJ(r) in S 1 . The idea is to work as much as possible on the Lie algebra level. For = 0 and odd, we use a trick to obtain S; S is con of and is found to be C) . Since tained in the normalizer in C) has no outer automorphisms, S must conjugate so by inner automorphisms. Then it is easy to see that S = so . The results are as follows.
n n
n
nL G0 s, n s
n
= 0:
s
=
n odd '
o �
n
s
even #- 0 :
=
(� � �). (� g -�· (�0 � �). 0
c
-a
0 -a
s!(2,
=
so ,
s �
SO,
S
S = so
�
{1, w0} .
s!(2,
104
A.
W.
KNAPP AND GREGG ZUCKERMAN
These results are compatible with the theorems of this section and the facts that
n n
= 0 or odd : even '# 0 :
=
= {I},
= { 1 },
=
W' Z2, R W' R Z2.
4. Irreducible tempered representations. We continue to investigate the repre sentations Up(f;, v) discussed in Here f; is a discrete series representation of M and v is imaginary on W. We need a parametrization for the discrete series, and we use Harish-Chandra's. Let r- be a compact Cartan subgroup of M, let t- be its Lie algebra, and let ZM be the center of M. Each discrete series representation of M is determined by a nonsingular linear form A on it- and a character X on ZM. The form A satisfies the integrality condition that A p (with p equal to half the sum of the positive roots of M in some order) lifts to a character on r-, and A and X satisfy the compatibility condition that and X agree on their common domain. We write f; = f;(A, C, where C is the unique Weyl chamber of it- with respect to which A is dominant. Two such f;'s are equivalent if and only if X = and there is some in the Weyl group M) = NormM(T-)/CentM(T-) with WA = A' and C = C'. Representations that are "limits of discrete series" are discussed in [18]. We can parametrize them in the same fashion, writing f; = f;(A, C, except that A is allowed to be singular and there is more than one choice of C that makes A dominant. These representations are irreducible or zero, and the criterion for equivalence of nonzero ones is the same as for discrete series. A is an induced representation Up(f;(A, C, x), v) , with f;(A, C, a limit of discrete series on If A is nonsingular, so that f; is in the dis crete series, then we say that the basic representation is A basic representation Up(f;(A, C, v), has if for each root a of (m, t-), (A, a) = implies that is not in M). A representation induced from discrete series automatically has nondegenerate data. Nondegeneracy accomplishes several things. For one thing, it eliminates the 0 representation from consideration. For a second thing, it assigns a definite par abolic to the data for a basic representation. For example, in R), there is a single limit of discrete series representation, and it imbeds as a full principal series. The nondegeneracy assumption requires that this representation be viewed in the principal series. In general, it requires that a basic representation be attached to as small a parabolic as possible. A third thing that nondegeneracy accomplishes is to allow the whole discussion of and R in to extend to basic re presentations with nondegenerate data. The theorem from [8], [9] is as follows.
§2.
-
x).
w w
el-p
el-p
x'
W(T-,
x).
basix) c representation series. 0
M.
x).sa
inducede datfrom discrete nondegenerat a W(T-: SL±(2,
W, W', L1', §2 basiblce ifrepresent atiifointswithR-groupnondegenerat e data is necessarily tempered,Twoandirreduci it is birlreduci and onl y i s t r i v i a l. edatbasia carerepresent ationsif and only ifx),thereandis an element wx')in, G wicarryith nondegenerat e equi v alent nEvery g M toirM'reduci, btole tempered to t', andrepresentationtois basiC'c, x'and, can be writ en with nondegenerate data. THEOREM 4. 1 . (a) A
(b)
Up(f;(A, C,
A
(c)
A', t
(A, C, X• v)
(A',
v)
UP' (f;(A', C',
v')
v') .
The R-group can be used to point to those basic representations with non degenerate data that are needed to exhibit the reducibility of a representation
NORMALIZING FACTORS AND L-GROUPS
105
induced from discrete series. The constituents of a given representation induced from discrete series, or even of all induced from an L-class of discrete series, are L-indistinguishable. A precise description of how this reducibility may be read off from the data, from the R-group, and from the set of orthogonal roots in Theorem 2.3(c) is given in [8]. REFERENCES 1 . J. Arthur, On the invariant distributions associated to weighted orbital integrals, preprint. 2. A. Borel, Automorphic L-functions, Automorphic Forms, Representations, and £-functions, preprint ; these PROCEEDINGS, part 2, pp . 27-6 1 . 3 . F . Bruhat, Sur /es representations induites des groupes de Lie, Bull. Soc. Math. France 84 ( 1 9 56), 97-205. 4. Harish-Chandra, Harmonic analysis on real reductive groups. Ill, Ann . of Math. (2) 104 (1 976), 1 1 7-201 . 5. A. W. Knapp, Commutativity of intertwining operators. II, Bull. Amer. Math . Soc. 82 (1 976), 27 1 -27 3 . 6. A.W. Knapp a n d E . M . Stein, Intertwining operators for semisimple groups, Ann . of Math. (2) 93 ( 1 97 1 ), 489-578. 7. , Singular integrals and the principal series. III, IV, Proc. Nat. Acad. Sci. U.S.A. 71 (1 974), 4622-4624 ; 72 (1 975), 2459-2461 . 8. A. W. Knapp and G. Zuckerman, Classification ofirreduCible tempered representations of semi simple Lie groups, Proc. Nat. Acad. Sci. U.S.A. 73 (1 976), 2 1 78-21 80. 9. , Classification theorems for representations of semisimple Lie groups, Non-Com m utative Harmonic Analysis, Lecture Notes in Math . , vol. 587, Springer-Verlag, New York, 1 977, pp. 1 3 8-1 59. 10. R. A. Kunze and E. M . Stein, Uniformly bounded representations and harmonic analysis of the 2 x 2 real unimodular group, Amer. J. Math . 82 (1 960), 1 -62. 11. , Uniformly bounded representations. Ill, Amer. J. Math. 89 (1 967), 3 85-442. 12. R. P. Langlands, On the classification of irreducible representations of real algebraic groups, mimeographed notes, Institute for Advanced Study, 1 973. 13. , On the functional equations satisfied by Eisenstein series, Lecture Notes in Math . , vol. 544, Springer-Verlag, New York, 1 976. 14. , Notes on the Knapp-Zuckerman theory, mimeographed notes, Institute for Advanced Study, Oct. 1 976 ; supplementary letter to A.W. Knapp dated May 25, 1 977. 15. G . Schiffmann, Integrates d'entrelacement et fonctions de Whittaker, Bull. Soc. Math. France 99 ( 1 97 1 ) , 3-72. 16. A. Silberger, The Knapp-Stein dimension theorem for p-adic groups, preprint. 17. N. R . Wallach, On Harish- Chandra 's generalized C-functions, Amer. J. Math. 97 (1 975), 386-403. 18. G. Zuckerman, Tensor products of finite and infinite dimensional representations of semi simple Lie groups, Ann . of Math. (2) 106 (1 977), 295-308 . --
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--
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CORNELL UNIVERSITY YALE UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part I , pp. 1 07- I I O
ORBITAL INTEGRALS FOR
G L2(R)*
D. SHELSTAD
(Coo)
We report briefly on the characterization of orbital integrals of smooth functions of compact support on following [3]. A similar argument ap plies to [3]. We begin by recalling some well-known properties of these integrals in a form convenient for the characterization, indicating the proof after wards ; a more elegant formulation is given in [3]. We fix an invariant 4-form on G = If T is a Cartan subgroup of G we take Wy to be the form where f l , are the eigenvalues of r under some order (prescribed by a diagonalization of T) and Cr is a constant as follows : Cr = 1 if T is split, and Cr = i otherwise. If f E C�(G) and E Treg • the set of regular elements of G lying in T, we set
GL2(R),
GL2(C)
we GL2(R). Crdndrzlnrz rz
r
= se ;yf(gyg.-1) ��
dg, dt Coo
=
r
we. wr
-
)
,
rz
)
gl2(C)
)
AMS (MOS) subject classifications ( 1 970) . Primary 22E30. *This work has been partially supported by the National Science Foundation under Grant MCS76-0821 8 . 9 1 9 7 9 , American Mathematical Society
1 07
1 08
D. SHELSTAD
cient to compute XrA}(z) for each X: XrA}(z), which we will not need explicitly, is the appropriately defined integral of Xf over the conjugacy class of (6 D and if T is not split then XrA? (z) ccXf(z), where cc is a constant depending only on our choice of Haar measure on G. We recall the proof. It is sufficient to consider the Cartan subgroups the diagonal group, and
=
B=
A,
{( - � �) : a2 + b2 o}. #
We find it more convenient to write an element of Breg as
O =(
r(it, )
)
it cos 0 it sin 0 - it sin 0 it cos 0 •
it
>
0, 0
t=
Proceeding formally, we may choose we so that
O(n").
(2)
where
= {(b f)}. = {(-�:g ����)}. = 1 Joo v = J /(
Ko
and
(3)
C/Jf (r(it , 0) )
where
J/
fo(x) = K (kxk-1) dk,
dn = dx,
N
4"
0
dk = dO
(Hie1 0 , e- 1 0) + H).( - etO, - e-tO)) (et - e-t) dt
H).(u, )
K
itk exp
(
_
� g)k-1) dk.
We obtain from the Iwasawa decomposition of SL2(R) and from the Cartan decomposition. The function on the right-hand side of can be analyzed as in [1]. It is easy then to see that this function can be expanded as on the right-hand side of The proof is now straightforward. To compute XrA}(z) note that IDI 1 12 C/J)a = Xr
(2) (1).
C"'
C"'
(4)
for r E N. n Treg ; the functions A•{z, properties :
(3) (3)
=
T T. T'
T
) T
) are assumed to have the following two
109
ORBITAL INTEGRALS
(a) AO ( , ) = if Tis split and (b) for each X in the center of the universal enveloping algebra of g12(C) the re striction to Z n N. of XrA 1 (z, ) is independent of T. Then Lemma of [3] asserts that there exists fe C';'(G) such that f/Jf f/JT for each T. We sketch the argument. Let G, = G - Z and Y = {(x� o x2) e R2 ; x2 :¢: Define 7C : G, -+ Y by 7C(g) (trace g, det g) ; 7C is submersive and each fiber is a conjugacy class in G. Let Then we define a function ¢ on Y - S by rp(7C(r)) S = {(2x�o xi) ; x1 :¢: i D(rW 1 2f/JT(r), r e Treg• allowing T to vary among the Cartan subgroups of G. If ¢ vanishes near S, that is, if each (/JT vanishes in a neighborhood of Z, then it is easy to find f e C';'(G) such that f/Jy = (/JT for all T (via the coverings Treg x G/ T -+ Tf.g). Suppose now that ¢ extends to a smooth function on Y. Since cjJ has compact support we may apply a partition of unity argument on Y and assume that ¢ has support in some neighborhood (to be specified) of a point in S. Fix a e S and g e 7C-I(a). We can find a neighborhood N1 of g in G, with a coordinate system Y�o · · · , y4 such that y1 = x1 7C, y2 x2 7C ; we may as well assume that (y;) maps N1 to a cube in R4. Set N2 7C(N1) and assume that ¢ e C';'(N2). We lift the form ix2i-31 2dx1dx2 to N1 • Using this and the invariant form lVG we construct a G-invariant measure on each fiber of '/C. It is easy to find fe C';'(N1) such that f"- 1 (..-) f rj;(x), X E N2 . On the other hand, suppose that r E Nl n T and that x = 7C(r). Then we find that f" - 1 (..-J h = I D(r) l l / 2 f/Jf(r) . h e C';'(N1). This is a straightforward computation with coordinates. Hence rj;(x) ID(r) j 1 1 2f/J} (r) and our argument is complete in the case that ¢ is smooth. We observe next that ¢ extends smoothly to Y when the functions AO(z, ) at tached to the Cartan subgroup B satisfy
0
4.1
=
0}.
0}.
o
=
=
=
o
=
=
=
lim (I D{r(). ' O) ) I 1 12 AO( + - ). ' r(). ' O))) 0-0; (0-,) � dOn
=
o
for each n or, more simply, X8A0(z', z) = for each X, z' and z. As before we suppress z' and write just X8AO(z). To compute X8A0(z) in general we resort to rapidly decreasing functions and their orbital integrals. On G, or any real reductive group, we may introduce the space of rapidly decreas ing (Schwartz) functions, as defined by Harish-Chandra [2] ; [2], together with earlier papers listed there, contains an extensive analysis of the "F/' (normalized orbital integral) transform on this space. If now f is rapidly decreasing on G then f/Jy has the properties listed earlier, except that in place of the statement about the support of f/J} we have that 1Di 1 12f/Jy is "rapidly decreasing on Tr.g" [2]. For the characterization we take {f/JT} as before, but allow IDI 1 1 2f/JT to be just rapidly decreasing on Treg· The argument is straightforward (since there are many rapidly decreasing functions with computable orbital integrals [2]), but lengthy. Here is our procedure. Consider the rapidly decreasing function IDI 1 1 2f/JA on A . Using wave-packets [2] we find f1 such that Ffi = IDI 1 12f/JA ; then f/Jj1 = f/JA. Consider sin 0 f/JB(r()., 0) ) - FR(r().. 0)) . From ( I) we see that this function is c� on B (and rapidly decreasing as a function of ).) . Then using essentially matrix coefficients of the discrete series representations of { x e G : ldet xi = we find /2 such that F� = sin 0 f/JB - FR and Ffz = Iff = /1 + /2 then f/J} = rpr for all T ( . . . for a
0
0.
1}
1 10
D . SHELSTAD
general group this argument characterizes only stable orbital integrals). We refer to [4] for details. Returning to our original family {1/JT} , where 1/JT vanishes off some set relatively compact in T, we can find a rapidly decreasing function f such that 1/Jf = (/JT . Then X8AO(z) = ccXf(z), z E Z. Multiplying f, if necessary, by a suitable function of det, we may assume that {x E Rx ; x det g, f(g) t= 0} is relatively compact in R x . This allows us to find in C;'"(G) a function f1 which coincides with/ on a neigh borhood of Z. Then Xf(z) Xf1 (z) for all z and X, and the function cjJ attached to {1/JT - 1/JfJ is smooth on Y. We now argue as earlier and the proof of the characterization is complete. Finally, fix a quasi-character X on Z. Suppose that fe Coo(G) satisfies f(zg) = x(z)f(g) for z E Z, g E G and has support compact modulo Z. Then 1/Jf is well defined and has the properties listed earlier, with the necessary modifications con cerning support and transformation under Z. To characterize {1/Jf} we can argue as follows. Let {1/JT} have those properties. We can easily find 1/Jt;, coo on Treg • invariant under the Weyl group, vanishing off a set relatively compact in T, satisfying (4) and such that 1/JT(r) = Jz x(z-1)1/.l'{;{zr) dz, r E Treg· For example, we may take 1/Jt; as (/JT multiplied by a suitable function of jdetj . We pick f0 E C;'"(G) such that I/Jf0 = 1/J'{; for each T. Then f defined by f(g) = Jzx(z-1)f0(zg) dz, g E G, satisfies 1/J} = (/JT for each T and is of the desired form. =
=
REFERENCES 1. Harish-Chandra, Some results on an invariant integral on a semi-simple Lie algebra, Ann . of Math. (2) 80 ( 1 964), 457-508 . 2. , Harmonic analysis on real reductive groups. I, II, III, J. Functional Analysis 19 ( 1 975), 1 04-204 ; Invent . Math. 36 ( 1 976), 1-55 ; Ann. of Math. (2) 104 ( 1 976), 1 1 7-201 . 3. R. P. Langlands, Base change for GL(2) (to appear) . 4. D. Shelstad, Characters and inner forms ofa quasi-split group o ver R, Compositio Math. (to appear). --
CoLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 1 1 1 - 1 5 5
REPRESENTATIONS OF p-ADIC GROUPS : A SURVEY
P. CARTIER Introduction. The aim of this article is to (partially) survey the present state of knowledge about the representations (mostly infinite-dimensional) of reductive algebraic groups over a local field. This includes the familar p-adic groups like G L n( Qp ) , Sp 2 n(Qp), . . . This theory evolved slowly and lately. The first steps were taken around 1 960 by Mautner and his students who concerned themselves with a detailed study of the particular group GL2 (Qp). The first general results were obtained by Bruhat [8] who imitated the 'real' methods of his thesis [7] and by Satake who determined the spherical functions [38] . But the next developments had to await the deep results of Bruhat and Tits [10], [11], [12] and [13] about the structure of p-adic reductive groups. In their reference work in which they were basically concerned with the group GL2 , Jacquet and Langlands [34] introduced the important notion of an admissible representation. They thus opened the way towards a purely algebraic theory of these representations. The basic results about induced representations were soon after obtained by Jacquet [32], who considered the case of GLn only, but used per fectly general methods . These results have been generalized by Casselman and Harish-Chandra. The main goal of this article will be the description and study of the principal series and the spherical functions. There shall be almost no mention of two impor tant lines of research which are still actively pursued today : (a) Plancherel theorem and detailed harmonic analysis on p-adic Lie groups. Here Harish-Chandra is the uncontested leader. We refer the reader to Harish Chandra's own description of his results [26], [27] and also to my Bourbaki lecture [14] for more recent results due to Harish-Chandra and Roger Howe. (b) Explicit construction of absolutely cuspidal representations (the so-called 'discrete series'). Here important progress has been made by Shintani [40], Gerar din [21] and Howe (forthcoming papers in the Pacific J. Math.). One can expect to meet here difficult and deep arithmetical questions which are barely uncovered. Let us give a brief description of the contents of these notes. In §I, we describe the various classes of representations in a very general framework. Following Harish-Chandra [25], we give the definitions for totally disconnected locally com.
A MS (MOS) subject classifications ( 1 970) . Pri mary 22E50 ; Secondary 1 4L l 0, 43A90. © 1 9 79, A merican Mathematical Society
Ill
1 12
P.
CARTIER
pact groups. Number theory and automorphic functions provide us with a host of such groups and their representations. Special attention is paid to various forms of and various notions of Our exposition is based on an apparently novel method using tensor products (over rings without unit element, alas !). In §II, we build up the machinery which enables us, after Jacquet, to reduce the classification of the irreducible admissible representations for a p-adic group to the two following problems : (a)
Frobenius reciprocity
induced representations.
G Constructs oftheitsabsolut elyc cuspi dal irasreduciwell.ble representations for G and the Levi component paraboli subgroups Study the representations inducedfrom parabolic subgroups to the whole group.
(b) Our presentation of Jacquet's fundamental construction (the two 'Jacquet's functors') is based on our previous description of the induced representations and is slightly more symmetrical than usual. After specializing the previous results to the "classical case of GLno we turn to the relation between unitary representations and admissible representations. Here the basic results are due to Harish-Chandra [28] (generalizing earlier results of R. Howe [29] for GLn) and Bernshtein [1]. They show that any reductive algebraic group over a local field is of type I in the sense of von Neumann-Murray classification. The foundations are thus laid down for Plancherel theorem. §III i s devoted to the These representations are par ametrized by the so-called of the Levi component of a min imal parabolic subgroup. Let us mention that those characters provide the crux of the applications to Langlands theory of L-groups (see Borel lectures in these PROCEEDINGS) . In general, the representations in this series are irreducible. They have always a nonzero vector invariant under a special maximal compact subgroup. The main result concerns the explicit construction of an equivalence between and where is an element of the (relative) Weyl group. In the case of real Lie groups, this question led to the introduction of singular integral operators (Stein, Knapp). The deep analytical problems involved in the construc tion ofthese operators are bypassed by a very ingenious trick of Casselman, making full use of Jacquet's construction V � VN. An important role is played by the so called Iwahori subgroups and their geometrical interpretation via buildings. In §IV, we culminate with the theory of spherical functions. Using the results expounded in §III, one recovers Macdonald's formula [36], [37] for these spherical functions. This theory has been highly developed in the case of real Lie groups. From the point of view of representation theory, the spherical principal series is quite special, but plays a prominent role in the applications to L-functions a Ia Langlands. The whole approach leading to spherical functions through §§III and IV has been developed by Casselman in a still unpublished paper [18] . I borrowed extensively from this paper as well as from a preliminary paper [17] by the same author de veloping the foundations of representation theory in the p-adic case. The reader will have to consult these papers for the details of proofs and for numerous gen eralizations. It is hoped that they shall appear soon. I have to thank several friends. Anna Helversen-Pasotto tape-recorded my lectures and made out of her tapes a transcript of the spoken words. This ungra-
unramifi ncipal serierses. unramifiedeprid charact I(x) I(x) l(wx) w
REPRESENTATIONS OF p-ADIC GROUPS
113
tifying task proved very helpful to me when transforming the sketchy notes dis tributed during the conference into the present report. Serge Lang and John Tate allowed me to use freely their notes about spherical transforms. My treatment in §IV has been influenced by them. Also John Tate corrected my misinterpretations about Frobenius reciprocity (see § 1 .7). I. Let A be a ring with unit element I . Let Ax be the set of elements a in A for which there exists in A with = I . Endowed with ring multiplication, Ax is a group, the of A . 2 . Let M be a module over a ring A without unit element. One says that M i s if every element i n M can b e written i n the form a 1 m 1 + · · · + · mk with a l > · · · , ak in A and m 1 , · · · , mk in M. l One says a sequence of elements m l > · · , mk in M generates M if every element in mk with a l > · · · , ak in M is of the form a 1 · m 1 + · · + ak · mk + n 1 · m1 + · · · + · · ·, A and integers (of either sign). If M is nondegenerate, we can omit the terms 1 · mh · · · , · mk. The module M is if there exists a finite sequence generating M. By a we mean a nonarchimedean local field with finite residue field. Such a field F comes equipped with a subring D F, whose elements are called the in F. There exists in DF a unique nonzero prime ideal PF · Moreover there is in D F a w F such that P F = D F · wF . Every element x in px can be uniquely written as wJ!. · u with some integer n and some u in Dj. = D F\PF · We set ordF(x) = in this case. By convention, ordF(O) is put equal to + oo . The index q = (D F : P F) is finite and a power of a prime number p. We set lxi F = q- o rd F Cxl for x in p x and IOI F = 0. The index 'F' may be omitted in DF, P F, Dj., wF, ord F(x) and i xi F when no confusion can arise. There are two possibilities : (a) If F is of characteristic 0, then is a finite algebraic extension of the p-adic field Q p and D F consists of the elements integrally dependent on the ring Zp of p-adic integers. (b) If F is of characteristic p, then F is the quotient field of the ring D F = of formal power series in one indeterminant with coefficients in the Galois field (also denoted as GF(q ) by various authors) . 4. We use the standard notations : Z for the ring of integers, Q for the field of rational numbers, R for the field of real numbers, for the field of complex numbers. 5 . Let be a topological group, whose unit element shall be denoted by I . By a of we mean any continuous homomorphism X : --> ex . We say X is in case is a complex number of modulus I for every i n that is
Notations and conventions.
"multiplicatb ive group"ab
nondegenerate
·
·
nk finitely generated
·
n nhnk nk 3. localfield integers prime element n
ak
·
F
Fq
C G ·unicharact tary er Gx(g)
Fq[[t]
t
x(g) x(g)-1 =
= x
Gg G,
2
'We make the convention that any uni tary module over a ring with unit element is nondegene rate. •some authors cal l 'quasi-character' what we call 'character' and 'character' what we call 'uni tary character' .
114
P.
G Cc(G)
CARTIER
Cc(G) G. G, modular function L1c Jaf(g)dg
Assume that is locally compact. We use the symbol to denote the space of continuous compactly supported complex-valued functions on If 1-' is a (left invariant) Haar measure on we denote usually by the integral of a function f in w.r.t. !-' · The is characterized by the in tegration rule
The group is called in case L1c = I. One has sometimes to integrate over homogeneous spaces Suppose that / is a continuous function on and there exists a function .fi in such that f( = f1 ( ) for any in Choose an invariant measure 1-' on The integral of/1 w.r.t. 1-' shall often be denoted by Jc; nf( ) dg. If is a Lie group (real or p-adic) we use the corresponding German letter g to denote its Lie algebra. Let X be any set. The in X shall be denoted by Ix. If A is any subset of X, the IA : X --> 0, I} is defined by
G unimodular g) gH g G. G g G 6. dentitiyonmap { characteristicifunct 0 A
G/H. Cc{G/H) GJH.
IA(x) = I for x in A , = for x in X\ A ,
where we denote by X\ the set-theoretic difference. If X is finite, its cardinality shall be denoted by IX 1 . 7. We use the abbreviations 'iff' for 'if and only if' and 'w.r.t.' for 'with respect to'. I. Totally disconnected groups and their representations.
1.1. Let be a topological group. We say is of if every neighborhood of its unit element I contains a compact open subgroup. Such a group is a locally compact Hausdorff space. Moreover it is totally disconnected (hence that is there is no connected subset of with more than one element. Let be a group of td-type. If X1 and X2 are nonempty compact open sets in there exists a compact open subgroup K of such that X1 and X2 are unions of . . . , x1 n 1 K for X1 and x2 , 1 . . . , x2, n2 K for X2 . We set finitely many cosets xu , ( 1)
Groups .of td-type. G Gtd), K,
G td-type
G
G,
G K,
For instance, if X1 and X2 are compact open subgroups and X2 is contained in XI > then (X1 : X2) is the index of X2 in X1 • The chain rule holds, namely (2)
Let 1-' be a left invariant Haar measure on
(3)
G.
The following formula is obvious :
Hence if f-t(X) is rational for some compact open set X every such set. In this case one calls the Haar measure 1-'
-1:
0 , the same is true for
rational.
REPRESENTATIONS OF jJ-ADIC GROUPS
1 15
1. 2. Examples of groups of td-type.
(a) Let G be of td-type. Then every open subgroup, every closed subgroup of G is of td-type. A factor group of G by a closed invariant subgroup is of td-type. (b) If Gb · · · , Gn are groups of td-type, so is their direct product G1 x · · · x Gn endowed with the product topology. (c) Let (G,.),."'1 be any infinite family of groups of td-type, and let K,. c: G,. be a compact open subgroup for each i in In the direct product n i E/ G,., let G be the subgroup consisting of the families (g,.),.E1 such that the set E 1/g; ¢: K,. is finite. Then K = fl i E I K,. is a subgroup of G. We endow K with the product topology. A set in G is open iff g U n K i s open in K for every g in G. Then G with this topol ogy is a group of td-type and K is a compact open subgroup of G. The group G i s known as the G,. K,.. (d) Let F be a local field. Then /x - y / F defines a distance in F, hence a topology. Then F as an additive group is of td-type, with DF as a compact open subgroup. Moreover px is open in F, hence, as a multiplicative group, it is of td-type with D)j. as a compact open subgroup. (e) Let n � be an integer. The linear group GLn(F) is the open subspace of the n2 -dimensional space over F with coordinates x 1 1 , x 1 2 , · · · , Xnm defined by det(x,.j) #- 0. It is a group of td-type. A compact open subgroup is GLn(DF), the set of n-by-n matrices with entries in DF, and determinant in D)j.. (f) Let G be a subgroup of GL n(F) defined as the set of common zeroes of a set of polynomials in the coordinates x,.j with coefficients in F. For short, G is an GL n(F) (more precisely, the set of F-rational points of an algebraic group defined over F). It is a closed subgroup of GLn(F), hence a group of td-type on its own merits. A neighborhood basis of the unit matrix In in G is given by the subgroups
I.
U
{i
}
restricted product of the groups w.r.t. the groups 1
algebraic subgroup of
Let G' be an algebraic subgroup of G Ln , (F) for some integer n' � Assume that the group homomorphism p : G -> G' is rational, that is, there exist polynomials � such that Pkt in F[X1 1 , · · · , XnnJ and an integer
1.
m 1
for g = ( ,.j) in G. Then p is continuous w.r.t. the topologies defined on G and G' by their embedding in the linear groups GLn(F) and GLn , (F) respectively. In par ticular if p is a biregular isomorphism, i.e., p is a group isomorphism and p, p- 1 are both rational, then p is a homeomorphism. In more intrinsic terms, the algebraic structure on G is defined by the ring F[G] of polynomial functions 3 and the topology defined above is the coarsest for which the elements of F[G] are continuous mappings from G to F (F is given the topology defined in (d)) . (g) By (b) and (f), the product of finitely many algebraic groups defined over the
g
'This ring consists of the functions g F[Xu , , x ] and an i nteger m ;;; 0. · · ·
••
>-->
u(g 1 1 ,
• • •
,
••
g )/(det g)m for a polynomial
u
in
1 16
P. CARTIER
same or distinct local fields is of td-type. Similarly, by (c) and (f), adelic groups without archimedean components are of td-type. (h) Let F be a local field and Fsep a separably algebraic closure of F. Let Funr be the maximal unramified extension of contained in Fsep and (}" the Frobenius au Gal(Fsep/F) be the Galois group of Fsep over tomorphism of Funr over F. Let F endowed with the Krull topology. It is a compact group of td-type. Let c be the subgroup of automorphisms rp of Fsep which induce some power (}"" of (}" in the of F, as a group of td-type Funr · In a unique way we can consider in which Gal(FsepiFunr (with Krull topology) is a compact open subgroup of Let be a group of td-type. If K is any compact open sub group of we denote by K) the complex vector space consisting of the complex-valued functions/ on which satisfy the following two conditions :
GF = F WF GF WF, Wei/ group ) WF. 1. 3. HeekeG, algebra. G.Yt'{G, Ghat isf(kgk') = f(g)for gin G and k, k' in K. (a)fi s hi i n vari a nt under t (b)/vanishes offafinite union ofdouble cosets KgK. .Yt'(G, G. {4) .Yt'(G, C. G {5) == f-L0 { 0 u0 eK . = eK( ) == 0 u Yf'(G, u=eK= I: K,
Moreover, let us choose a (left invariant) Haar measure fL on One defines a bilinear multiplication in the complex vector space K) by the customary con volution formula
This integral makes sense since, as a function of x, the integrand is locally constant and compactly supported. For this multiplication K) becomes an associative algebra over the complex field Let us choose a set of representatives {ga} for the double cosets of modulo K, i.e., G is the disjoint union of the sets KgaK. For any index a, let ua be defined by ua(g)
K) 1 if g E KgaK, otherwise. -
In particular we may assume that g0 1 for some index function shall be denoted by Hence (6)
g
f-L(K) -1
and the corresponding
if g E K, otherwise.
The family { a} is a basis of the vector space K). Moreover, is the unit element of this algebra and the multiplication table is given by Ua * � 7ca� r ur where the coefficients Ca�r are computed as follows. The group Ka K n gaKg;;1 is compact and open, hence of finite index in K. There exist therefore elements xi > · · · , Xm of K such that K is the disjoint union of the sets x1Kao · · · , xmKa. Then KgaK is the disjoint union of the sets x1gaK, · · · , xmgaK. Define similarly K� and the elements YI > · · · , Yn · Then caflr is the number of pairs such that g;:-1x;gayigfl belong to K (see Shimura [39]). When K' is a compact open subgroup of K, then K) is a subring of K') but with a different unit element if K =F K'. Define K) where K runs through a neighborhood basis of consisting of compact open subgroups. Then For the convolution product defined by is an associative algebra. It has
(i, j) Yf'(G,£'(G) = U K Yf'(G, .Yt'(G, £'(G) is the space of locally constant(4),1 and£'(G)compactly supportedfunctions on G.
REPRESENTATIONS OF p-ADIC GROUPS
117
no unit element unless G is discrete, and it is commutative iff G is commutative. The algebra £(G) is called the of G and £(G, K) is called the Heeke algebra of G w.r.t. K.4 For later purposes, we need a generalization of £(G). Namely, let Z be a closed subgroup of the center of G and X a character of Z. We denote by £iG) the set of complex-valued functions which satisfy the following conditions :
Heeke algebra
lyzg)constax(nt;z)-1f(g)for z in andg in (a)fiones local hasf( (c)fis compactly supported modulo
(b)
=
Z G; Z. More explicitly, assertion (c) means that/vanishes off a set of the form [) · Z where [) is compact in G. Choose a Haar measure v on G/Z. The convolution product is defined in £'iG) by
J
/1(x) fz (x-1g) dx xz z 1.4. Smooth representations. representation nv v v h H, 1.1. equirepresent aytiifonV(n-, UKV) ofVK where is smootrunsh overthe sttheabicompact lizer of every vector i n Vi s open, v alent l open subgroups(n-, of V. coefficient v v* g) (8) n-(f) * (9) ( v , f v) J f(g)n-v (g) dg. v K vE V cJg, K ( K) n-(f) f-t f n-(f) n-(eK) VK ; v V (f *x fz) (g) 1
(7)
=
GIZ
(notice that the integrand takes the same value for x and if belongs to Z). This product is bilinear and associative. There is no unit element in the algebra £iG) unless Z is open in G. When Z = { 1 }, this construction brings us back to £(G). By a of G, we mean as customary a pair (n-, V) where V is a complex vector space and a homomorphism from G into the group of invertible linear maps in V. If H is a subgroup of G, we denote by V H the space of vectors in V such that n-(h) · = for any in that is, vectors whose stabilizer in G contains H. DEFINITION A G iff = K ' G. Let V) be a smooth representation of G and V * the space of all linear forms n-v, v• of n- (for in V and on The in V * ) is defined by n" v v•( = ( v * , n-(g) · v) . ,
It is a locally constant function on G. For fin £(G) there exists a linear operator n- (
)·
=
G
acting on V and such that
'
v•
It is computed as follows : given in V there exist a compact open subgroup K of G, constants c!> " ' • em, elements gb " ' , gm of G such that and/ = I: 7'=1 · . ) n-(g,. v ,. is equal to Then · 1.,; �1 c ·v Using standard calculations one checks that ,__. is an algebra homomor phism from J'C'(G) into Endc( V) ; hence we may consider V as an £(G)-module. For every compact open subgroup K of G, the operator is a projection of V onto hence every vector in satisfies I' = n-(ex) · v for a suitable K. As a corol lary, V is a nondegenerate £(G)-module. The following facts are easily proved : 'For G GL,(Qp) and tors attached to p. =
K
=
GL, (Zp), this algebra is just the classical algebra of Heeke opera
118 Vis stfianbl£(G). e under the operators TC(g)for gin G iff it is invariant under Athsubspace e operatorsV1TCo(ff)for ( V1) V) irreducibleV) finitely generatg edG 1 � i � m .. ) (v1, V. liUTCnear(/)forLetmap.everyThenV)finuand£(G). sat(isfiesV'7e') be(g)usmootuTCh (represent g) for everyationsg iofG, n G iffandit satilet u:sfiV---> es 7e'(V'f)ube a V' phi7C'sm.(g)u UTC(g) g G inter twining mapu:V---> G-homomor any irreduci representintertwi ationninofg map G. Assume tscalar. opol(c)oLetgy of G V)hasbea countabl e5 basibles.smooth 6 Then every u : V--->thatV isthae Schur's lemma G (G: G; £(G, C. V, 1; f £(G)TC(f) U K£(G,£(G) u V) C. C,C(x). A C (x C, Z(G) G. G) ( ) TC(z) ( ) g Z(G.G). centralatcharact eunir ofque smooth representa Any nondegenerate £(G)-modul e is associ e d to a tion ofG. the category of nondegenerate £(G)-modules is identical to theca tegory of smooth representations ofG and intertwining maps. V) G. G. TC(e ) ( smooth dualK V. v*, v V. Uic K G (10) (ic(g) v, v) < v, (g 1) v) P. CARTIER
(a)
Otherwise stated, the subrepresentations 7C I > of (TC, correspond to the submodules of the £(G)-module V. In particular, the representation (TC, V) is (algebraically) iff V is a simple £(G)-module. The representation (TC, is said to be if there exist finitely many vectors vb · · · , vm such that the transforms TC(g) v; for in and generate · vm the complex vector space V. By (a), this amounts to the assertion that generates the £(G)-module ' (b) (TC, 7C , ·
,
=
=
A map
or a of £(G)-modules. (TC,
such that
for every in is called an By (b), it is nothing else but a homomorphism =
This version of is proved as follows. Since has a countable basis, the index K) is countable for every compact open subgroup K of hence K) has a countable dimension over Moreover, there exists a countable basis of neighborhoods of hence K) has a countable dimen sion. For any v #- 0 in the map �--. · v from to V is surjective ; hence the dimension of V is countable. Let A be the algebra of intertwining maps from V into V. For any v #- 0 in V, the map �--. u(v) of A into V is injective because (TC, is irreducible ; hence the dimension of A is countable. But A is a division algebra over the algebraically closed field If A #there exists a subfield of isomor phic over to the field of rational fractions In this field, the uncountably many elements - ;o - 1, for .A running over are linearly independent. Con tradiction ! [This proof is due to Jacquet [33].] For instance, let z belong to the center of Then TC z commutes to TC(g) for every in Hence there exists a character w" of Z( such that w" z · 1 v for every z in One refers to w" as the TC. (d) =
=
To summarize,
To conclude this section, we define the contragredient representation to a smooth representation (TC, of Let K be a compact open subgroup of Denote · v) v) for by V * (K) the space of linear forms v* on V such that (v * , to In V there ex every in The space V = V * (K) is called the ists a smooth representation of characterized by the relation =
·
=
TC
-
·
5 A finite set is countable ! ' This condition is satisfied for the algebraic groups over a local field .
1 19
REPRESENTATIONS OF p-ADIC GROUPS
in V and v in V. The representation (it, V) i s called the representation to V). We come now to a more restricted class of representations. DEFINITION V)
g G, vent (n-, contragredi 1. 5 . Admissible representations and their characters. 2. A represent n (n-, K isoffiniGteis-dicalmlensied admi e if it compact is smoothopenand thesubgroup space K ofof1.G.vectors invariaanttiounder onalsforsiblevery (n-, G K G. tff(K) 1 L Ktff(K) K/L L. v nK n-K/v.(kL.1) kv, n-, (kkmm) v K/L, K. b n- K K V i s ot y pi c component o f cl a ss b n-; K K, v n-(k) x(k) v G and K is(n-,a compact openssibsubgroup ofis EB ation ofMoreover G,finitteh-endIfim(n-,viensis othnalise difora smooth recteverysumbirepresent V) i s admi l e iff b E b n tff(K). K E* E. Assume is admissible. K-Thein varirestarntictbiiolninearof tform he paionring between and V v defi(n-nes, V)a nondegenerate is admissibolenandconttrhagredi e pairienntg tbetween Venabllfe(n-,s one toisidadmientifsysi(n-,ble, thenwith the representati o and represent ation of G. Then n-is admissible !tf the operator n-(j) isLetoffini(n-, te rankbediaforssmooth every fi n £'(G). G £'(G) tribution (n-, G G (11) f) f £'(G). charact er ly n-.of admissible and irreducible representations of I , V Let (n)aE be a fami a a that n-a, hence is inequimutvualent lG.inearlAssume y independent ally ditsotin-n13ct.for �- Then the characters are for
in
VK
Fix a smooth representation V) of and a compact open subgroup of Let be the set of (equivalence classes of) continuous irreducible finite-dimen sional representations of K; every neighborhood of in K contains a subgroup of which is compact, open and invariant in K. It follows that every representation in factors through the finite group for such a subgroup Let be a vector in V. Since is smooth, there exists a subgroup L as above fixing Let h · · · be coset representatives for K modulo L. The subspace spanned by · is stable under and affords a representation of the finite group • · · ·, It is therefore the direct sum of subspaces affording irreducible representations of hence of In other words, the restriction of to is a semisimple re presentation. For any in tff( K), let Vb be the subspace of V generated by the minimal K-invariant subspaces of V affording a representation of of class b. The space b is called the of if c is the one-dimen sional representation of given by c:(k) = I for k in K, then V, = VK. More gener ally for any character X of the space Vx consists of the vectors in V such that · for every k in K. ·v = We state now the elementary properties of admissible representations. (a) V) b E @' (K) Vb .
vb
Let now in tff( K) and let 5 be the representation of contragredient to b . Since the space of is finite-dimensional, the space of 5 is the dual of (b) V to vb X b vb X Vb. Since Vb is finite-dimensional, so is V6 and each of the spaces Vb and Vii can be identified to the dual of the other. So we get from (b) the following result : (c) V) (it, V) V V) (it, V). We come now to the characters. (d) V)
By definition a (in the sense of Bruhat [9]) on is a linear form on the space of "test functions" . According to property (d) above, we can associate to any admissible representation V) of a distribution 8, on by
8,(!) = Tr (n- ( ) for in
One refers to 8, as the (e)
of
a =1-
8,.
120
P. CARTIER
For the of e" we may proceed as follows. Assume to simplify matters that G has a countable basis of open sets, hence a countable basis (Km)m�l for the neighborhoods of 1 consisting of a decreasing sequence of compact open sub groups. Let Vm be the space of vectors in V invariant under n(Km). Since 11: is ad missible, each Vm is finite-dimensional, and V1 c V2 c · · · c Vm c · · · with V = U m Vm. Let g be in G. For every m, the operator n(eK,.) · n(g) · n (eKJ maps Vm into itself. Since Vm is finite-dimensional, this operator has a trace, to be denoted by e" . m(g). Notice that as a function of g, e".m(g) is hi-invariant under K, hence locally constant. Let f be in JIC'(G). There exists an integer � 1 such that f belongs to JIC'(G , Km0) and then one gets
calculation
( 1 2)
e"(f) =
for every integer
m
�
m0 •
J
m0
c
e tr.m (g)f(g) dg
This fact can also be stated as
e" = mlimoo etr. m (weak limit in the space dual to JIC'(G)) . ....
(13)
For each m, let Bm be a basis of Vm over assume that B 1 c B2 c . . . c Bm c Put B = { va } a E I ; hence Bm+l c . . Then B = mBm is a basis of V over Bm {va}a E im for some finite subset lm of Define the matrix (na 13(g) ) by · .
=
U
C; I.
(14) (15)
n(g) . l'J3 = .E 1T:aig) va. aE[
( 1 6)
e_ = t...J "" na . a aE/
C.
Then e",m can be calculated as
Hence we get the series expansion ••
which converges in the weak sense. REMARK. The definitions of the convolution product by and of n(f) by (9) depend on a Haar measure fl. on G. To free them from this dependence, we can proceed as follows. Let Coo(G) be the space of locally constant complex-valued functions on G. By C�(G) we denote the space of linear forms T on Coo( G) which satisfy the following two properties : (1) K G wheref'(g) = .f(k 1 gk2) . K. Namely, forf in Coo(G) and k l > k2 in K, then = namely there exists a compact open subset Q of G such that = for every f in coo(G) which vanishes identically on Q. By a generalized function on G we mean a linear form on C�(G) ; they make up a vector space c-oo(G) over We can embed in a natural way Coo(G) as a subspace of c-oo( G). The on C�(G) is defined as follows : for T; = I , in C�(G) choose D; as in above and let Q = Q1 U Q2• For fin coo( G) there exist functions fi , . . . ,J ', , f'{ , . . . , J;, in Coo(G) such that
(4)
There exists a compact open subgroup ofT(f) suchT(fth'at) Tis hi-in variant under (2) Tis T(f) compact0ly supported, C. convolutio(2)n product (i 2)
121
REPRESENTATIONS OF jJ-ADIC GROUPS
; ; f f(glgz) f (gl) (gz) gb gz Q. (T1 * T2)(f) ; ; f f Tl( ) Tz( ). n < v* , n(T) · v) T n v v* Brc(T) n(Tn u f cf(g)u(g) dg Absolutely cuspidal representations. A x-representation n (n, n n(f) n (20) < v* , n(f)·v) J < v* , n(g)·v) f(g) dg v v* m
= "£.
( 1 7)
Then
for
·
z'=l
in
is defined by
m
"£.
( 1 8)
·
£=1
For T in C':(G), the operator
(T) in V i s defined in such a way that
=
( v, v*) for in V, in V* (notice that nv, v• belongs to Coo( G)) . If is admissible, its character is the genera lized function defined by = Tr( )) for T in C':(G). If we choose a left invariant Haar measure f.t on G, we get an isomorphism of £(G) onto C'�(G) which associates to E £(G) the linear form ,_. S on Coo( G). By duality, one gets an isomorphism of c-oo(G) with the space of dis tributions. This brings us back to our previous constructions. 1 . 6. In this section, we denote by Z a closed subgroup of the center of G. We fix a character X of Z and a Haar measure on GjZ and assume that G is unimodular. of G is a representa tion V) of G such that (z) = x(z) · 1 v for every z in Z. If is smooth and ir reducible, this means that the restriction to Z of the central character w, is equal to X (at least, when G has a countable basis of open sets). Let be a smooth x-representation of G. For f in £x{G), the linear operator in V is defined in such a way that ( 1 9)
=
GI Z
holds for in V and in V*. It is then easily proved that the category of smooth x-representations of G is isomorphic to the category of nondegenerate £' i G)-modules. Moreover the ir reducible smooth representations correspond to the simple £; /G)-modules. DEFINITION 1 . 3. X Z. (n, V) G (or supercuspidal, or parabo]jc according to some authors) nv, v ( in V, in V) Z (hence belongs to £'x-1(G)). . A representation is called absolutely cuspidal (w.r.t. Z) if it is an absolutely cuspidal x-representation for some character X of Z. One of the main properties of absolutely cuspidal representations is embodied in the so-called
Let be a character of A x-representation of is calleifd absolut e l y cuspi d al it is admissible and each coefficient v v is compactly supported modulo Schur orthogonality relations. Letthe (charact n. beerany oirfreducicanblebeabsolut ely cuspi dcharact al x-representati oThen n of Assume that extended to a e r of G. there exists a constant d(n) 0 such that the following identity J
G.
(2 1)
1.1.
V)
>
lxl
Z
=
G/Z
v
v.
122
P. CARTIER
The assumpti on about X is satisfied if X is unitary or else if is a connected reductive algebraic group over a local field. The number is called the It depends on the choice of the Haar measure on Indeed, multiplying the Haar measure by a constant > amounts to replacing by More invariantly, to is associated a Haar measure v,. on such that
G
) d(nG/Z. formal degree of n. G/Z d(n) d(n)fc.
c 0
n
in V and i\ , in V. We denote by the integration w.r.t. v,.. holds for the number v,.(K) is well defined and may If is a compact open subgroup of be called the formal degree of w.r.t. K. For instance assume that is induced from a finite-dimensional representation A of K. Then v"(K) is the degree of A. Theorem has a number of interesting corollaries. Generally speaking, let V) be any admissible irreducible x-representation of For any linear map V ---> V, the following conditions are equivalent : (a) There exists a function f in such that = rc(f) holds. (b) There exists a compact open subgroup of such that u = n(k) · u . n(k') holds for k, k' in K. in v and linear forms Vj , . . (c) There exist vectors Vj , . . · , in v such that
K vb v2 1.1
d,.g
v2 G/Z, n
n
G. u: (n, .lt/G) K u G vm · , vm m
i=l
holds for any in V. The set .P( ) of all such operators is a subalgebra of the al gebra of all linear operators in V. Assume now that is absolutely cuspidal. Denote by the two-sided ideal in consisting of the functions such that n f) = The vector subspace of generated by the functions of the form g ,_. is then and Hence a two-sided ideal in and we get an isomorphism ,_. n(f) of the algebra with the algebra .P( ) . is given by u ,_. (/Jrr. u where (/J", u (g) =
for
in and u in .P(n). Notice also that we have an isomorphism V ---> .P(n) given by
(J : V
(25)
for
(n) Tr (u · n(g- 1 ))
v,
v
'
·
in V and
v
'
=
in V. x
·
V
V, .P(n)
f x Let 8,. be the character of n. The projection A,. of Jf'(n) is given by A,.(f) = (/Ju C f l ; hence more explicitJy7 7
We abuse notation by treating e. as a function !
d(n)
onto .l:1'(n) with kernel
123
REPRESENTATIONS OF !J-ADIC GROUPS
g d(�) J
A "( f) ( )
(26)
=
GI Z
8,..(xg- 1 )j(x)
dx Z {1}.
for g in G and f in .Yt C G). To simplify assume that = Then we can rewrite formula (26) as a convolution (of a distribution and a compac.tly supported func tion !)
x
(27) where 8� is the distribution on G deduced from 8" by the symmetry >-+ g- 1 of G. From the decomposition .Yt G) %(�) EB d(�) one deduces easily the fol lowing theorem, due to Casselman [16] :
g / irreduciof nondegenerat ble absolutelye .Ytcuspi(G)-dalmx-represent ation of Then Vis projLet(�, ective inV)thebecatanegory odules. x Any absolutdal ex-represent ly cuspidalax-represent aticount on ofed iswitah difinirectte summultofi iplirreduci b l e absolut e l y cuspi t i o ns, each city. =
G.
THEOREM 1 .2.
There are two important corollaries. CoROLLARY 1 . 1 .
G
The following is a converse of Schur's lemma and follows immediately from Corollary 1 . 1 .
ely cuspi Assume every intertwiLetningc�.map u:beV---+anyViabsolut s a scalar. Then�dal isx-represent irreducible.ation of COROLLARY 1 .2.
V)
G.
So far we considered one absolutely cuspidal representation at a time. We give now the second half of Schur's orthogonality relations.
V') iareon holinequi x-representati1.o3ns. Assume of Thenc�. thV)e folandlowi(�'ng, relat ds valent absolutely cuspidal (28) S ( ii, �(g)·v) ( ii' , �(g- l) . v' ) dg whatever be v in v' in V', ii in and ii' in (�of C � cuspi d al part ChangqJe of groups andxFrobenius reciprocity. V) V V' (�'�(g), V')rest�' (qJri(cgt))ion functor qJ* extension functor qJ* THEOREM
G.
GIZ
V,
=
V
0
V'.
Let A be a complete set of mutually inequivalent irreducible absolutely cuspidal x-representations of G. From Theorem 1 . 3, it follows immediately that the sum of the two-sided ideals d ) (for running over A) is direct in .Yt/G). This sum i s called the .Yf' G) to be denoted by .Yf'xC Gt. 1 .7. Let G and G' be two groups of td-type and a continuous homomorphism from G into G'. Denote by Yc (Yc·) the category of smooth representations of G (G'). The from YG ' to Yc takes the smooth representation of G, where = and of G' to the smooth representation (�, for g in G.s = from YG to Yc • , we take the view that YG To define the consists of the nondegenerate .Yf'(G)-modules and similarly for YG ' · We consider .Yt(G') both as a left JIC'(G')-module in the obvious way and as a right .Yf'(G)-module 8
If
P. CARTIER
1 24
f' * "'f (f'*"' f)(g') L f' (g'· rp(g)-1) f(g) Llc� (rp(g)-1) dg. rp* c. rp* rp* f� (f� ® (f� * f�) f� c rp* rp* u: rp* EJ(u) f' ® v f' v Frobenius reciprocity rp' . (n, generalized vector as follows : for in Jf'(G') and / in Jf'(G), we define the convolution /' function in Jf'(G') whose values are given by
as the
=
(29)
We define now via tensor products. Indeed for any V in !I' we put V = Jf'( G') ® JF (Gl V and define the action of Jf'(G') on V by the rule v) =
·
(30)
® v
forf�, in Jf'(G') and v in V. Let V be in !I' and V' in !I'G' · Using the universal property of tensor products, we produce a canonical map (3 1 )
as follows : for Jf'(G'), in V).
V')
8 : Home( V,
V -->
....
Hom c , (
V', the linear map
V, V')
takes
into
· u(v)
(f' in
i s the assertion that e is an isomorphism. Unfortunately, as John Tate pointed out to me, this is not true in general due to the lack .of unit elements in our rings. We have to introduce another functor First of all let us define the notion of a in a representation space V) for G. One defines in Jf'(G) the translation operators by (32)
g, g1 u : f v-oo n(g)u n v v-oo n-oo(g)(u) f n f · v. v-oo, n-oo(k)(u)(n' , u u , -oo n'(n'-oo-(oorp(g)). rp' rp* G. (n, The map extends to an isomorphism rp rp ) Hence thefunctor rp'from !I' to !I'c is a right adjoint rp* to rp* . In orderandthatsufficibeentanthatisomor is necessary rp* phism forforeveryeveryin in and every in it {I} rp1 V' (n'V', -oo
for of generalized vectors consists of the in G and in Jf'( G). The space linear maps Jf'(G) --> V such that uLg = for every g in G. We identify any vector in V to the generalized vector �--+ ( ) The representation of G into V is extended to by = uRg- 1 · It is easy to show that V consists of the 'smooth' vectors in that is the generalized vectors for which there exists a compact open subgroup K of G such that = for every k in K. We apply this construction to a smooth representation V') of G'. We get a representation V' ) of G', hence a representation of G on the same space by the operators One defines rp' V' as the set of smooth vectors for G in the space V'-oo. It is clear that V' is a subspace of V' and the carrier of a sub representation for Let V) be a smooth representation of G. One establishes easily the following facts : (a) 8 (3 3)
8' : Homc( V, G'
'
V'
-->
Hom c . (
*
V, V' ).
By general functorial results it follows that is a right exact functor and rp ' a left exact functor. V ' !I'G' • (b) e V !I'c V' = rp1 V' V' !I'G' · In order to get a counterexample to Frobenius reciprocity, it is enough to find a smooth representation V') of G' such that V'-oo # V' and to consider the group G= since in this case. Consider the left translations acting on =
125
REPRESENTATIONS OF p-ADIC GROUPS
.Yt'(G'). The identity map is a generalized vector in .Yt'(G') - oo and is not of the form */0 for a fixed fo in £'( G') unless G' is discrete. Hence £'( G') # .Yt'(G')-oo in >-+ this case. G" (c) G'
ff
[I'c
o
[I' G"
This follows from the associativity of tensor products. that is is open in G' for every open set in G. Then, for any module V' in [I'G'• a generalized vector in V' - oo is smooth for G iff it is smooth for G', that is belongs to V'. Hence one has lfJ' V' = V' and by prop erty (b) above, Let V) be any smooth representation of G. We can describe more explicitly V as follows. Let H be the kernel of and let V(H) be the subspace of V gener ated by the vectors v - v for in H and v in V. Moreover, by Frobenius reciprocity, there is a G-homomorphism c : V ---+ V such that 8(c) is the identity map in V. Finally, let aEA be a set of representatives for the cosets G) in G' (notice that is open in G') . (d) c V V the c ( V) EB a EA G' V' .Yt'( (e) H (for instance H is a unipotent algebraic group over a local field, or
·
·
For every compact open subgroup K of G, the operator (see § 1 . 3, formula (6)) is a projection of V onto the set V K of vectors invariant under K, with kernel V(K). Hence V V EB V(K). Under the assumptions made in (e) one has =
H
=
U VK
=
U ker
where K runs over the compact open subgroups of H. Hence, for every G-invariant subspace of V, one gets W(H) = n V(H), hence the exactness of There are two special instances of the previous results. Assume first that G is an open subgroup of G' and is the injection of G into G'. Then we can identify · V. V to its image by c in V and then V = EB aE A = G'. Then defines an isomorphism of topological Assume now that groups from G/H onto G'. Moreover, c defines an isomorphism of the linear space VH = V/ V(H) onto V. 1 .8. In this section, we denote by G a group of td-type and by H a closed subgroup of G. Let V) be a smooth representation of H. We denote by the space of func tions : G ---+ satisfying the following assumptions : = · (a) G. (b) K G ch = G The group G acts on "f�" by right translations, namely (34)
The representation (()
" "Y") '
of G is smooth. It is called the representation induced
1 26
P. CARTIER
from 11: and usually denoted by Ind� namely an isomorphism
11:.
8* "Y"
(3 5)
:
dual Frobenius reciprocity, c( 11:) "Y" G 8�(g) 11: "Y�. g
One has a kind of
Homy(Res� A, 11:)
__:::____.
Hom A, Ind}j
G.
for every smooth representation (A, W ) of The proof is trivial. The functions in are locally constant, but assumption (b) is usually stronger the subspace of consisting of the func than just local constancy. Denote by tions which vanish off a subset of the form Q where Q c is compact. For in the translation operator induces an operator in The repre sentation is called the c-induced representation from and is usually denoted by c-Ind� 11:. If is compact, there is no need to distinguish between y" and and c-Ind� = Ind� The adjoint of a composite functor being the composite of the adjoints in reverse order, one deduces from Frobenius reciprocity the possibility of Namely, if L is a closed subgroup of there is a canonical isomorphism
"Y� H 8"(g)
G, (8�, "Y�) G/H "Y�, 11: 11:.
inducing by stage.
H,
Ind� Indf A __:::____. lndf A
(36)
L. H rp rp* o H o(h) iJy(h)/llc(h) h H. H, twisted representation (11: V)H (11:, V) h H. Letrepresent H be aaticloonsed subgroup ofrpG* Vandis isomorphi the injectic toon tohfe Hc-iinnduced to G. Forrepresent everyatsmooth V) of H, ion c (11: b-1) . Thefunctor c �from !/y to!/cis exact. 124). rp* P": rp* V -> "Y�tZlo-1 "(f v)(g) s f(g-1 h) o(h)-1 11:(h) dh £'(G), v g Properties of algebraic groups. n G GLn(F).
for any smooth representation (A , W) of A similar property holds for the c-induced representations. Let be the injection of into G. We want to compare our functor to the induced representations. Define a character of by (35)
for
=
in
If X is any character of and a smooth representation of the acts on the same space as 11: via the operators ® x, (11: ® x ) (h) = x(h) 11:(h) for
(36)
THEOREM 1 .4.
-Ind�
CoROLLARY 1 . 3.
rp
(11:,
®
-
in
Ind
is right exact (see above, p. It is clear that c-Ind� is a We know that left exact functor, hence the corollary. An explicit isomorphism is given as follows : p
(37)
for f in
in
V
{8)
and
=
H
·
V
i n G.
II. The structure of representations of p-adic reductive groups. We summarize here a few properties of al 2. 1 .
gebraic groups. For a more complete exposition we refer the reader to the lectures by Springer [41] in these PROCEEDINGS. As usual, F is a local field. Let � 1 be an integer and let be an algebraic subgroup of We say :
127
REPRESENTATIONS OF j:l-ADIC GROUPS
unipotent torus split torus reductive semisimple
G is if it consists of unipotent matrices (all eigenvalues in some algebraic closure of F equal to 1 ) ; G is a i f i t i s connected, commutative and any element o f G can b e put in diagonal form in some extension of F; G is a if it is a torus and the eigenvalues of every element of G belong to F; G is if there exists no invariant connected unipotent algebraic subgroup of G with more than one element ; G is if it is reductive and its center is finite. A connected reductive algebraic group G is called if there exists in G a maximal torus which is split. Then every maximal split torus in G is a maximal torus. G Any split torus in G is contained in some maximal split torus of G. Any two maximal split tori in G are conjugate by an element of G, their common dimension is called the of G. There exists in the center of G a largest split torus Z. We do not repeat the definitions of a parabolic subgroup of G, a Borel subgroup and a quasi-split group (see [41]). A (P, A) consists of a parabolic subgroup P of G and a split torus A subjected to the following assumption : If N is the unipotent radical of P (its largest unipotent invariant algebraic sub group), there exists a connected reductive algebraic subgroup M of G such that P = M · N (semidirect product)9 and A is the largest split torus contained in the center of M. Any parabolic subgroup P can be embedded into a parabolic pair. Given P, the split torus A is unique up to conjugation by an element of N. Given (P, A), the group M is the centralizer of A in G. One says the parabolic pair (P, A) the parabolic pair (P', A') in case P ::J P' and A c A' hold. There exists then a parabolic subgroup P 1 of M such that P' = P 1 · N and (P1 , A') is a parabolic pair in M. This result is used very often in proofs by induction on the dimension of G. Let be given G, P, A, M and N as above. Define two homomorphisms
split From now on, we assume is reductive and connected.
split rank
parabolic pair
dominates
2.2. Jacquet's functors.
p
� �M
G
mn m c first Jacquet's functor Jc. M n(n)J c v-v n v
n (n,
were c is the injection of P into G and p( ) = for m in M and in N. The group P is thus a kind of link between the groups G and M, and will be used to define functors relating the categories of G-modules and M-modules. V) be a smooth The is = p * c * : .? --+ .?M. Let representation of G. Since c* is simply the restriction Res�, the space of the repre sentation . M 1t' is VN = V/ V(N) where V(N) is generated as a vector space by the elements for in N and in V. The representation of M on VN is obtained from the restriction of 1t' to M, which leaves V(N) invariant since M normalizes N. ·
' The equation P
=
•
M N is called the Levi decomposition of P.
P. CARTIER
1 28 THEOREM 2. 1 .
V) (n-N,(n-,VV)N) iiss anan admi admisssisibblleefinifinitteellyy generat generateedd represent representaattiioonn ooff G. Then JG, M(n-, Suppose
M.
=
This theorem is a deep result essentially due to Jacquet [32] (see also [17]). CoROLLARY 2. 1 .
VN is finite-dimensioAssume nal. G is quasi-split and P is a Borel subgroup of Then G second Jacquet's functor JM. G W) !I'M !I'G(n-,;· , W n f(mng) ).(m)· f(g) g G G g, g1 G at ().s,siW)ble finiis anteladmi ssibleefinid represent tely generat of Then (n-,;, Assume ,;) is anthadmi y generat ationedofrepresentation G. W) G G principexact al series W) V, (n-, V) G. The main theorems. A)G; A') G. N G A' A a A' (a G.
If is quasi-split and P a Borel subgroup, M is a maximal torus in G, hence is commutative. It is then easy to check that any admissible finitely generated repre sentation of M is finite-dimensional, hence the corollary. = Ind� p * : The is --+ More explicitly, this functor takes a smooth representation (jl, of M into "Y,;) where "Y,; is the space of functions f : G --+ such that =
(1)
The group
for
in , m in M and
in N.
acts via right translations, namely
(2)
for in and f in "1""1 . The following result follows easily from the compactness of GfP. THEOREM 2.2.
M.
"Y
G.
This construction is especially interesting when is quasi-split and P is a Borel subgroup of We may take for ()., a one-dimensional representation corre sponding to a character of the maximal torus M. The corresponding representa tions of comprise the (see also §III). The Jacquet's functors are by the results quoted in § § 1 .7 and 1 . 8. They are adj oint to each other, giving rise to a canonical isomorphism HomM(JG,M
W) _____:::____, HomG( V, JM , G
for any smooth representation ()., W) of M and any smooth representation of 2.3. Let G, P, A, M and be as before. The parabolic pair (P, dominates a parabolic pair (P', where P' is a minimal parabolic subgroup of hence A' is a maximal split torus in Let l/J be the root system of w.r.t. and L1 be the basis of l/J associated to P'. There exists a subset e of L1 such that is the largest torus contained in the intersection of the kernels of the elements of e (we view the roots a in l/J as homomorphisms from to px). For any real num ber e > 0, let A-(e) consist of the elements in A such that l a ) l F � e for every root a in L1\8. Moreover, let P be the parabolic subgroup of G opposite to P. The roots associated to P are the roots - a where a runs over the roots associated to P. Let fit be the unipotent radical of P. Let now be any admissible finitely generated representation of G. Let (it, V) be the representation contragredient to Using first Jacquet's functor we get representations and of M.
(n-, V)
(n-N, VN) (7rfl> VR) (n-, V) .
REPRESENTATIONS OF p-ADIC GROUPS
129
The following theorem is due to Casselman [17] . It plays a crucial role in the representation theory of reductive p-adic groups.
2.3. Therewithexithsetfoll s a ouniwinqguepropert M-invariy : ant nondegenerate pairing < ·, · )N between V Nand in with canoni thereGivexien svtsina Vrealandnumber such thcatal images u in VN and in respectively, holds for every a in (ii:N, (1CN, VN). any irreducible admissible representation of G. The followi(7C,ng V)assertis absolut ioLetns are(1C,elyequiV)cuspivbealent: dal. P G of G, with unipotent radical we have For every paraboli c subgroup VN = Z Z(G) G. lO Z £\(Gt Jf'/G) G. (a)fbel f beloongsngs ttoo Yl'x;.:;(Gt; (G) and THEOREM
f1fl
v
V,
c
>
u
0
f1fl
(4)
A-(c).
As a matter of fact, the previous pairing identifies f1R) to the representation of M contragredient to The following criterion is due to Jacquet [32] and results easily from Theorem
2.3.
THEOREM 2.4.
(1)
i=
(2)
0.
N,
Note that 'absolutely cuspidal' is with reference to the maximal split torus in the center of An alternate formulation of Theorem 2.4 is as follows. Choose a character X of and recall that is the subspace of generated by the coefficients of the irreducible absolutely cuspidal x-representations of Then the following condi tions are equivalent : (b)
Jlt'
(5)
holdsfor g in G and every subgroup as in Theorem G/Z. Letpair(1C,(P,V) bewianyth associ irreduciatebdleLeviadmidecomposi ssible represent at=ion of· G.andTherean eirxreduci ist a paraboli c t i o n P phibyc ble absolutatieolyn ocuspi dali\represent a1tiiosnth(i\e,extW)ensiofon ofsuchi\ tothPat=1CM·is isomor ti\o1(mn) a subrepresent f lnd� 1 where i\ gi v en ( = i\(m)). G M (P,G G. N 2.4. The next result is again due to Jacquet [32] . It is easily proved by induction on the split rank of THEOREM 2.5.
A)
M N
M
N
In principle, the classification problem for irreducible admissible representations is split into two problems : (a) Find all irreducible absolutely cuspidal representations for and for the groups which occur as centralizer of A for some parabolic pair A) in of
10 I t is well known that Z(G)/Z i s a compact group. I t makes no essential difference to take Z o r Z(G) a s central subgroup. The choice o f Z is quite convenient however.
P. CARTIER
130
il1
(b) Study the decomposition of the induced representations lnd� as above, in particular look for irreducibility criteria. Needless to say, a general answer to these problems is not yet in sight. The con struction of some absolutely cuspidal representations has been given by Shintani [40] for GLn(F) and in more general cases by Gerardin in his thesis [21]. The idea is to induce from some compact open subgroups. As to problem (b), let us mention the notion of Two parabolic pairs (P, A) and (P ', A') are associated iff A and A' are conjugate by some element in G. Let f/(P, A) be the set of irreducible admissible representa tions of G which occur in a composition series of some induced representation Ind� il1 where il is an irreducible absolutely cuspidal representation of M and M is the centralizer of A in G. If (P, A) and (P', A') are associated, then f/(P, A) = f/(P', A'). 2.4. A Take for instance the group G = GLn(F). For any (ordered) partition = of let Pn�o ... , n, consist of the matrices in block + ··· + form = (Gk1)1�k. t� r where Gk1 is an nk x -matrix and Gk1 = if > I. Let for -=1- I An�o ... . n, be the set of matrices which in block form are such that Gk1 and Gkk is a scalar matrix · lnk · For Mn �o · · ·, n, take the matrices in diagonal block form, i.e., Gk1 = for -=1- I and let finally Nn �o ... , n, be the subgroup of Pn�o .. . n, defined by the conditions G11 = lnl · · · , G, = In, · Then (Pn�o .... n,• An�. .... n) is a para ' bolic pair, with associated Levi decomposition Pn�o ···, n, = Mn�o ···, n, · Nn�o ···, n, · Up to conjugation, the pairs (Pn 1 ; , n,• An�o · ·· , n) comprise all parabolic pairs in GLn (F). The pairs corresponding to partitions = n1 + · · · + and = + ··· + are associated iff = and · · ·, is a permutation of · · ·, Notice that the group Mn�o ···, n, is isomorphic to GLn1(F) x · · · x GLn,(F) . The well-known operation a o {3 o n characters o f the various finite groups GLn{q ), introduced by Green [24], may now be generalized to the case of a local field. Namely, let = + be an admissible representation of GLn,(F) let (n ' , " and (n , an admissible representation of GLnn(F). Define a representation of Pn ' , nn acting on the space by
associated parabolic subgroups.
g
nnexampln1 e.
n, n, 0 ka
n1
0 0k k =
.
.•.
r s nl > n, n
V")n n' n", V' ® V"V') il1 ( � g��) G
=
n'(Gu)
n, ml >n mm1 •.
n
"
m. il1
(G ) .
® zz
By induction from Pn ' , nn to GLn{F) one gets a representation n ' o n " of GLn(F). This product is commutative and associative. In particular, given characters a1 o · · · , an of px one gets a representation a1 o • • • o an of GLn(F) ; these representations com prise the Given a partition = + · · · + the associated to the parabolic group Pn�o · · ·, n, consists of the representations of the form n 1 o • • • o 12: , where ni is an irreducible absolutely cuspidal representation of 1 , · , Finally the is the family of irreducible GLn/F) for absolutely cuspidal representations of GLn(F). From the general results summarized in §2.3, one infers that any irreducible admissible representation of G Ln(F) is contained in some representation of the form = n1 + · · · + n o • • • o 12:, where and ni belongs to the discrete series of GLn/F). The question of the irreducibility of the representations n 1 o • • • o 12:, has now been completely settled by Bernshtein and Zhelevinski [3].
principal series. j r. =
1
n
n n1 n, intermediate series discrete series
· ·
n,
131
REPRESENTATIONS OF p-ADIC GROUPS
Square-integrable representations. G. rff (G, (n-, V)2 G n-(gz) = x(z) n-(g) z g G, S j (u \ n-(g) v) \ 2 dg (ui v) V. l 1 j (u \ n-(gz)·v) \ 2 = j (u \ n-(g)·v) \ 2 ; (7) d(n-) li uu l 2 l v l v.2 d(n-) 0, formal degree ofn-, square-integrable. G (n-, V) G, (n-: nGiven and as above, one gets d(n-) (n-: and in particular (n-: for every square-integrable irreducible representation n- of G. (n-, V) rff2(G, VK (n-: V n( K). Voo = U K VVK G. V (n-00, V G.n-(G). oo Jlt'(G), n-(f) = Vfcf(g)n( g) dg V K ntegrable irreduci f) = (n-(8",f))a J't'(G,ble representation hasEMa character every square-i distributiTheonfundament on G. al estimate. (P, G. N G; P
2. 5 . For the results expounded in this section, see Barish-Chandra and van Dijk [28, especially part 1] . Let again Z be the maximal split torus contained in the center of Fix a unitary character X of Z. We let x) denote the (equivalence classes of) irreducible unitary representations of which satisfy the following two conditions :
for
(6)
(7)
GI Z
Here
·
E
E
Z,
<
+
oo .
denotes the scalar product in the Hilbert space
(6), one gets
Due to assumption
hence we can integrate over G/Z in formula (7). The integral is equal to where the constant > the is independent from and These representations are called as usual Fix now a compact open subgroup K of and a class b of irreducible continuous representations of K. Every representation of class b acts on a space of finite dimen sion, to be denoted by deg b. Moreover, for any unitary representation of denote by b) the multiplicity of b in the restriction of to K. The following theorem is easy to prove (see [28, p. 6] ) . K, b
THEOREM 2. 6.
(8)
�
1r E C2 (G,
(9 )
X)
X
b) � deg b/meas(K/(Z n K))
deg b b) � d(n-) · meas(K/(Z n K))
Let be in x). If b is the unit representation of K, the integer b) is the dimension of the space of vectors in invariant under Let where K runs over the compact open subgroups of It is easy to check that is dense in and stable under By the previous theorem we get an admissible x-representation 00) of Moreover for any function/ in the operator in the Hilbert space has a finite-dimensional . range (contained in if / belongs to Tr K)) , hence a trace To sum up, 2.6 . Fix a parabolic pair A) where P is a minimal parabolic subgroup of hence A is a maximal split torus in Let be the uni potent radical of and N the unipotent radical of the parabolic subgroup P op11
The scalar product (u J v) is assumed to be linear in the second argument
v.
P. CARTIER
1 32
posite to P. Hence there exists a basis L1 of the root system of G w.r. t. A such that N (resp. is associated with the roots a with positive (resp. negative) coefficients when expressed in terms of LJ. Let be the set of elements in such that i a ) i F � I for every root a in LJ. According to Bruhat and Tits [13], there exist a compact subgroup of A and a finitely generated semigroup in such that Moreover there exist finitely many elements , in G and a compact open subgroup of G such that G ('Cartan decomposition'). Let be a compact open subgroup of G. We make the following assumptions : (a) (b) n n N)
N) A-
a A (a L S A A=ZLS. gio K0 gm K0 =K U 1;;;;;,;m K0 SZg, Koneis hasinvariK a=nt (Kin K0 ;P)·(K and a-1(K N)a K N, a(K P)a-1 K P for every a in K P). g aga-1 K K. openthsubgroup Kmofple module is as above. Then tK)hereeitexiherstiss aitificonstnite-adntimensi=onalThe(overcompact K) C) orsuch at every si over else has a dimension bounded by Z Kt K. (n-, V) VK K); VK There[ is an i!nteger p such that the higher commutator ft . · · · . = ·· vanishesfor arbitrary elementsf�o in Kt. Properties of unitary representations. Letrepresent K be aaticompact openTheresubgroup ofconstandant be a classK,of irredu cisuchble tconti n uous o ns of exi s t s a hat (n-: for every irreducible unitary representation V) of KK. 12 · · ·
c
n
c
n
n
n
s (hence the inner automorphism f-> of G expands n N and contracts n It is known that every neighborhood of the unit element in G contains such a subgroup The following estimate is due to Bernshtein [1]. THEOREM 2.7. N
N(G,
>
G
0
Yf'(G,
N.
be the subalgebra of Yf'x (Gt Let X be a unitary character of and Yf'.jG, consisting of the functions invariant under right and left translation by an element of If is an irreducible absolutely cuspidal x -representation of G, then � N by Theorem 2.7. By a well is a simple module over Yf'(G, hence dim known argument due to Godement [23], one infers from this bound the following corollary : CoROLLARY 2.2.
�2
� sgn{o-)/.. m
p]
( 1 0)
aESp
· · ·
,
Jp
·
1.-
Yf'x (G,
In his lectures [28], Harish-Chandra 2.7. was unable to prove Corollary 2.2, and had to assume it 12 in order to establish the following result : THEOREM 2.8.
b) �
N
K.
In the proof, one may assume that
b is the unit class of representations of
G
b
N
=
(n-,
b)
N(G,
G.
>
0
is as in Bernshtein's Theorem 2. 7 and that
As before (see end of §2.5), one deduces from Theorem 2.8 the following corol laries :
" Harish-Chandra assumes apparently the stronger 'Conjecture I' (p. 1 6 of [28]). But the proof (p. 1 8, end of first paragraph) uses only Corollary 2.2 above.
REPRESENTATIONS OF jJ-ADIC GROUPS
133 Let in stabebleanyby irreduciforblsome e unitacompact ry represent atisubgroup on of Letof beThenthe spaceis dense of vectors open of The representinatioandn stable under is admishence sible. affords a smooth representation Letthe operatober anyf irreducif bgle unig tadgry irepresent afinitionteof-dimensi Foroanynal functi o n fi n n has a range, hence a trace B.Jf). 8" character CCR Some other results. 8" COROLLARY 2.3.
G.
Vco
V
CoROLLARY 2.4.
Jlf(G)
(n, V) V
(nco, Vco)
(n, V)
G.
n(K)
K
G,
n( ) = cf( )n( )
Vco G. l'rco
G.
V
The distribution is called the of n. By an easy limiting process one deduces from Corollary 2.4 that n{f) is a compact ( = completely continuous) operator in the Hilbert space V for every integrable function/ on G. Hence the group G belongs to the category of Kaplansky. In particular (see Dixmier [19]) every factor unitary representation of G is a multiple of an irreducible representation, there exists a Plancherel formula, . . . . 2.8. Much more is now known about the characters of the irreducible unitary representations of G. The character is for instance repre sented by a locally integrable function on G, which is locally constant on the set of regular elements (see Harish-Chandra [26] and my Bourbaki report [14]). Moreover it is now known that Conjecture II in Harish-Chandra's lectures holds true. More precisely, if the Haar measure on G/Z is suitably normalized, the formal degree of any irreducible absolutely cuspidal representation of G is an integer. As a corollary (see [28, part III]), the algebra Jlf .x_CG, Kt is finite-dimensional and its dimension is bounded by a constant depending on G and K, but not on Also for any character X of Z, any compact open subgroup K of G and any class b of irredu cible continuous representations of K, there exist only finitely many irreducible absolutely cuspidal representations 1r of G such that w" = X and (n : b) =f. Let (n, V) be a smooth representation of G. We say (n, V) is if there exists a hermitian form tll on V such that > for =f. in V and
X·
( 1 1)
0. preuni t a ry tll(v, v) 0 v 0 tll( g · v, g v') tll(v, ' n( )
n( ) ·
=
v)
for in V and g in G. We can then complete V to a Hilbert space V and extend by continuity n(g) to a unitary operator fr(g) in V. Then (fr, V) is a unitary representa tion of G. If (n, V) is admissible, V is exactly the set of smooth vectors in V, that is V U K VK where K runs over the compact open subgroups in G. It follows from Theorem 2.8 and these remarks that G G.
v, v'
=
theofclathssifie preuni cationtaoryf therepreir b l e uni t a ry represent a t i o ns o f amount s t o t h e search reduci sentations among the irreducible admissible representations of 3.1. Preliminaries about tori. multiplicative group in one variable, k k. k,
. III. Unramified principal series of representations. For this section only, we denote by a (com mutative) infinite field. Let Gm be the considered as an algebraic group defined over To a connected algebraic group H defined over we associate two finitely generated free Z-modules, namely X * (H) = Homz(X * (H), Z).
In these formulas, Hom k-gr (resp. Homz) means the group of homomorphisms of algebraic groups defined over (resp. of Z-modules). We denote by (q>, .1.) (for
k
P. CARTIER
134
in X* (H) and it in X*(H)) the pairing between X* (H) and X * (H). We can as well use this pairing to identify X * (H) to Homz (X * (H), Z) . Let now S be a split torus defined over k. A sequence (ii. l > · · · , An) is a basis of the free Z-module X * (S) iff the mapping s ...... ( ii. 1 ( s), · · · , An (s)) is an isomorphism from S onto the product (Gm)n Gm x · · · x Gm (n factors). Moreover we may identify X * (S) to Hom k-gr (Gm , S) in such a way that the following relation holds rp
=
(1 )
A(rp(t))
=
t<
for rp in X * (S), it in X * (S) and t in k. By construction, the elements of X * (S) are polynomial functions on S and it is easily shown that they form a basis of the k-algebra A of such functions. Otherwise stated, S is the spectrum of the group algebra A k[X * (S)] of the group X * (S) with coefficients in k. For any commutative k-algebra L, the £-points of S corres pond therefore to the k-algebra homomorphisms from A into L, hence an isomor phism S(L) � Hom(X * (S), U ) . From the duality between X * (S) and X* (S) we get another isomorphism S(L) � X* (S) ®z L x . Unramified characters. Let H b e a connected algebraic group defined over our local field F. There exists a homomorphism ordH : H --> X* (H) characterized by =
3. 2. (2)
for h in H and it in X * (H). In the right-hand side of this formula, ordF (ii.(h)) is the valuation of the element it(h) of p x . We denote by o H the kernel and by A (H) the image of the homomorphism ordH. By construction, one gets an exact sequence 1
(S)
_____,
o H _____, H � A(H)
_____,
1.
We can also describe o H as the set of elements h in H such that ii.(h) e D� for any rational homomorphism it from H into F x . Therefore o H is an open subgroup of H. A character X of H is called unramified if it is trivial on o H. Otherwise stated, an unramified character is of the form u o ordH where u is a homomorphism from A(H) into ex . Introduce the complex algebraic torus T = Spec C[A(H)]. By definition, one has A(H) = X * (T) and T(e) = Hom(A(H), ex). Thus, there exists a well defined isomorphism ...... X t between the group T(e) of complex points of the torus T and the group of unramified characters of H. If H is a torus, one has
t
(3)
X e(rp(wF))
=
rp( t )
for t in T (e), rp in X * (H) = X * (T) and any prime element w F of the field F. Let again be a connected reductive algebraic group defined over the local field F. We fix a maximal split torus A in and denote by M its centralizer in We let N(A) be the normalizer of A in and W = N(A)/M be the corresponding Weyl group. We choose also a parabolic group such that A) is a parabolic pair. Hence is a minimal parabolic subgroup of and = M · N where N is the uni potent radical of We denote by (/) the set of roots of w.r.t. A and by A the group A(M). Hence (/) is a subset of X*(A) and defines a basis L1 of (/). Let LJ- be the set of roots opposite to the roots in L1. The basis LJ- of (/) corresponds to a parabolic subgroup
G
P
GG P (P, G P P. G P
G.
REPRESENTATIONS OF jJ-ADIC GROUPS
1 35
P - = M · N- of G. Let n be the Lie algebra of For any m in M, the adjoint re presentation defines an automorphism Ad n ( ) of n. We set (4)
A AtA
o(m)
m N.
= j det Ad n ( ) j F for m in M.
m A
A
A
The group o (resp. oM) is the largest compact subgroup of (resp. M) and o is equal to oM n A . Thus the inclusion of into M gives rise to an injective homo morphism of into M;oM. We do not know in general if this map is surjective (see however Borel [5, 9. 5]). More precisely, the inclusion of A into M gives rise to a commutative diagram with exact lines
(D)
l
------+
l
------+
o
------+
A A 1
1
ord A X (A) *
1
oM ------+ M ordM A
A A
------+
1
------+ 1
A
Indeed, since is a split torus, ordA is surjective. The inclusion of into M enables us to identify XA ) to a subgroup of finite index in X* (M), hence the relation X* ( ) c A c X* (M). We denote by X the group of unramified characters of M. We may (and shall) introduce as before a complex torus such that X * (T) = A and an isomorphism �--> Xt of onto This isomorphism enables us to consider X as a complex Lie group. The subgroup of acts on M, oM, o via inner automorphisms. Using diagram (D) above, we may let the Weyl group W = N(A)/M operate on X* (M) so as to leave invariant the subgroups X * (A) and A of X* (M). The group W acts there fore on X and by automorphisms of complex Lie groups. For instance, if X is any unramified character of M and w any element of the Weyl group W, the trans formed charaCter wx is given by
A
t
T(C) X. N(A) G T
T
A, A
(5) where is any representative of w in The unramified character X of M is called if wx of. X for every element w of. I of W. For the applications to automorphic functions, one has to examine the case where G is that is the following hypotheses are fulfilled : (a) G (b)
xw ar N(A). regul F,F. iTheres unramifi quasiexis-tssplaneidt over over splits over F' . unramified extension F' ofF, of.finitF.e degree d, such that G T', X. g' g'" T' t' h a-conjugate gi gz g2 h-1 gih". Any semisimple element is a-conjugate to an element of T'. Let a denote the Frobenius transformation of F' over In this situation, the L-group associated to G is defined. It is a complex connected reductive algebraic group L Go endowed (at least) with a complex torus an automorphism ,_. such that T'" = and a homomorphism �--> x ;, of T' onto We say two elements and of L Go are if there exists in L Go such that = The following theorem has been proved by Gantmacher [20] and Langlands [35]. THEOREM 3 . 1 . (a)
in L Go
P. CARTIER
1 36
� and t� of T' are a-conjugate iff the unramified characters x'1,1 Two element s t and X;'z of Mare conjugate under the action of the Weyl group W. L G0• The unramifi 2. er of We define the re P Leted priinncipalbeserianyes.unramifieod1 1charact presentTheatiospace n I(x)I(x))consiof Gstass offoltlhoews:locally constant functions/: G --> C such that f(mng) o(m)1 12 x(m)f(g) form in M, n in N, gin G. The group G acts by right translations on I(x), namely (v/g) -f)(g') f(g'g) for fin I(x), g, g' in G. (7) Yl'(G). Px : Yl'(G)I(x)--> J(x) Pxf(g) J J o1 12(m)x-1(m)f(mng) dm dn. dm M dnYl'(G) and N P I(x). (b)
Otherwise stated, the orbits of W in the group X of unramified characters of M are in a bijective correspondence to the a-conjugacy classes of semisimple elements in For more details, we refer the reader to Borel's lectures [5, §6, 9.5] in these PRO CEEDINGS. This series shall presently be defined via in 3 .3 . duction from with the slight adjustment of DEFINITION 3 . 1 . X X M. (vx, (a)
(6)
=
(b)
=
It is important to give an alternate description of Indeed one defines a surjective linear map (8)
=
M
as a factor space of by
N
(See formula ( 37) in § 1 . 8 .) The groups M and are unimodular, hence the Haar measures on and on N are left right invariant. The map x intertwines the right translations on with the representation vx acting on the space From the general results described in §II, one gets immediately the following theorem . THEOREM 3 .2. (a) (b) (c)
X
s admiofssiI(x).ble. Fortion every isn isomorthe prepresent aetiocont n ragredi I (x))eontf G(/i(x))1 The represent a I(x) i hi c t o t h preuniIfxtary.is a unitary unramified character of M, the representation I(x)) ofG is o1 12 J(x) J /(o1 12) G J(P0vz f) L f(g) dg f Yl'(G) f I(x) f' J(x 1) I(ol l2) f') J(ff ') . f�>I(x) J(x) /1 /2 J(o1 12) X (vx,
X,
(vx,
One of the reasons for inserting the factor in the definition of is to get assertions (b) and (c) above. We state them more precisely : there exists a linear form on invariant under the right translations by the elements of and characterized by
(9)
=
for in
(see Bourbaki [6, p. 4 I ] for similar calculations) . For in function//' belongs to and the pairing is given by ( 1 0)
(f,
Similarly, for unitary and the unitary scalar product in
.h in the functi on is given by
( I I)
and
in
-
, the
=
belongs to
and
REPRESENTATIONS OF p-ADIC GROUPS
1 37
We now state one of the main results about irreducibility and equivalence (see also Theorem 3. 1 0 below). X
Letry andbereanygulunrami.fi ed charactateiornofM. is irreducible. i s uni t a a r, t h e represent atiareons irreducible.and have the same charactTheLeter,.Yf(G)-modul hencebe inareW.equieThevalentrepresent if t h ey is of.finite length. semisimplijied W, Structure of Jacquet's module M (m, z) ...... m any smootsmh representation of G. For any unramified character of M,Letone(g:n:et, s anbeisomorphi THEOREM 3.3. (a) If x (b) w
(vx,
(vx,
(c)
I(x))
I(x))
(vwx•
l(wx))
I(x)
In general, if V is a module of finite length over any ring, with a Jordan Holder series 0 V0 c V1 c . . · c Vn - 1 c Vn V, the semisimple module V cs> 83 7= 1 V;/ V;_ 1 is called the form of V. According to Jordan-Holder theorem, it is uniquely defined by V up to isomorphism. Let J(x) cs> be the semisimplified form of I(x). It exists by Theorem 3.3(c) above. the It is clear that I(x) and J(x) cs> have the same character. Hence for any w in representations l(x) cs l and J(wx) cs l are semisimple and have the same character. By the linear independence of characters, they are therefore isomorphic. I(x)N. For any unramified character X of M, 3.4. let Cx denote the one-dimensional complex space C 1 on which acts via X • viz. by x( ) z. Frobenius reciprocity takes here a simple form, namely (see §2.2) : =
=
=
·
THEOREM 3.4. x
V)
Homc ( V, I(x)) --""-> HomM( VN, Cx0 1 12).
The proof is obvious. Indeed the relation ·
W(v) (g) (rp, :n:(g) v) =
( 1 2)
v
(for g in G, in V)
expresses an isomorphism (/j +-> rp of Homc( V, I(x)) with the space of linear forms on V such that
rp
(rp, :n:(mn) . v) v m:n:(n)Mv-v n n v
( 1 3)
=
o 1 12 ( )x( )
m m (rp, v) rp
for in V, in and in N. Recall that VN VI V(N) where V(N) is generated by the vectors · for in N and in V. Any solution of ( 1 3) vanishes on V(N), hence factors through VN· The previous theorem exemplifies the relevance of Jacquet's module I(x)N in the study of the intertwining operators between representations of the unramified principal series. We know by Theorems 2. 1 , 3.2 and 3. 3(c) that I(x)N is a finite dimensional complex vector space. The following basic result is due to Casselman [17] . =
X
er oftM,he gtroup he semiMsimactplifis terdivform the M-module For ianys unrami.fied charact Moreover, ial y onof Weyl group The dimension of over is equal to the order I WI of the Assume regular. Then as an M-module is isomorphic to THEOREM 3 . 5 .
I(x)N
I(x)N ·
EB w e w Ccw r.J ·o l /2 ·
CoROLLARY 3. 1 .
I(x)N
W.
CoROLLARY 3.2.
EB w e W C (w r.) · ol l2 ·
X
0
C
I(x)N
P. CARTIER
1 38
For the proof of Corollary 3.2, notice that M acts on through the commu tative group isomorphic to A. By Schur's lemma, the semisimplified form of is therefore of the form EB :=l Cx; for some sequence of unramified of by a well-known lemma, each representation Cx; occurs characters as a subrepresentation of Hence for w in W, Ccw xl · o i/2 occurs as a subrepre sentation of by Theorem 3 . 5 . Corollary 3.2 follows at once from thi s remark as well as the following corollary (use Frobenius reciprocity in the form of Theorem 3 .4) :
I(x)N M/" M I(x)NXI > · ··, X M; s l(X)N. I(x)N LetThereX beexianysts aunramifi edicharacter ofoperator M and w be anyI(x)element of tIfheX Weyl n tertwi n i n roup W. nonzero --> l(wx). g g is regular this operator is unique up to a scalar. (n-, I(x), wE (n-, I(w· x) T COROLLARY 3 . 3 .
Tw :
Tw
A similar argument shows that if V) is an irreducible subquotient of then there exists W such that V) is isomorphic to a subrepresentation of (see 6.3.9 in [17]). We shall describe more explicitly the operators w in §3.7. We sketch now a proof of Theorem 3 . 5, a streamlined version of Casselman's proof of more general results in [17]. Basically, it is Mackey's double coset tech nique extended from finite groups to p-adic groups. The case of real Lie groups has been considered by Bruhat in his thesis [7], it is much more elaborate. (A) We remind the reader of Bruhat's decompositi on = w where means For w in for any g in N(A) representing the element w in W = N(A)/ W, the set is irreducible and locally closed in the Zariski topology, hence has a well-defined dimension. Set - dim(P). For any integer � 0, = dim let be the union of the double cosets such that < The Zariski closure of any double coset is a union of double cosets of smaller dimension, hence is Zariski closed, hence closed in the p-adic topology. We let be the subspace of consisting of the functions vanishing identically on We have then a decreasing filtration
Fr
PgPPwP
Fr I(x)
Ir
d(w) (PwP)PwP PwP
G U PwP,M. PwP d(w) Pw'r.P r Fr.
( 1 4)
I(x)The P-nextstablstepe subspaces. is to prove that anyfunction on Fr+l which satisfies the relation f(mng) ol l2(m)x(m)f(g) m n Fr+l is the restriction of someG function belonP\G) ging to I(x). f(g) IM IN o1 12(m)x-l(m)q>(mng) dm dn q> Frf+l· q> q>' G). P .,_ q> ' I(x) Fr+ l· Irllr+l Fr+l by
of
(B)
( 1 5)
fo r
=
local cross-sections of
fibered over
in M,
in N, g in
Indeed, one proves easily (using that such a function is of the form
=
for a suitable locally constant and compactly supported function on Extend to a function in Jll'( Then belongs to and restricts to
(C) From this it follows that
satisfy the following conditions : (a) f is locally constant ; (b)/ vanishes on
Fr;
is the space of functions f on
in
which
REPRESENTATIONS OF p-ADIC GROUPS
1 39
relation (15). d(w) r, open (16) Jw f(mng) o1 12(m)x(m)f(g) n (16) .Yt(M)-modules I(x)N and ffiwEW(Jw)VN haveVN isomorphic semisimplifiedforms. M CJw)N. w N(A)N(w) w) (w) M·N(w)Jw (mn) w-I(ol i2X) (m) (17) M, n N(w). M4, a (mn) m M n 1. aAaw · ow) (18) V V a)*(aw·ow).o a* o a)a* P(w) M CJw)N(w)N o awowi M. o N(w) (24c) o M ol l2(w-Io)-1 12 Cial y on I(x)N. I(x)N oM actE8 wEW s t r i v compact M. I(x)N M I(x)N, o) I(x)N. 3. 5 . Buildings and Iwahori subgroups. 3. 5 . (c) Moreover, F,+J is the union of F, and the various double cosets PwP such that = which are in F,+ J ; hence one gets an isomorphism
Here
is the space of functions/ on PwP such that
for m in M,
=
in N, g in PwP,
and which vanish outside a set of the form PQ where Q is compact. Since Jacquet's functor =:> is exact, one infers from that the
Here we (D) It remains to identify the representation of on the space are paid off the dividends of our approach to induced representations via tensor products. Choose a representative Xw of in and put P( = p n x-;;} Pxw ; hence P = with a suitable subgroup of N. lt is then easy to show that, as a P-module, carries the representation c-Ind� (wl aw where the character aw of P(w) is defined by aw
for m in
=
in
Consider the group homomorphisms P(w) ..!!., P L and � = for m in and in N. By Theorem where the character Ow of P(w) is defined by
where is the injection one gets c-Ind� (wl aw =
Since � * is Jacquet's functor =:> N and �* one gets that (Jw)N is = (� the carrier of the representation (� Since � is the projection of onto with kernel and the characters aw and Ow are trivial on one It remains to prove the gets an isomorphism � c, where A = to be able to conclude formula (see formula below) ;;;1 1 = ( 1 9)
<w xl-1!112 have (E) From (C) and (D) above, we know that and isomorphic semisimplified forms. It remains to show that But oM is a subgroup of Hence its action on and its semisimpli fied form are equivalent. Since any unramified character of (including is trivial on oM, this group acts trivially on the semisimplified form of hence on This concludes our proof of Theorem
Our aim i n this section is mainly t o fix notations. For more details, we refer the reader to the lectures by Tits in these PROCEEDINGS [42] or to the book by Bruhat and Tits [13]. Let P4 be the building associated to and let .9/ be the apartment in P4 associated to the split torus We choose once for all a special vertex x0 in d. Among the conical chambers in .9/ with apex at x0 there is a unique one, say, enjoying the following property : for every in N, the intersection n contains a translate
A.
G
n
({? n({? ({?
1 40
P.
CARTIER
for one of its of 'tl . There is then a unique chamber C contained in 'tl having it is called by vertices (see Figure 1). The stabilizer of in shall be denoted by Bruhat and Tits a of The interior points of C all have the same stabilizer in called the attached to c.
x 0 G K; x 0 special, good, maxi groupsubgG.roup of G B mG,al compact sublwahori
FIGURE 1
Any element takes the apartment d to itself, it fixes every point of d iff it belongs to We may therefore identify the group = called the to a group of affine linear transformations in d. The group can be represented in two different ways as a semi direct product : (a) Let Q be the subgroup of consisting of the w's taking the chamber C to itself. The walls in d are certain hyperplanes and to each of them is associated a certain reflection. These reflections generate the invariant subgroup w.ff of Since acts simply transitively on the set of chambers contained in .s:J, the group is the semidirect product Q · = (b) Any element w of has a representative in and We may therefore identify the Weyl group = to the stabilizer with the group of transla The intersection of of x0 in tions in d is which we identify to A by means of the exact sequence (D), p. 1 35. Then is the semidirect product A where A is an invariant subgroup. The fundamental structure theorems may now be formulated as follows.
gioM.n N(A) W1 modified Weyl group,
W1 N(A WM,
W1
W1 . WWa1H WaH· w(w) K n N(A) oM W K(Knn M.N(A))j(K n M) W1 . WW1 N(A)/M WM/0M W· 1 and, more the sets Pw(w)Bfor runningGover tPKhe Weyl W. sely, G is the disjoint union of grouppreci over the modified Weyl group WG1 • is the disjoint union of the sets Bw1Bfor w1 running Let of beopposi the subset of intconsi selftingThenof thGe element ssjoiofnt tuniakionngofthteheconisetscalK·chamber ·Kfor). t e to o i t s i s t h e di running over tion). Moreover one has B n1 MB (BoMnandN-)·(B n M)·(B n N) (unique factoriza m(B n N)m- B n N, for any m in M such that G A. a(x0 ) , a) =
lWASAWA DECOMPOSITION.
w
BRUHAT-TITS DECOMPOSITION. CARTAN DECOMPOSITION.
ordA}().)
'tl-
Ad
'tl
A
A
A.
=
lWAHORI DECOMPOSITION.
=
c:
ord M (m) E A-.
As before, we denote by fP the system of roots of w.r.t. We identify the ele ments of 1> to affine linear functions on sif in such a way that + ). = ( ).
REPRESENTATIONS OF p-ADIC GROUPS
a
141
for in r!J and ). in A. The conical chamber '?? is then defined a s the set o f points x in s1 such that (x) > 0 for every root in the basis L1 of r!J associated to the para bolic group We denote by r!J0 the set of affine linear functi ons a on d, vanishing at x0 and such that the hyperplane is a wall in s1 iff the real number is an integer. An affin e root is a functi on on d of the form + where a belongs to r!J0 and is an integer ; their set is denoted by r/Jaff · For any affine root the reflection in the wall is denoted by Sa. The reflections Sa for running over r!J0 (resp. r!J.H) generate the group W (resp. w. H) . For in r/J0, there exists a unique vector ta in A such that
P. a
a-1(0) )
a
a-1(r)
a
Sa(X) = x -
(20
a(x)
a k a
·
r
a,
k
ta for any x in d.
We denote by aa any element in M such that ta = ordM(aa). The set r/J 1 is obtained by adjoining to r!J 0 the set of functions /2 for in r/J 0 such that (Bw(Sa)B : B) # o(aa) -I IZ . The sets r/J, r!J0 and r/J 1 are root systems in the customary sense (see for instance [37, p. 14 sqq .]). When the group G is split, the sets r/J, r!J0 and r/J 1 are identical. In the nonsplit case, all we can assert is that, for any a in r/J, there exists a unique root in r/J0 proporti onal to and that any element of the reduced root system r!J0 is of the form for a suitable in r/J. Let w 1 be any element of W1 . Since w 1 is a coset modulo the subgroup a M of B the set Bw 1 B is a double coset modulo B. We put
.A.(a) (2 1
a a
a, a
.A.(a)
)
Since K = B WB, one gets
(K: B) = 2: q (w). E
(22)
w W
Let L1 1 be the set of affine roots in s1 which are positive on the chamber C and whose null set is a wall of C. The group W is generated by and the reflections Sa for in L1 1 • The value of q (w ) is given by
a
(23)
1
Q
1
where w 1 = wSa1 . . . sam is a decomposition of minimal length m (w in a[ , · , a m in Ll l ). To each root {3 in is associ ated a real number q� > 0. This association is characterized by the following set of properties
Q,
0 0
r/J1
qw ·� = q� for {3 in r/J 1 and w in W, IT q (w) = q�, w �>O ; - 1 . �<0
o(m) = IT qj�( m ·xo) for m in B>O
M.
In the previous formulas {3 is a variable element in r/J 1 and the notation {3 > 0 means that {3 takes only positive values on '?? . We make the convention that q a 1 z = I for a in r/J0 if a/2 does not belong to r/J 1 • Two corollaries of the previous relations are worth mentioning
P. CARTIER
1 42 (24.)
for any > 0 in f/)0 • When is split, q� is equal to the order of the residue field D FIPF for any root {3 i n = r/Jo = The structure of the Heeke algebra has been described by Iwahori and Matsumoto [30], [31].
a
q f/J G f/J1 . .Yt'(G, B) eristic function of the double coset BwThe1fami B. ly ForC(wW1)1}winEWhw1 isleta basiC(ws lof) bethethecomplcharact ex vector space .Yt'(G, B) (Bruhat 1 Tits decomposi t i o n! ) . ... , am in L11) be Let w be any element o f W and let wSa1"· S am (w i n ah 1 1 a decomposition ofw1 ofminimal length m. Then For each a in L11 , one has whereIfq(Sa)a andhas arebeendidestifinnedct elbyementformuls inaL1h thereabove.exists an integer such that THEOREM 3.6. (a)
{
0,
(b)
(25)
(c)
(26)
(2 1)
(d)
rna� � 2
{3
(27)
factors ma{3factors Moreover the relations and are a complete set ofrelations among the C(Sa)'s. Action ofthe Iwahori subgroups on the representations. projection of V ontLeto VN defiV)nesbeananyisomoradmiphissismbleofrepresent VB onto a( tVioNntMof. G. The natural VB (VN)"M P G. M VB (VN)" , v a V(N)A VB, l a(a)I F 0 a f/J C(a)) . v P.0 v q(Sa) en any I(x)(N). unramified character of M, one has a direct sum decomposition I(x) GivI(x)B oM I(x)N. G I(x)B. w(w) openM B Pw(w)B (woM W), GM xo1 12. mnw(w)b, f/Jw. x(g) 0o1 12(m)x(m) g Pw(w)B. rna �
(26)
(27)
3.6. due to Casselman [18] and Borel [4].
Here is the main result,
(n,
THEOREM 3.7.
One proves first that maps onto using Iwahori decomposition of B and the methods used in the proof of Theorem 2.3. There are some simplifications due to the fact that is a minimal parabolic subgroup of To prove that maps injectively in one first proves that, for any given = vector in n there exists a real number e > such that n( for every in satisfying � e whenever the root e is positive on But this relation implies = 0 by Theorem 3 . 6, since one has > 0 there. CoROLLARY 3.4.
=
X
®
This follows from Theorem 3 . 7 since A direct proof acts trivially on can also be obtained using the methods used in the proof of Theorem 3 . 5 . Using the decomposition of into the pairwise disjoint subsets Indeed, normalizes and n = in one gets easily a basis for Hence the following function is well defined on lies in the kernel of (28)
=
=
if = if g rt
REPRESENTATIONS OF p-ADIC GROUPS
143
The family {W.,,x}w EW is the sought-for basis. The next result is due to Casselman [18] (see also Borel [4]).
(n , V)
bealeanynt: admissible irreducible representation of The follo wiThere ng assertare iionLetns arenonequi v ero vecteodrscharact invariaentr under Bsuch(thatthiats (7r, is isomorphic zunrami.fi There exi s t s some o f to a subrepresentation of THEOREM 3 . 8 .
G.
V
(a) (b)
X
(vx, I(x)).
VB
M
i= 0).
V)
B y Theorem 3 . 7 , assertion (a) means that ( VNrM i= 0. B y Frobenius reciprocity (Theorem 3.4), assertion (b) means that there exists in the space dual to VN a non zero vector invariant under oM which is an eigenvector for the group M. Since oM is compact, acts continuously on V N and M;oM is commutative, the equivalence follows immediately. X
)B
I(x) as a G-module.Let be any unrami.fied character of The space I(x generates Ix B I(x). I x I x = u I(x B) = Ix I x)) I x I(x-1) w we assumew theW,W,unrami.fied character to be Intregulertawr,ining operators. Ix mn · f) 1 1 2 m wx m (29) m n I(x), J f(w(w)-1n) dii -l)). w \N w F, (rI x d(....w. I(wx) Ix w, w(w) w. PK K= I(x) K WKjmnk) (j1 12(m)x(m) n CoROLLARY
3.5.
M.
We prove Corollary 3 . 5 by reductio ad absurdum . Assume that ( ) does not generate Since ( ) is finitely generated, there exist an irreducible admissible representation (n, V) of G and a G-homomorphism u : ( ) --> V which is nonzero 0. and contains ( )B in its kernel. Since B is compact, one gets VB ( ) By duality, one gets an injective G-homomorphism u : V --> ( ( - . Since ( ( )) by Theorem 3 . 2, it follows from Theorem 3 . 8 that VB i= 0. is isomorphic to the finite-dimensional spaces VB and VB are dual to each other and this is But clearly impossible. In this section, 3.7. that is the characters wx, for running over are all distinct. X Corollary 3.2 may be reformulated as follows : given in there exists a linear form Lw i= 0 on ( ), such that
Lw(vx (
)
for in M, in N and f in normalize Lw by (30)
Lw(f) =
= o
( )
( ) Lw(f)
unique up to multiplication by a constant. We
N (w) \N
= The Haar measure for a function / whose support does not meet is chosen in such a way that N( ) ( ) · (N n B) be of measure 1 . To Lw we associate an intertwining operator Tw : ( ) defined by (3 1 )
Twf(g) = Lw(vx(g) · f) for f in ( ) and g in G.
It is easily checked that Lw, hence Tw , depends only on not on the representative for Since G = (Iwasawa decomposition) and the character o112 X of M is trivial on M n 0 M, there exists in a unique function WK .x invariant under and normalized by W K . /I ) = 1 . Explicitly, one has
(32)
=
for m in M,
in N and k in K.
P. CARTIER
144
If the Haar measures on and N are normalized by JM nK = J NnK = 1, one may also define (/)K x a s Px(h) where Px i s defined a s o n p. 1 36 and IK is the characteristic function of K. The space I(x ) K consists of the constant multiples of (/)K.r The next theorem again is due t o Casselman [18] .
,M
dn
dm
I(x) K
The operator Tw takes into where the constant cw(X) is defined by cw(X) (34) THEOREM 3 . 9 .
l(wx) K .
(33)
More precisely, one has
C ( ), a aX
= IT
(35) ljq,
a
(/)0
Thebut product such thatin is ines gextatievnded e overover the affine roots in which are positive o1•er -- - a (34)
ljq,
wa
When G is split, formula (3 5) takes the simpler form 1 C a) ca(x ) - - I q--x1 x(aa) - · _
(36)
·
-
The bulk of the proof of Theorem 3.9 rests with the case where w is the reflection associated to a simple root {3 in (/)0. In this case, there exists exactly one positive root a in (/)0 taken by w into a negative root, namely a {3. The general case follows is equal to when the lengths of w1 and w2 add to the length then since of w1w2•
= Tw1 Twz Tw1wz tertwining map Tw is an isomorphism from onto cw(x) and Theareinnonzero. cw(Xkw-i(wx) Tw- 1 Tw Assume Xisiisrreduci any unrami.fi ed regular character offor erery Thenposithe treire present a t i o n of b l e root in positive over where is the unique element in which takes into I(x )
COROLLARY 3 .6.
Uf
C w-i( wx)
I(wx )
Indeed is multiplication by by Theorem 3.9. One may strengthen the irreducibility criterion (Theorem 3 . 3) . THEOREM 3 . 1 0 .
ljq-.
a
(/)0
(v x , I(x))
G
ijq,
w0
iff ca (X)
=1-
0, ca(w0x)
=1-
M.
0 W
CC
IV. Spherical functions.
Some integrationformulas.
We keep the notation of the previous part. 4. 1 . If F is any of the groups G, M, N, K, then F is unimodular. We normalize the (left and right invariant) Haar measure on F by Jr nK dr = I . The group P M · N is not unimodular. One checks immediately that the formulas (I)
(2)
J f( ) dip = JM J J d,p J J f( P
P
p
f(p)
N
=
define a left invariant Haar measure
M
dip
N
=
dm dn, ) dm dn
f(mn) nm
and a right invariant Haar measure
REPRESENTATIONS OF p-ADIC GROUPS
(P K)P. (3) d,p on n
145
Since aM = M n K is the largest compact subgroup of M, one gets (M n K) (N n K), hence the normalization
=
·
The Haar measures are related to the Iwasawa decomposition by the formulas
sG f(g) dg = JK JPj(p k) dk dip, L f(g) dg = JK L f(kp) dk d,p
(4) (5)
for f in Cc(G). Let us prove for instance formula into Cc(G) by the rule h >-+ uh from Cc(K x
P)
(6)
ui pk- 1 ) =
(4).
One defines a linear map
JPnK h(kp1 , PP1) d1PJ .
is then a left invariant Haar measure The linear form h >-+ Jc uh(g) dg on Cc(K x on K x hence by our normalizations, one gets
P)
P;
(7)
sG uh(g) dg = s K sPh(k, p) dk d1p.
It suffices to substitute f(pk- 1 ) for h(k, p) in formulas From the definition of a (see p. one gets
135),
(8)
(6)
and (7) to get formula (4) !
for any function / in Cc(N). As a corollary, we get the alternate expressions for the Haar measures on
P
JPj(p) dip s fN o(m)- 1/(nm) dm dn, JP f(p ) d,p = s sN o(m)j(mn) dm dn .
(9)
)
(10
=
M
M
Otherwise stated, the modular function of
Llp(mn) = o(m) - 1
(1 1)
P
is given by
for m in M, n in N.
The group N is unipotent and M acts on N via inner automorphisms. It is then easy to construct a sequence of subgroups of N, say N = N0 ::J N1 ::J • • • ::J N,_ 1 ::J N, = 1 , which are invariant under M and such that Nj_I fNj is isomorphic (for j = 1, to a vector space over on which M acts linearly. Putting . . .
( 1 2)
, r)
F
for m in M, one gets by induction on ( 13)
r
the integration formula
JN f(n) dn = Ll(m) JN f(nmn- 1 m-1 ) dn
for f in Cc(N) and any m in M such that Ll(m) #- 0 (see [28, Lemma 22]). Let us define now the so-called Let m be any element of M
orbital integrals.
P. CARTIER
146
such that #- 0, let be the centralizer of in G and Gm the conjugacy class of in G. Then A c M and n M has finite index in since is a compact group, the same is true of Moreover, the conjugacy class Gm is closed in G; hence the mapping >-+ is a homeomorphism from onto Gm . Therefore, the mapping A from G/A into G is proper and we may set after Harish-Chandra
m Ll(m)
Z(m) Z(m) m 1 m gZ(m)Z(m)/A. g g g ...... gmg-1 Fj(m) LJ(m)J f(gmg-1) dg
G/Z(m) (14)
=
Z(m); M/A
GI A
for any function / in Cc(G). The Haar measure on A is normalized in such a way that JM I A = }.
dffz f beisanyequalfunctioNn f(innm) dn. K) and m an element of such that LJ(m) ThenLetFj(m) J u(g) dg J JN u(nm1) dm 1 dn Fj(m) u(g)u f(gmg-1), F1(m) J h(m1) dmb LEMMA 4. 1 . #- 0.
to
Yf(G,
J
M
From formulas (2) and (5), one gets
( 1 5)
=
G
M
for any function in C,(G) which is invariant under left translation by the elements = one gets the following representation for in K. Putting =
( 1 6)
MI A
( 1 7)
= and set We get therefore
(m)m1 L1(m2). m2 m1mm11 . h(m1) L1(m2) JN j((nm2 n-1 m21)m2) dn (13). JN f(nm2) dn m2 mK K, f(nm2 f(fnm) K) n Nf(nm) dn / m1 a ake isomorphism. Fix Ll
=
From the definition ( 1 2) of Ll, one gets
=
by ( 1 7)
=
by
Notice that the group Mt M = M/(M n is commutative. Hence we get E and since the function is invariant under right translation by the is elements of one gets ) = for any in N. It follows that equal to J for any in M . The contention of Lemma 4. 1 follows from formula ( 1 6) since M A is of measure 1 . 4.2. S t The construction we are going to expound now is due to Satake [38]. It is the p-adic counterpart of a well-known construction of Harish Chandra in the set-up of real Lie groups. For any A in A, let ch(A) be the characteristic function of the subset ordA}(A) of = I , one gets M. Since JMn K (1 8)
dm
ch(i!) * ch(A')
=
ch(A
+
h(m1)
A')
for A, A' in A. Moreover the elements ch(A) (for A in A) form a basis of the complex algebra Yf(M, aM), which may be therefore identified to the group algebra C[A]. We define now a linear map S : Yf(G, Yf(M, 0M) by the formula
(19)
K) --> Sf(m) o(m)1 12 JN f(mn) dn o(m)-1 12 JN f(nm) dn =
=
147
REPRESENTATIONS OF p-ADIC GROUPS
f .Yf'(G, K)
for in and m in M. The two integrals are equal by formula (8). It is im mediate that the function on M belongs to Cc (M). That it is hi-invariant under oM follows from the fact that is hi-invariant under K and that oM M n K. The following fundamental theorem is due to Satake [38].
Sf f f ionconsiS sistinang ofalgthebrae invariisomor antspofhisthme from Weyl .Yf'groupTHEOREM (G, K)W. onto4.1.the ThesubalSatgebraake transfoormat S S is a homomorphism of algebras. .Yf'(G, K) .Yf'(P) G {3 P. ({3u)(m) u(mn) dn f)(m) f(m)o(m)1 12 ; The image of Sis contained in W Sf(xmx-1) Sf(m) m M mx n(m - In) F M; m (21) Sf(m) D(m) J f(gmg-1) dg -1 1 2 ; D(m) .d(m)o(m) D(m)2 I - 1nF) D(m) (m) D(xmx-1) D(m) m x G A. G A, =
C[A]W
C[A]
Here are the main steps in the proof. By construction, (A) three linear maps �
is the composition of
L .Yf' (M) L .Yf'(M).
Here a is simply the restriction of functions from to It is compatible with convolution by an easy corollary of (4). The map is given by JN and one checks easily that it is compatible with convolution. The map r is given by (r = since o is a character of M, it is compatible with convolution. (B) C[A]W . Since (N(A) n K)rM, this pro perty is equivalent to =
=
(20)
=
for in and in N(A) n K. The function ,_. det ( Ad ) from M to is polynomial and nonzero. The elements of M which do not annihilate this function are therefore dense in they are called the regular elements. Hence by continuity it suffices to prove (20) for regular. From Lemma 4. 1 , one gets =
G/A
hence = j det(Adn(m) - l n) j} - j det Adn(m) j:F1 = jdet(Adn(m) - 1 n) l p · l det(Adn(m- l ) 1 n)
for m regular in M. Here
is equal to
-
= jdet(Adn(m) - l n) lddet(Adn-(m)
F
) IF ·
The last equality follows for instance from the fact that the weights in 11 ® F P (F an algebraic closure of are the inverses of the weights in 11- ® F P of any maximal torus of M. Since g = 11 ffi m ffi 11-, one gets = !det(Ad gt nt (22) - l gt m) l ¥ 2 and therefore
(23)
=
for
in M,
in N(A).
On the other hand, the compact group N(A) n K acts by inner automorphisms on and It leaves therefore invariant the Haar measures on and hence
P. CARTIER
148
G/A. m M G;
the G-invariant measure on Let in be regular, X in f in One hasf(xgx-1) = f(g) for any g in hence
.Yt'(G, K). J
GI A
J = J
f(g(xmx-1)g-1) dg =
GI A
GI A
N(A) K n
and
f((x-1 gx)m(x-1 gx)-1) dg f(gmg-1) dg.
The invariance property (20) follows from the representation (2 1), the invariance property and the just-established invariance. (C) -> C[A]W We let as before A - denote the subset of A consisting of the translations in d which take '?!- into itself. For ). in A - , let lfJ;. be the characteristic function of the double coset ordA{().) by Cartan decomposition, the family { lfJ;.};."',�- is a basis of Moreover, any element of A is conjugate under to a unique element in A - . We get there fore a basis {ch'().)}J.r,.1- of C[A]W by putting . I ch'().) = ch( w . ).), (J.)i W
The(23)linear mapS: .Yt'(G, K)
is bijective.
W
(24)
where (25)
W().)
.Yt'(G,KK). ·
·
K;
w�W
I
is the stabilizer of ). in W. Define the matrix { ).' ) ch'().)
= I: ).
S!p;.- c().,
·
where )., ).' run over A - . To calculate c(A, A'), choose representatives Then we get (J..t is the Haar measure on
c().,
).')} by
and m' respectively of A. ).
in M.
G) 1 m S!p.,(m) o(m)- 12 f..t(Km'K NmK). mK NmK mK; 1 (27) o(m)- 12. non negative realKmcoeKfficieNmK nts ofisthempt e posiytiunlve erss s is linear combination with (27), S!p;.- , 4.1. The algebra .Yt' G is generated over Z. .Yt'(G, K) Mf(m)x(m) Mo odmM) .Yt'(G, K) c(). ,
(26)
).')
n
It is clear that K
=
n
=
=>
'
hence
c(A, A1) �
Moreover
'
n
oo
).'
-A
a
Using a suitable lexicographic order ing � , we conclude that c(A, A' ) = 0, unless 0 � ).' � A. Since c(A, ).) =f. 0 by this remark shows that the elements for ).' in A, form a basis of C[A]W, hence our contention (C) . CoROLLARY
C.
(
t
, K)
.
commutative an d finitefy
The algebra C[A] is commutative and generated by ch(). 1 ) , · · · ch().m), ch(A11 ), Since the group W is finite, it follows from well-known results in commutative algebra 1 3 that C[A]W is finitely generated as an algebra over C, and that C[A] is finitely generated as a module over C[A]W. We determine now the algebra homomorphisms from to C. Let X be dm any unramified character of M. Since J = I, the map / >--> J to C, and we get in this way all is a unitary homomorphism from .Yt'(M, such homomorphisms. Define a linear map w x : -> C by ,
· · ·, ch().;;;1 ) if {A 1 . · · · , Am} is a basis of A over
13 See for instance N. Bourbaki, Commutative algebra, Chapter V, p. 3 2 3 , Addison -Wesley , 1 972.
149
REPRESENTATIONS OF p-ADIC GROUPS
(J.Jx(f) f Sf(m) . x(m) dm. 4. . Anyed charact unitaryerhomomor pMoreover, hism fromoneJft'(hasG, K)(J.Jx into(J.JCix' s otfheretheexiformsts some unramifi o f M. (J.Janx forelement in such that = 4.1 Jft'(G, K) M. * T) T 4.3. DetJft'ermi(G,nK)atiKongKof the sphericalfunctions. Jft'(G, K), Jft' ( G, K) G, K: Jft'(G, K), (J.J(f) L t(g)F(g) dg (30) F(g) (J.J(IKgK) jfKgK dg1 g G. (JJOneis ahashomomorphism ofalgebrasfrom Jft'(G, K) to C. (31) for gbForg2 anyin G.function fin Jft'(G, K), there exists a constant such that (28)
=
COROLLARY w
2
x
W
'
X
w
M
=
· X·
iff
This corollary follows from Theorem and the classical properties of invariants of finite groups acting on polynomial algebras (see previous footnote). Otherwise stated, the set of unitary homomorphisms from to C is in a bijective correspondence with the set X/ W of orbits of W in the set X of unramified characters of We refer the reader to §3.2 for a discussion of this set X/ W. Notice that since X is isomorphic to a complex torus such that X ( = A, then X/ W is a complex algebraic affine variety and Satake isomorphism defines an isomor phism of with the algebra of polynomial functions on X/ W. Since the characteristic functions of the double cosets form a basis of the complex vector space one de fines as follows an isomorphism of the dual to the space onto the space of functions on hi-invariant under (29)
=
for f in
=
for
in
I claim that the following conditions are equivalent : (a) (b)
(c)
}..(f)
(32)
The equivalence of (a) and (b) follows from the following calculation
(JJ(/1 * /2)
ff F(g1g2) /1 (g1) /2(g2) dg1 dg2 = f /1 (g 1 ) dg 1 f f2(g2) dg2 f K F(g 1 kg2) dk G G =
(gh g2) JKF(gK1kg2) dk G G (JJ(/1 * /2) f/2(g)(j1*F)(g) dg f f1(g)(F*f2)(g) dg (J.J(j). /1 , /2 Jft'(G, K) f f(g -1) .
and the fact that the function ,_. on x under left and right translations by elements of x K. The equivalence of (a) and (c) follows from the following formula : =
for
in
where (g) =
=
Hence A.(f) =
is invariant
P. CARTIER
1 50 DEFINITION 4. 1 .
n on Gngw.thre. t.equiivsalent a .functipropert on ronies G, hi-inandvariant above. under A such thatspherical functandioenjoyi M. mn ) x(m)o1 12(m) M, n g) J G. Theunramifi spheriecdalcharact functioensrs onof M.G w.Ther.t. spheriare ctalhefuncti functioonsns r and Let and x' be are equal iff there exists an element win the Weyl group W such that x' (zonal)
K,
=
K
I
(a), (b) (c) We may now translate our previous results in terms of spherical functions. Let X be any unramified character of Recall that the function (/JK . x is defined by
(/J K , x(
(33)
F(I)
k
=
for m in
in N and k in K.
We put
Fx(
( 34)
=
(/J x ( for g in K K, kg) dk
THEOREM 4.2. (a) (b) X
x· Fx = W · X·
K
Fx·
It is clear that Fx is hi-invariant under K, and Fx( l ) = I . Theorem 4.2 follows from Corollary 4.2 and the formula w .x_(.f) =
(35)
J
c
Fx ( ) ·
This in turn is proved as follows :
s
G
F x( ) ( )
g f g dg
= = = =
g f(g) dg
for f in
Ye(G,
K).
s (/JK, ig ).f(g) dg JK J J (/JK , x (mnk)f(mnk) dk dm dn s x(m)o(m) 1 1 2 dm sN .f(mn) dn s x(m) S.f(m) dm = (f) G
M
N
M
w 'X
M
We used the integration relations ( 1 ) and (4). 4.4. DEFINITION 4.2.
The sphericalA prirepresent ncipal aseritionesofofGrepresent aspheri tions. cal i s called irreducible and contains a nonzero Gvector invariant under f G g ) cl > ..
.
K) if it is smooth,
(w.r.t. K. Let F be a spherical function on (w.r.t. K). We denote by Vr the space of func tions f on of the form (g) = 1: 7=1 c;F( g; for , cn in C and g l > . . · , gn in G. From the functional equation of the spherical functions (formula one de duces ·
(31)),
(36) dk J G (37) ion ant multiipslesspheriofr.cal and that the elements of Vr invariant under the represent are theatconst K
We let
.f(gkg')
= F(g ) · f(g ')
for f in Vr and g , g ' i n G.
operate on Vr by right translations, namely
I claim that
nr(K)
(nr, Vr)
Indeed , it is clear that for any
REPRESENTATIONS OF p-ADIC GROUPS
151
function/in Vr there exists a compact open subgroup K1 o f G such thatfis invariant under right translation by the elements of K1 ; hence the representation (11:r, Vr) is smooth. Let / "# 0 in Vr and choose an element g' in G such that f(g') "# The functional equation (36) may be rewritten as
0.
(38) Any vector subspace of Vr containing / and invariant under 11:r(G) contains there fore hence is identical to Vr. Finally, if a function in Vr is invariant under 11:r(K), one gets/ = f(I ) · Fby substituting g' I in the functional equation (36).
F, f = Let ion Fsuchbe tanyhat spheriicsalisomorphi representc taotion of There exists unique sphericalfunct 11:(/) simple : v v' = v, v' C= 11:(/) v. K) C. C C 11:(/)-v =w(f)v v v ii (ii, v) = = (ii, 11:(g)-v) f w( = F ) = cF(g)- f(g) F v' THEOREM 4.3.
(11:, V)
G. (11:r, Vr).
(11: , V)
·
a
As usual, let V K denote the subspace of V consisting of the vectors invariant under 11:(K). If/ is any function in .1t'(G, K), the operator takes VK into itself; hence VK is a module over .1t'(G, K). I claim that this module is indeed, let "# 0 and v' be two elements of VK. Since the £(G)-module V is simple, there exists a function f in .1t'(G) such that · The function fx lx * f •lx be longs to .1t'(G, K) and 11:(/K ) · substantiating our claim. The algebra .1t'(G, K) over the field is commutative and of countable dimen sion. By the reasoning used to prove Schur's lemma (see p. I I 8) (or by Hilbert's Zero Theorem), one concludes that any simple module over .1t'(G, is of dimension I over Hence VK is of dimension I over and there exists a unitary homomor phism w : .1t'(G, K) ---> such that =
(39)
for any f in .1t'(G, K) and any
in VK.
Let (it, V) be the representation of G contragredient to (11:, V). The space VK of vectors in V invariant under it(K) is dual to VK, hence of dimension 1 . Choose a vector in VK and a vector in VK such that I and define the function r on G by F(g)
(40)
for g in G.
From (39) and (40) one deduces dg for any fin .1t'(G, K). It is J obvious that F( l ) I and that is hi-invariant under K. Hence is a spherical function. The map which associates to any vector in V the coefficient 11:v ' , ii defines an iso morphism of (11:, V) with (11:r, Vr). Moreover, for any spherical function F' on G, the representation (11:, V) is isomorphic to (11:r ' • Vr, ) iff the following relation holds (4 I )
F' F
This holds for only. Q.E.D. The definition of spherical functions as well as the results obtained so far in this section depend only on the fact that K is a compact open subgroup of G and that the Heeke algebra .1t'(G, K) is commutative. We use now the classification of sphe rical functions on G afforded by Theorem 4.2. Let X be any unramified character of M; when the spherical function F is set equal to Fx , we write (11:x , Vx) instead of =
15 spherical principal series ofrepresentations of G. represent ationare the constis irareduci blipe,leadmi ssible, and the only functions in invariAny a nt under nt mult s o f Let areandisomorphi be unramifi ed exicharact erseloefment Thein represent agroup tions such thatand c iff t h ere s t s an t h e Weyl x' = w·x e represent antiI(x)on defineI t))heifunct n theiounram!fi ed principalf(serikg)esdk.is iThen rreducitAssume bhelmap e. ForfthatanyPthfunct i o n! i nP by P(g) with the repre sentatIniongeneral, let is an= isomorphism of the representV,ati=onI(x) I(x)) be a thatJordan-Holderr seriandesthatof tthhee .Yt'represent (G)-modulatioen I(x).of G There exi s t s a uni q ue i n dex such in is spherical. Then this representation is isomorphic to I(x)= I )I)( ) (w provided it is irreducible. 144). bounded G P. CARTIER
2
(nor, Vr). The family of representations {(n-x , Vx) } x ccx is called the
We summarize now the main properties of the spherical principal series ; they are immediate corollaries of the results obtained so far. (a) (n-x, Vx) Vx
(b)
(n-x , , Vx,)
X
x
n-x(K) '
Fx .
M.
(n-x , Vx) W
w
·
(c)
(x
().! x ,
=
,_.
(vx,
JK
(n-x, Vx).
(d)
0
V0
c
V1
c
·· ·
c
V,_1
c
j
Vj/ Vj- I
(n-x, Vx).
I �j �
The last two statements come from the fact that (j)K.x is, up to constant multiples, the unique function in invariant under vxCK) and from the relation (34) which can be expressed as rx (j)k x · REMARKS. ( 1 ) It is easy to show without recourse to Satake's Theorem 4. 1 that the representation (vx, x) is spherical Since then the representations ().!x, (x and (vw · x• l · x ) ) are equivalent for any w in W, this provides another proof of step (B) in Satake's theorem (see criterion 3 . 1 0, p. (2) We refer the reader to Macdonald [37, p. 63] for a characterization of the spherical functions, that is the spectrum of the Banach algebra of inte grable functions on which are hi-invariant under K. It does not seem to be known which spherical functions are positive-definite, or stated in other terms, which spherical representations are preunitary. The following result is due to Macdonald [36], [37].
4.5. The explicitformulafor the sphericalfunctions. he unramifi in the val4.u4e. ofSuppose the spherithatcaltfuncti onI eisd gicharact ven asefolr loofws: is regular. For any m = o(m) Iz w x) x(m) where q(w)-i, THEOREM
M-,
(42)
(43)
X
Fx
Fx( )
Q- I
I; c (
wEW
·
M
111
w ·
Q = I; wEW
a
forwhicanyh areregulposiartivunramifi er of (product extended over the roots in (j)0 e on theedconicharact cal chamber A
M
'6').
We sketch the proof given by Casselman [18], which rests on the properties of the intertwining operators. We use the notations of §3.7.
REPRESENTATIONS OF j.J-ADIC GROUPS
1 53
For each w in the linear form Lw on I(x) transforms by M according to the character o l 1 2(w · and is trivial on I(x)(N) Since X is regular, these characters of M are distinct ; hence the linear forms Lw are linearly independent. The vector space I(x ) 8 is supplementary to I(x )(N) in I(x) and its dimension is equal to Hence there exists i n I(x) B a basis Uw .x}w E W characterized by
x)W,
.
I WI .
LwUw• , z) = I if w' = w, = 0 if w' #:- w .
(45)
As a corollary of Theorem one gets WK x = .E wEW cJx) ·fw. ·;: On the other hand, one has Fz (m) = fK WK. z (km) dk. Sincefw. z is invariant under ).i z (B ) and B is a subgroup of one gets from these remarks the relation
3. 9
K,
(46)
.
where 1 4 gw = fJ.(BmB) - 1 n(I8m8) fw. z · By the methods used in Theorem 2. 3, one proves, in general, that, for any admissible representation (n, of G and any m in M, the operator n(lB mB) fJ.(BmB)n(m) maps into (N) From (45), one infers (47) gw = o (m) 1 1 2 w · x (m) w . •
V V V) .
-
f z From (46) and (47), one deduces that, on M-, the spherical function Fz agrees 2
with a linear combination of the characters o l l (w · ) of M. Taking into account the invariance property rw· x = rx , i t suffices t o calculate one o f these coefficients, for instance the coefficient of o 1 2(Wo · X ) where Wo is the (unique) element of which takes the conical chamber C(l into its opposite ((5- (or any positive root in r/)0 to a negative root) . In this case, one proves without difficulty that fwo. x is equal to the function r/)wo. x defined by formula (28) in § 3 . 6 . The sought-for coefficient is obtained by multiplying cwo<wo · x) by
x
1
sK fwo,
x
(k) dk
=
JK Ww0, x
(k) dk
=
W
f.J.(Bw0B) .
It remains to show that the measure of Bw0B, that is the index (Bw0B : is equal to Q- 1 . This follows easily from the formulas (2 1) to (24) in § 3 . 5 . Q.E.D. Note added i n proof As I was told by the editors, my conventions about alge braic groups differ slightly from those of other authors in these PROCEEDINGS. The local field F being infinite, the set G(F) of F-points of any of the algebraic groups G used in the previous paper is Zariski-dense in G and I allowed myself to identify G to G(F). " The Haar measure fJ. on G is normalized by fJ.
(K)
K),
=
1.
REFERENCES 1. I. N. Bernshtein , All reductive p-adic groups are tame, Functional Anal . Appl. 8 ( 1 974), 9 1 -93. 2. I . N. Bernshtein and A.V. Zhelevinskii, Represelltations of the group GL(n, F), Uspehi Mat. Nauk 31 (3) ( 1 976), 5-70 Russian Math . Surveys 31 (3)(1 976), 1 -68. 3. , Induced representations of the group GL(n) over a p-adic field, Functional Anal. Appl. 1 0 ( 1 976), 74-7 5 . 4. A. Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an lwahori subgroup, Invent. Math. 35 (1 976), 233-259. =
--
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P.
CARTIER
5. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-6 1 . 6. N. Bourbaki, Mesure de Haar ; Convolutions et representations, Integration, Chapitres 7, 8 , Actu. Sci . Indust. n o . 1 306, Hermann , Paris, 1 963. 7. F. Bruhat, Sur les represen tations induites des groupes de Lie, Bull. Soc. Math. France 84 ( 1 956), 97-205 . 8. -- , Sur les represenlalions des groupes classiques 'p-adiques. I, II, Amer. J. Math. 83 ( 1 96 1 ) , 3 2 1 -3 3 8 ; 343-368. 9. -- , Distributions sur un groupe localement compact et applications a ! 'etude des represenla tions des groupes 'p-adiques, Bull. Soc. Math. France 89 ( 1 9 6 1 ) , 43-7 5 . 1 0 . -- , Sur une classe de sous-groupes compacts maximaux des groupes de Chevalley s u r u n corps 'p-adique, I n s t . Hautes Etudes Sci . Publ . Math. 2 3 (1 964), 46-74. 1 1 . --- , 'p-adic groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math . , vol . 9, Amer. Math. Soc . , Providence, R.I., 1 966, pp. 63-70. 12. F. Bruhat and J. Tits, Groupes algebriques simples sur un corps local, Proc. Conference on Local Fields (Driebergen , 1 966), Springer-Verlag, Berlin, 1 967. 13. --- , Groupes reductifs sur un corps local. Chapitre I , Donnees radicie/les valuees, Inst. Hautes Etudes Sci . Publ. Math. 41 (1 972), 5-25 1 . 14. P. Cartier, Sur les representations des groupes reductifs 'p-adiques et leurs caracteres, Semi naire Bourbaki, 28e annee, 1 975/76, expose 47 1 , Lecture Notes i n Math . , vol . 567, Springer Verlag, Berlin, 1 977. 15. -- , Representations ofGL, o ver a local field, lectures at the I . H . E . S . , Spring term 1 979. 16. W. Casselman, An assortment of results on representations of GL , (k) , Modular Functions of One Variable, II. Lecture Notes in Math., vol. 349, Springer-Verlag, Berlin, 1 97 3 , pp. 1 -54. 17. --- , Introduction to the theory of admissible representations of 'p-adic reductive groups (to appear) . 18. --- , The unramified principal series of p-adic groups. I (to appear) . 19. J. Dixmier, Les C*-algebres et leurs representations, Gauthiers-Villars, Paris, 1 969. 20. F. Gantmacher, Canonical representation of automorphisms of a complex semisimple Lie group, Mat . Sb. 47 ( 1 939), 1 0 1 - 1 46. 2 1 . P . Gerardin, Construction de series discretes p-adiques, Lecture Notes i n Math . , vol . 462, Springer-Verlag, Berlin, 1 97 5 . 2 2 . --- , Cuspidal unramified series/or central simple algebras o ver local fields, these PROCEED INGS, part 1 , pp. 1 5 7- 1 69. 23. R . Godement, A theory of spherical functions. I, Trans. Amer. Math . Soc. 73 (1 952), 496-556. 24. J. Green, The characters of the finite general linear groups, Trans . Amer. Math . Soc. 80 ( 1 955), 402-447. 25. Barish-Chandra, Harmonic analysis on reductive P-adic groups, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos . Pure Math., vol . 26, Amer. Math. Soc . , Providence, R . I . , 1 973, p p . 1 67-1 92. 26. -- , The characters of reductive 'p-adic groups (to appear). 27. --- , The Plancherelformula for reductive 'p-adic groups (to appear) . 28. --- (Notes by G. van Dij k), Harmonic analysis on reductive 'p-adic groups, Lecture Notes in Math . , vol . 1 62, Springer-Verlag, Berlin, 1 970. 29. R . Howe, The Fourier transform and germs of characters (case of GL. o ver a p-adic field), Math. Ann. 208 ( 1 974), 305-322. 30. N. Iwahori , Generalized Tits systems (Bruhat decomposition) on 'p-adic semisimple Lie groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math. , vol. 9, Amer. Math. Soc., Providence, R . I . , 1 966, pp. 7 1 -8 3 . 31. N. Iwahori a n d H. Matsumoto, On some Bruhat decompositions and the structure of the Heeke ring of the 'p-adic Chevalley groups, Inst. Hautes Etudes Sci . Pub I. Math. 25 (1 965), 5-48. 32. H . Jacquet, Representations des groupes lineaires P -adiques, Theory of Group Representa tions and Harmonic Analysis, (C. I . M . E . , II Cicio, Montecatini terme, 1 970), Edizioni Cremonese, Roma, 1 97 1 , pp. 1 1 9-220. 33. --- , Sur les representations des groupes reductifs P-adiques, C. R. Acad. Sci . Paris Ser. A 280 ( 1 975), 1 27 1 - 1 272.
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34. H. Jacquet and R. P. Langlands, A utomorphic forms on G L(2), Lecture Notes in Math. , vol. 260, Springer-Verlag, Berlin, 1 972. 35. R. P. Langlands, Problems in the theory of automorphic forms, Lectures i n Modern Analysis and Applications. III, Lecture Notes in Math . , vol. 1 70, Springer-Verlag, Berlin, 1 970, pp. 1 8-61 . 36. I. G. Macdonald, Spherical /unctions on a p-adic Chevalley group, Bull. Amer. Math. Soc. 74 ( 1 968), 520-525. 37. -- , Spherical functions on a group of p-adic type, Ramanujan Institute, Univ. of Madras Pub! . , 1 97 1 . 38. I. Satake, Theory of spherical functions on reductive algebraic groups over P-adic fields, Inst. Hautes Etudes Sci . Pub! . Math. 1 8 ( 1 963), 1 -69. 39. G. Shimura, Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press, Princeton, N.J., 1 97 1 . 40. T. Shintani, On certain square integrable irreducible unitary representations of some p-adic linear groups, J. Math. Soc. Japan, 20 ( 1 968), 522-565. 41. T. A. Springer, Reductive groups, these PROCEEDINGS, part 1, pp. 3-27. 42. J. Tits, Reductive groups o ver local fields, these PROCEEDINGS, part 1, pp. 29-69. JNSTITUT DES HAUTES ETUDES SCIENTIFIQUES, 9 1 440 BURES SUR YVETTE, FRANCE
Proceedings of Symposia in Pure M athematics Vol. 33 ( 1 979), part I , pp. 1 57-1 69
CUSPIDAL UNRAMIFIED SERIES FOR CENTRAL SIMPLE ALGEBRAS OVER LOCAL FIELDS
PAUL GERARDIN
1. Statement of the theorem.
1 . 1 . Let A be a central simple algebra over a nonarchimedean local field F. Call
E an unramified splitting field for A, and r the Galois group of E over A quasi character of Ex is called regular if all its conjugates by the action of r are distinct. For any quasi-character X of Fx , let XA I F and XEI F be respectively the quasi characters of Ax and EX defined by : XA I F = x o NAI F and XEI F = x o NEI F• with NAI F the reduced norm of A over and NE 1 F the norm of E over F. For any nontrivial character ¢ of F, the characters rf;AI F = rf; o TAI F of A and ¢E1 F = rf; o TEl F of E are nontrivial : here, TA 1 F is the reduced trace of A over F, and TE 1 F the trace of E over F. Let be the rank of A , that is the degree of E over F. 1 .2. We have the following theorem :
F.
F,
n the above notastsiiobns,le forrepresent any reagtiulonar quasiof-charactsucher()that:of there is a welthle-derepresent finedWitihrreduci b l e admi ationsif and onlyandif0 andassoci aconjugate ted to twounder regulF;ar quasi-characters and 0tharee restequiricvtalent 0 are ion of () to the center of is given by the quasi-character anyto quasi-character of the twisted representation ® is equivalforthente cont ragredion oefnt() represent ation of () is equivalent to i s 1; tthhee L-t-ffauncti Moreover, whenctor oisfnot() a diisvision algebra, the representation() is cuspidal. THEOREM.
0
(b)
X E Fx
(c)
(d) (e) (f)
AI F
0AI F
(a)
I->
0
AI F ( - 1 ) Cn-l) val x ()(x) ; (() X EI F) A I F ;
A
()AI F
X
Fx
EX,
Ax
Ax
()A I F
FX ,
AI F AI F t( 8 AI F• cp )
AI F
=
XA I F
(()-1)A 1 F ;
( - 1 ) Cn- l ) ord ¢ c ( 8 , rf;EI F) . AI F
Here, cuspidal means that the coefficients of the representation are compactly supported modulo the center. 1 . 3 . Recall how the £-function and t-factor are defined [6]. The reduced trace A I F defines a nondegenerate bilinear form on A by its value on the product of two the mapping elements in hence, for any nontrivial additive character rf; on
T
A;
A MS (MOS) subject classifications ( 1 970) . Primary 22E50, 20G25.
F,
© 1979, American M athematical Society
1 57
158
PAUL GERARDIN
a, b e A
,_.
¢A1 p(ab) = cp (TA 1 p(ab)) ,
identifies A with its Pontryagin dual group ; the corresponding self-dual measure on A will be written dA .
Let .w'(A) be the set of classes of irreducible admissible representations of the multiplicative group A x of A . Godement and Jacquet have proved in [6] that the integrals SA x (x) n (x) dx, f e sP(A), dx any Haar measure on A , where n is a representation of A x with class in can be defined as meromorphic functions of s e from the integrals JA x f(x)c(x) l x i �1F dx where c is a coefficient of n, and the convergence holds for Re(s) large. They proved too that there is an eulerian factor L(n) and a factor <: (n, cp) which is an exponential in s, such that
.JQ'(A),
C
f
SA x /q, (a)n (a) -l laiA:/F l )/Z dA.q, a JAx f(a) n(a) laiA:/�+l )IZ dA,
1
---> E x
--->
WE I F ---> r --->
1
a quasi-character () of E x defines an n-dimensional representation of WE1 F by in duction ; but WE1 F is a quotient of Wp , so this gives an n-dimensional representa tion of Wp, denoted Indk 0. Moreover, this representation is irreducible if and only if all the transforms of () by F are distinct, that is for () being regular. In general, let F0 be the stabilizer of () in r, and E0 be the corresponding subfield of E; there
1 59
CUSPIDAL UNRAMIFIED SERIES
are = [E : £0] quasi-characters 0' of E0x such that 0 = OF:tEo , and each of them is regular with respect to F; so the representation Indk 0 is the direct sum of the n0 representations Indf0 0', each one being irreducible. This gives part (a) of the fol lowing proposition. For part (e), we use the property of inductivity of the £..func tion ; for part (f), there is only a degree zero inductivity for the e-factor, and the factor ).(E/F, ¢) is computed from the identity e(IndfO, ¢) = ).(E/F, cp)e(O, ¢E 1 F) by taking for 0 the trivial character, so that Indfl is the regular representation of F, that is the sum of the n unramified characters �-+ 'val x where ' are the n different nth roots of 1 ; so :
n0
IT e('val , ¢) c
=
x
).(E/F, ¢) e(l , ¢E l F) ;
but e('val , ¢) = (' q l iZ)ord ¢ and e(l , ¢E I F) = n ord ¢12 , if q is the order of the residue field F of F. This gives the formula }.(E/F, ¢) = ( - l)
q represent ations of induced by the quasi-characters of satisfyththeerepresent followinThegatpropert i e s: ionsy if andand0 are conjuigate nducedunderby tr;wo quasi-characters and 0 are equithvealent if and onl one-dimensional representation is given by the quasi-character: -character of the representation twisted by is equival(c)for ent tthoeanycontquasi tthhee L-fe-faurctornctiagredion ent representis tahteioL-fn ofunction is equivalent to r, r a WF
PROPOSITION 1 .
(b)
1J
X E px
(d) (e) (f)
IndfO
IndfO
(a)
1-+
( - l)
Indf(O XE I F) ;
Ex
0
det IndkO
X
px ,
X
IndfO
IndH0- 1 ) ;
Indff)
L(IndfO) L(O) ; e (IndfO, ¢) is ( - t )
3. Construction o f the representation. 3 . 1 . For any quasi-character 0 of E X , we have the n characters Ojf) r , r E with the Galois group Gal(E/F). They are trivial on P . Let (O , r) = a(r) be the con ductor of OjO r ; it is a nonnegative integer, and the following relations hold : a(r) = a(r-l ) for any T E F, a(rr') � Max ( (r) . a(r')) , for any r. r' E r, equality for a(r) "# ( r ' ) , (r) � 1 for any r =f. 1 , if and only if f) is regular. For each integer � 0, let be the subgroup of formed by the elements r with a(r) � and E, be its fixed points in E ; call A, the algebra End E, E; we have the inclusions :
a
a
r, r
F
F,
E
c
A0
c
a
A
c
A2
1 r s r s: TA,tA, :
c
· · · A,
c
· · · A.
Observe that A, = A . for � � Max r, a(r). The algebra A, can be seen too as the algebra over E of the group r, ; this leads to A.-linear projections for � A , --+ A. and decompositions in direct sum of A.-submodules : A, = A s + A: = l:: r,ti', rAs where A: = l:: r�. As = Ker due to the partition of r, in r. and its complement, written r:, more over,
TA,IA,•
TA,tA,(xax-1) x(TA,tA, a)x-1 , x a r s, =
for
E
A;,
E
A"
�
the group A;< acts by conjugation on A , and leaves invariant each term of this decomposition.
1 60
PAUL GERARDIN
The natural filtration on E by its ideals Pl induces filtrations on each of the algebras A , : the lattice A,(pm) is the set of A, such that pl (!}8, the ring of integers in it is too the set of functions from r, to !Jl. As the extension E is unramified over F, these filtrations are compatible with the maps The residue field of E, is written E., and A, = A,((!}8)/A,(p8) is the algebra Ende, E. The finite group A: acts by conjugacy on each £,-algebra A ,(pm)JA ,(pm+l) A ,(pmJpm+ 1 ), and leaves invariant each of the subspaces rA.(pmJpm+l), r r,;r., for � 3.2. Assume now that (} is a regular quasi-character of E x , i.e., that Ao is E. For each � 0, define a subgroup K, of A; as follows ; we denote by r ' (resp. the largest (resp. the smallest) integer such that 2 ' � � then K0 px , K1 is the isotropy subgroup in A{ of the lattice in and for r > K, = l + A,_1(p•") + A�-1(p•'). For each � 1 , the set K: = + p£ + A�(p•") is an in variant subgroup of K,, normalized by each of the groups K., � r, moreover, by conjugation K, acts on K., and sends K, in K: for r > � 2. The factor groups K,/Kt are the following :
aE
£;
a E TA,tA,·
=
r s. r
(!}8r £,r 1 2r"; 1, s s
r
A f, Kt!Kt K,/Kt for > l is the group ( l + Pk)/( 1 + A, :;6 A ,_1 where it is the central extension : =
r
p£)
� 'Pk/P£ , unless
r
=
E r")
is odd and
0 ---- Pk/P£ ---- K,/Kt ---- A �-1 (p•"ft;J") ---- 0 ,
defined by the 2-cocycle
TA,;E(ab) E PF1/p£. r O,x'£J8,. r r (} £,_1 PF1 x) f/l'i18,jx) x E p 1 £ f/l'iJ8,_1( TA,tEab) r', r" a, bE Forl on eachand such thattheretheis resta uniricqtueioncltoass of irreduciis isbotleypirepresent aby(},; tions ,;de�fiofLEMMA t r i v i a c , v en gi a representonlyaton(}, ion andofis.fixbyed by!C,(txhe) actix5fon:toEf,(anyx),;;(x),s Er. then, the classneofth,;en, depends r" r r a, bE
A �-1(p •'j p•")
�--+
3.3. For each � l , choose a character (}, of Ex with conductor equal to Max7 e r,a(r), such that (}j(}, is fixed by the group r, : this means that there is a For l , the character (} 1 defines a quasi-character x<• l of E,x with (} character of E" regular with respect to .£1 • For > 1 , the restriction of , to l + such that is fixed by Fr - h so there is a character fjJ <•J of =
=
0 ,( 1 +
£- /p
for
=
;
moreover, the mapping defined by A �-1 ( p'' /p'") �--+
for
as in 3 .2,
is a self-duality ; this gives the following result : K,
l.
Kt
r >
l,
A:,
E x n K,
K,
=
K,
X
K, ;
<
For even or with A, A, - I > this is clear. In the remaining cases, we use r ' and from as above ; we introduce the subgroup H(r) of K,/Kt defined by the central extension by p£- 1/p£ of A �-1(p• 'Jp•") given by the same cocycle ; it is a Heisen berg group, and if K(r) is its pull-back in K., then 1 + Pk' normalizes K(r) and we have K, = (1 + Pk)K(r) . The first part of the lemma comes from a well-known property of Heisenberg groups ; for the second, we observe that the restriction of =
CUSPIDAL UNRAMIFIED SERIES
161
� to K(r) factors through any group H5(r ) , 1 � s < r, central extension by !JE:-;-1 /PE:, of A�-1 (p''/p'") obtained from H(r) by the map TE!E, on its center :
11:
0 --> p�;;J PE:, --> H, (r) --> A �- 1 (p''/p'") --> 0
defined by the 2-cocycle a, b E A �-1 (p''/p'")
>-+
TA,!E,TA,!A,(ab) E PE:-;-1/ PE:,
c
p£';/ !JE:, ;
it suffices now to observe that the action of A; by conjugation on A, fixes the form u E A, >-+ TA,IE,TA,IAP · 3 .4. More precisely, for r > s > 1 , the action of on gives a trivial action on H.(r) . For s = 1 , the group H1(r) is a central product of the group's H1(r) inverse images of the subspaces rA1(p''/p'") + r- 1 A1(p''/p'"), the group H1(r) is a cen tral extension of this subspace by lJk/lJE: 1, the 2-cocycle being given by a, b E rAl( P'' /p'") ...... TA 11E1(ab) for r rl = r- 1 rh a, b E rAl(lJ''fp'") + r- 1 Al(J.1''/p'") >-+ 1;E1(a'b" + a"b')
K. K, TA
, with a = a' + a", b = b' + b", a', b' E
rA1(p''/p"'), a", b" E r- 1 Al( P''/p'"), in general ; the action of K1 on K, gives an action of KdKt = A [ on each of these subgroups H1(r) ; for rF1 = r- 1 rh the above 2-cocycle is a unitary form on the £1 -vector space rA1(p'' f p'") with respect to the field E� of fixed points of r in E1 o and A 1 preserves this form ; from [5, Corollary 4. 8 .2] , there is a canonical extension to A[ >4 H1(r) of any irreducible representation ��:; of H1(r) which is given on the cen ter b� c/J'k{IE r - 1 ; for a r E r;-1 with rr1 '# r- 1 rh the group A [ leaves stable each of the A1 -submodules of the quotient of H1(r) ; hence any irreducible representation 11:7 of this group which is given on lJE:-;-1/pE;1 by ¢)!{1E , _ 1 has a canonical extension to the group A1 � H1(r) ([5, Proposition 1 .4], in fact, the character of the restriction to A [ is positive). The product of the representations 11:� defines a representation 11: � of as in the lemma, and the product of the extensions defines an extension of 11: � to the group Now, the representation of this group obtained from by twisting with x < r l as in the lemma is, up to equivalence, independent of the choice of cp < r l . For r = 1 , there is a unique class of irreducible representations 11:� of K1 which is trivial on and such that its tensor product with the pull-back of the Steinberg representation of A [ is the representation induced by the one-dimensional repre santation tx >-+ {} 1 (t), t E it is the representation x E Kt, of the subgroup RE1(01) of [2, Theorem 8 . 3], up to a sign ( - l)n 1-\ n1 = [E : Ed, and it is a cuspidal representation. Define then ��:1 as the twist of 11:� by x m :
i�i; K,
K1K,.
Kt
i,
@'i;Kt:
@�,
its class does not depend on the choice of x m . Finally, the representation ��:0 of = Fx is the restriction of {} to
Fx . PROPOSITION 2. A regular quasi-character {} of Ex defines canonically a class of ir reducible representations of by the formula Ko(xox1 · · · x, · · · ) = Ko(xo) ® 11:1(x1) ® · ® i rCx 1 )��: , (x,) ® · · ·
K8
K0
· ·
PAUL GERARDIN
1 62
for x, in Kn where K , is defined as abo ve, ii: , is its canonical extension to K1 for r odd > 1 with A, of. A ,_J, and if, is the trivial representation for the other r > 1 .
4. Commuting algebra.
4. 1 . Let 'JCo be the representation of A x induced by the representation teo of K0 in functions on Ax with compact support modulo the center px . The commuting al gebra is identified with the set of applications from Ax to the space of teo which transform on both sides under the action of K0 according to the representation te0 . For a quasi-character (} of E x such that any character (}j(}r, for r E r, r #- I ' has conductor I , the representation teo is obtained by twisting from a cuspidal ir reducible representation te ' of A. x ; if T is a maximal diagonalizable subgroup of A x such that A = K0 TK0, then a function ¢ in the commuting algebra i s determined by its values on T ; for a t E T not in K0, there exists a unipotent radical U of a proper parabolic subgroup of A x such that t(U n K)t-l is contained in the kernel of K0 --> A x ; we have then, for u E n K: rp(tu) = Ko ( t )rp(u ) = rp(tut -1t) = rp(t) ; hence, from the cusp property of te ' , ¢(t) 0. This shows that the commuting al gebra is reduced to the scalars, and .,.8 is irreducible ; its coefficients being com pactly supported modulo the center, the representation .,.8 is cuspidal too. Let now r be such that r = M axr ,a( r) > 1 . Let ¢ be a nonzero function in the commuting algebra of the representation 'JCo ; choose a in A x with ¢(a) of. 0. Sup pose we know that a E K0a,K0 for some a, in A;' ; as the values of ¢ are determined by its values on a set of representatives modulo K0 on both sides, we replace a by a,. From the definition of te0, its restriction to the subgroup 1 + A ,(p'") of K, (r " is the smallest integer with 2r " ;;;;: r ) , is a scalar operator, defined by the restriction of K n that is
4.2.
U
=
4. 3 .
1 + X E 1 + A ,(p'")
I-+
x5r/;E,( l + x) (} ,( l + TA,!Ex) .
As ¢(a) is nonzero, we have, for X E A,(p'") n a-1A,(p'") a, the identity : (} ,( 1 + TA,!Ex)
=
(} , ( 1 + TA,;E (axa- 1) ) .
Choose a character a of E such that a( t) = 8 , (1 + t ) for t E pjf' ; then the character of A, defined by x �---+ aA,IE (axa- 1 )aA ,;E (x) - 1 is trivial on the intersection a-1 A ,(p'")a n A,(p'") so it can be written as IJ(axa-1 )- 1 �(x) with two characters � and 1J of A , trivial on A,(p'"), so that aA,IE (x) � (x) = aA ,;E (axa- 1 )1J(axa-1 )
for any x E A , .
4.4. Fi x a character ¢ of E, with order - r ; the restriction to PE- l of the character a is trivial on PE, and its stabilizer in r, is r,_ 1 ; so, there is an element -r in the multiplicative pull-back of £x in Ex such that E,(-r) = £,_ 1 and a(t) = ¢E;E, (-r t) for t in p£1 . LEMMA 2 . With these notations, then
(a) any character of A, equal to aA,IE , on A,(p'") has a conjugate by an element of
K, which is equal to
x �---+
¢A,;E, (rx) on A ,_ 1 (p'") + A�-1 ;
(b) if an element of A;:' conjugates two characters of A , equal to x on A,_ 1 (p• ") + A�-1 , then this element belongs to A;:'_ 1 .
�---+
aA,IE, (-rx)
The proof uses the identification of A, with its group of characters obtained from
CUSPIDAL UNRAMIFIED SERIES
I63
is the conjugation. the application a, b E A , ,__. a A tE (ab) , where the action o , , Part (a) is easy, and for part (b), we have to show that an element b E such that b(1: + X) b- 1 = 1: + X' with X, Y in , 1 (l.Y' ) , r ' + r " = r, is in fact in as X and Y centralize 1:, taking the successive powers qn m of both sides, we get at the limit m infinite, the equality b 1: b- 1 = 1:, so b centralizes the center of hence is itself in A;_1 • We apply this lemma to the above situation : the characters of a Ar!E r ' and aA tE r;, are conjugate by suitable elements from K, to characters equal to x ..... , , part (b) of ¢ A ,td7: x) on A,_1 ( p'") + A;-1 ; as they were conjugate by a E Lemma 2 shows that a belongs to K,A;_1 K,. As for r large, = A , we have shown that the relation ¢(a) #- 0 implies a E KAtK. If A1 = E, then a is in K0, so ¢(a) = Ko (a) ¢ ( I ) , and ¢ ( 1 ) is an intertwin ing operator on the representation Ko ; the irreducibitity of Kn implies that this operator is scalar, and the representation is irreducible, and cuspidal for its coef ficients are compactly supported modulo the center. Suppose now that A1 is not £. Let a be an element of At which is not in K0, i.e. not in K1 Fx .Then, there is a unipotent radical U of a proper parabolic subgroup of such that a(U n K1)a- 1 is in Kt, and even in the kernel of any "' for any r � I , for the determinant i s trivial on any unipotent element. For each r E F with a(r) r, an odd number > 1 , the group U n K1 acts on the£1 -vector space r A1(�r' / p'") + r- 1 A 1 (p''f p'") ; its fixed points form a subspace which can be lifted as a commuta tive subgroup Wr of H1 (r) (notations as in and there is a subgroup Wr of unipotent elements in K, with projection Wr on H(r). For r even, or without r such that a (r) = r , define W, = I ; for r = I, put W1 = U n K1 ; for the other r ' s , W, is the product of Wr for r with a(r) = r, taken in a chosen order, and without repetition. Then, for any r, a w, a- 1 is in K;; and made of unipotent elements so is in the kernel of "o · For x = w 1 • · · w, · · · , w, E W, we have :
A, A: A:_1: £,_1 A,-h A, A;,
A_
4. 5 .
A,
A{ =
3.4),
From the next lemma, it results that the operators Jw, K,(w,) dw, are fixed by
K,(w 1 ), so that the average
is equal to
Sw r
..
W,
··
K 1( w1 ) ®
·· ·
® IC ,(wi)K, ( w,) ®
·· ·
dw1
•..
dw, . . .
which is from the cusp property of K 1 . The lemma is the following, with the same notations as in [5] :
0 3. Let be afinite dimensional vector space) nontrivial overfinite field, and be an irreducible representation of the Heisenberg group on the center; let U be the unipotent radical of a proper parabolic subgroup of G( and W the space of its fixed points in then W has a lift in as a subgroup, and the LEMMA
(a)
r;
V
H( V
V
x
V* ;
V),
H( V)
operator L: w r;(w) isfixed by the action of U through the Wei/ representation wv ; (b) let F be a K/k-unitary vector space over a finite filed, and r; be an irreducible
164 represent at tUionbeofthethunie associ atradi ed Heicalsofenberg groupparaboli H(F, ci)subgroup which is ofnontriU (tvhiealuniontathery cent e r; l e p ot e nt a proper group i)),asanda subgroup, W the corres pthondie operat ng tootrally is7Jot(rw)opiiscfixed subspaceunderin tF;he actthenionWofhasU athlrough ift inU(F,H(V) and the Wei/ representation W
I: w
The proof is easy for (a), and for (b) it suffices to use a realisation of 7J as func tions on W with values in the space of the representation associated to the Heisen berg group Finally, we have proved the following result : PROPOSITION 3 .
Ex.
Ko
n8
Ax.
5. Intertwining operators. 5. 1 . Let now fJ and be two regular quasi-characters of E x ; we denote with one (resp. two) dot(s) every object attached to fJ (resp. ic = n�. i< = K�, i = Kfh . . ; when the object is the same for () and we take off the dots. The intertwining operators between the two representations ic and if are identified with the applica tions ifJ from A x to the vector space of linear maps between the space of i and the space of ;e satisfying the relation : = i ) ( ( , E i<, y K. 5 . 2. When there is a r such that = fJr , then for any the groups f, and i", are the same ; we can take and = ()� ; moreover, for > I , the represen = tation i� and its extension i� have the required properties for the corresponding representations for so are equivalent to and i, respectively. The representa tations i1 and of K1 are trivial on Kt and their pull-back with a Steinberg repre sentation of Af is equivalent to the representation induced from l!J� Kt by the one dimensional representation = for l!JE , Kt, so they too are equivalent ; finally, on £X , fJ is so iio is i0 ; this shows that the representations ;e and iT of i< = K = K are equivalent, so too are the representations if and fer ; but the operator ic(r) sends the representation ic on fe r , so if and 7t are equivalent. We have proved that tWO regular quasi-characterS Of Which are COnjugate Under r give equivalent representations of A x . 5. 3 . Suppose now that there exists a nonzero intertwining map ifJ between 7t and if; choose E Ax such that =F 0. Consider the following property for a non negative integer for any � F1 = F1 and there i s r e r such that a e i< A � r k. We observe that implies i< = k, and that ifJ(r) intertwines ;e and iT so that = fJ on Fx ; on 1 + PE • the restriction of i8 is the scalar operator defined by the restriction of () to 1 + PE • so and {Jr coincide on 1 + PE ; this gives the equi valence between each of the representations ;e, and i� of K, for > 1 , and of the ex tensions �' and i� to K1 ; the isomorphism ¢(r) intertwines the restrictions of ier and ir to K1 ; as the character of �' restricted to K1 does not vanish, this implies that K t and ii are equivalent, having the same character as x {l) = { l ) on 1 + PE · this implies that the corresponding representations of Af are equivalent, so and are conjugate by rl : O t = Tl E rl ; as rl acts trivially on £X and on the restric tion of to 1 + PE• we obtain that = (Jrn : the quasi-characters () and are con-
rp(xay) (x rp a)ie y) r,x e 0 r x
a r: rp(a) P(r): P(O)t r, 0 0 0
O{Ti, 0
r x
0
01 OI
165
CUSPIDAL UNRAMIFIED SERIES
jugate under r. We note that ( ) is satisfied for large ; moreover ( ) is S � is any integer larger than all the conductors a(r) and ii(r) for 7 F,.
( ) if
Pr r PE r P s With these notations, we have the implication, for r P r) Pr a)P(r). r",r s", s KaK, a2r" r 2s" s. a) a a L r", s" a )a, r
5.4. LEMMA 4. ( - 1).
� 1:
(
=>
We may assume that Maxr, a(r) � = Maxr, ii(r). Suppose first that is not 1. As if>( is determined by the double class we may take in A; r, from the property Let resp. be the smallest integer such that � r, resp. � We keep the notations as in 3 . 3 . These integers are larger than 1 ; the nonvanishing of if>( implies that, on the lattice A,(JJ'") n -1 A ,( p'" the two scalar operators iC(x) and i ( x -1) are equal : =
=
0,( 1 + TA,IE x) x5r}1E,( l + x)
=
0 , ( 1 + TA,IE axa-1)x5r}IE, ( 1 + x) ;
the map X ...... x 5r}IE, ( 1 + x) x5r}IE, ( 1 + x)-l is a character, trivial on A,(p') n -1 A,( p ' ) ::::> · so this character factors through TA IE as a character ' of C EIE,(t)0, ( 1 + En trivial on P£, . Let now a be a character of E such that a ) for pi{, and let ii be a character o f E such that ii(t) for p';. 0,( 1 + The two characters iiA,IE and x ...... iiA,IE (axa-1) of A , coincide on so there are two characters � and � of such that � is trivial on A,(p'") and � trivial on A,(p'") satisfying the identity : ii A,IE(x) � (x) = aA,IE(axa- ! ) �(axa-1) for all X
a
tE
a E LL L,
t) t) t E L, X E A,. EE,. _ ' {t
=
=
A,
X 1 .E 2, )baxa-!b- ) , ,�1 bE K" X a b a-I)bab-1 qnm r 2 c E K, ) c c ),£ X A,_1(p'') a-Ib-l(f )ba + c ;_1 qn m , a b £ ba c c; 1 bac1 bac !rA,_1 . r r; a! K x rA[ a! E a E Ka!r 1 W K a r Wr-! a ! 1 fw iC(w) dw rp a r K a KFK: P(O), F, define ethunder e samethcle agroup ss of ir reducible representaTwotionsregul of Axar ifquasiand-charact only ifetrsheyofare conjugat
Applying now part (b) of Lemma the right-hand side can be written A , (p' ) , and f in the for some with ¢A IE ((t + 1 m �It{plicative pull-back of in E, primitive over The left-hand side is ¢ A IE (( T + X)x) for some T in p£' and X in A,(p'-'"). We have the relation : = T + X as the sequence of - I -\ t + powers of the left-hand side has a limit, xba, we must have = s; so, by part (b) of Lemma again, there is a such that =
¢A,IE,((f + X x -1 with X in and f in the multiplicative pull-back of in E, primitive over E,. The equality + = cl(f X ) gives by taking the succes = -1 f sive powers the relation - I - 1 t this proves that t and f generate the same subfield of E, hence that ,_ 1 = E,_ � o and that the conjugation by is an automorphism of Er - ! ; as it is F-linear, it is given by a r E r; finally fixing a primitive element of E,_ 1 over E, belongs to This proves the lemma for ¢A,IE, (( T + X)x)
>
I.
we may suppose that Suppose now = 1 ; we know that /', = f, for any r! ¥- 1 . Let be such that K. As in § 4.5 if ¢ ! F ' there is a subgroup c is contained in the kernel of i, and with such that = 0. But this implies that ( and in this 1 ) = 0, so a 1 is in is for we know already that the groups are the same. We have proved the following result. PROPOSITION 4.
Ex
r.
PAUL GERARDIN
I 66
6. Computation of the c-factors. 6. 1 . The £-functions of the representations 'll:o defined by regular quasi-charac ters 8 of Ex are identically I [6, Proposition 5. I I ] . If we apply then the formula of I .3 giving the c-factor of 'll:o to the subspace of type !Co, we get the identity :
J
�
I¢(X)K o (x)-1 j x J :A)F1 l 12 dA . ¢ x = c('ll:o , cjJ)
J
q
f (x)Ko (x) J x J:A)�Hl /2 dA ,
for the irreducibility of 'll:o ; representation induced by /Co means too that Ko occurs with multiplicity one in the restriction of 'll: o to K0. Taking in this formula for f the characteristic function of the subgroup I + A(pa) with a any integer larger than the conductor of (), the representation /Co is trivial on this group, and the formula gives c (7T:0 ,
cp) =
J
Ke
cjJ AI F(x) Ko (x) -1 jxj:A}�-llt2 dA . ¢ x,
scalar operator,
the meaning of the integral being by principal value for l x i A 1 F large, and by analytic continuation for l x i AI F small. 6.2. Let 8 be a regular character of Ex with conductor 1 . Then the representation K o of K0 comes from a representation ,. of K0/K+ = A. x , so we have : c('ll:o , cp) = I: + KetK
(J
K+
,,
'f' AI F(x ,
u) dA . ¢ u)IC0(x)-1 J x J AI (n+ll F /2 ·
The measure dA . ¢ gives to A((!)) the volume q-n2ord ¢12 , and K+ = 1 + A(p) , so the inner integral is 0 unless x belongs to p-ord ¢ - 1A((!)) where it is the measure q- n2 (Ho rd ¢/2) of K + . Our expression is now : c('ll: o , cp) = qn ord ¢12q- N I: cp A I F (7T:-1-ord ¢a)8(7T:-1-ord ¢a)-1 , )! X
with = n (n - 1 )/2 and a a prime element in E. The cusp property o f 'll:o implies that this sum is in fact on the semisimple elements of A_x where the character of ,. is known (3 . 5) ; the computations have been made by Springer [11], and give
N
c('ll:o , cp) = qn o rd ¢12( - 1 ) n- 1 2: c/JE! F(7T:-1-ord ¢ t)8(7T: -1-ord ¢ t) -1 ft X
6.3. Suppose now that 8 is a regular quasi-character of E x with even conduc tor 2m : the integral over K0 giving c (7T:0, cjJ) can be split with respect to the subgroup K ' = 1 + A(pm) : c('ll: o , cp) = I:
Ke!K'
(J
K'
c/JAI F(xy)Ko (Y)- 1 dA. ¢ Y)K o (x) - 1 jxJ5ri}1l 12 ;
choose an element ()
+
t) = c/JEI F(8¢t),
t E p� ;
then, from the definition of Ko, we see that the restriction of /Co to K' is given by the scalar
I 67
CUSPIDAL UNRAMIFIED SERIES
y = I +
UE
I + A(pm)
1-+
O(I + TA I Eu)
=
¢A I F(Oq,u) .
The inner integral above is the product by cp(x) o f the integral over A(pm) of the character ¢A1 p ( (x - Oq,)u) for the measure dA.
=
q-n' (m+ord ifi/2) J O q, J�tl) /2 cfJEI p(Oq,)O(Oq,)-1
= J Oq, J1tlF ¢E I p(Oq,)O(Oq,)-1
=
= e(no . cpE I p) .
JE X ¢EI p(t)O(t)-1 dE.¢ t
6.4. Take now for () a regular quasi-character of E x with conductor m, an odd integer greater than I ; call m' and m " the integers such that m " = m' + I and 2m' < m < 2m " . Choose an element Oq, in . E such that 8(1 + t) = ¢E1 p(Oq, t ) for t e �" ; the restriction of () to the subgroup I + p'Jf defines a second degree character of P'lf such that O(I + t) O ( I + t ') = () (I + t + t')cp E i p(Oq, tt ' ) . The application t e p'Jf ...... cp(Oq, t) 8 (I + t)-1 is trivial on p�", and the e-factor of the quasi-character 8 on Ex is
JE X cpEI p(t )O(t)-1 dE. ¢ t = J Oq, ll<'lF cfJEI p(Oq,)O(Oq,)- 1 I"
e (O , cpE I p) =
12
m'
m'
PE I�E
cpE/ p(Oq, t)O (I + t) -1,
where J A for a finite set A means (Card A)1 !: A The integral giving e (11:0, ¢) is split now with respect t o the subgroup K " A(pm") :
=
I +
call K' the subgroup I + A(pm' ) ; the inner integral is 0 unless x belongs to Oq,K', so the formula can be written :
But K /K is isomorphic to A(pm'/pmn) = p'e'/p'e" + A0(pm'fpmn) , with the kernel A0 of TA I E • so the expression splits in two terms and gives, using the formula for e(O , cpE/ p) : '
"
Call r the maximum of all the conductors a(r) of O ( O r for r ¥: 1 . We examine the situation in two cases, according to the respective places of m and r, the conductor of 8 . (a) Suppose first m > r . But 8 = (J < r l x'i)F as in 3 . 3. The restriction of Ko to K' is a scalar, given by
PAUL GERARDIN
18 6
1
+ x e K' =
I + A(llm')
�--+
8,(1
+ TAl Ex)x XM I + x) .
For X in A0(�') it is only x5rlF (I + x) = x (I + O'z (x) ), where O'z (x) is the trace of the action of x on A 2E, as an F-vector space. But now the application x ...... x ( I + a2 (x)) is a nondegenerate second degree cha_racter on the I )-dimen sional vector space A0(�' fllm") over E, so its sum J A• (pm'/pm"l x (I + a2(x)) is ( - I )n-1 ; this gives
(n -
e(11:o, rp) = ( - I )n -1 e(8, r/JEI
F)8(8rp)Ko(8rp)-l ;
8
O(Orp r )Ko(Orp8, )-1
from the definition of the representation we know that the operator depends only on the class of modulo I + lJE ; but the condition m > means that this class is fixed by F, so there is a representative in Fx ; but on Fx , " is so We have proved the formula : e(11:8, ¢) = ( e(O, ¢E1 F). &0 ( ) = (b) Suppose now that m = put = = m " . The group K' is contained in K;_1 K, (notations as in From the decomposition A0 = A�_1 + Ar- 1 defined by TA.1A,_ 1 , we get, for X E A 0 ( lJ' '), y E A�_1 (lJ''), z E A r -l(p''), x = y +
-1)n-1
z:
® x 5r;=.I[,Er - 1 ( I + y)&,( l + z) . ...... x5r;=.1{1E , _ p + y) = x ( I + a!i- 1 (y)) is a nondegenerate + x) =
As in (a) the map y second degree character on the vector space A�_1(p''/p'") over E,_1 and we have :
J,.
1 XX��� r 1 Er - 1 ( I + y)- = - (
-n 1 8(8rp ) K8(8rp ) �>o 2). e(11:8,i 8 J,. �>,(1 z)- 1 i,(8rp)- 1 . Orp( I 8rpz) 8rp:z 1-+ 8rpz8¢/ - H(r) i,(8rp)-1 , n F). er ofor ofand8, webe have: the representation constructed in8 §3.LetThen,8 beifa regul a(8) aisr charact the conduct A ' r - 1 (pr'/pr") _
I ) r- 1
where n,_1 is the dimension of A,_1 over Er - h that is the order of r,_1. The element 8rp of E is fixed modulo I + l'E by r,_h so the operator is simply I ® ,( rp) - 1 , by definition of the representation (Proposition This means that ¢) differs from e(8,¢ E l F) only by the scalar defined by the operator : Ar - Ipr'/pr•
+
In the semidirect product � EX , the image of + is conjugate to the image of this comes from the fact that if r E r fixes modulo I + PE then 1 r E r,_h so z is an automorphism of the £-vector space A r - (ll'' /p'"). To compute the above scalar, we compute its trace, which is the trace of equal to ( - I ) n r - 1 ([4, Theorem I , p. I 6], or [5]). This shows that in case (b), we have the formula : e(11:8 , ¢) = ( - I)n -1 e(8, ¢E1 6 . 5 . The discussion in the preceding section gave the following result : -
PROPOSITION 4.
( - I )
-
11:8
Ex
e (11:8 , ¢) =
l)a (8l e ( , ifJEI F).
This formula can be written
e(11:o , ¢) = ( - I)
r/JE I
e(8, F). (
Define a quasi-character 0 from by the formula 0 = - l )
8
F:
1 69
CUSPIDAL UNRAMIFIED SERIES
DEFINITION. The representation () AI F associated to the regular quasi-character () of E x is 'lrih where n0 has been defined in §3. Then, from Propositions 1 , 2 , 3, we get the theorem for the split algebra A = EndFE. 7. Remark. Keep the notations as above. For a nonregular quasi-character () of E x , the field Eo is strictly contained in E, and there are n0 = [E : E0] quasi-characters ()' of E0x such that () = ()�!Eo• and they are regular with respect to F. From the theorem, they define irreducible admissible representations ()� . IF of the multiplica tive group of the algebra A' = EndFE0. The direct product of n0 copies of A ' x is F-isomorphic to an Levi subgroup of an F-parabolic subgroup of Ax, and by Theo rem 3 of [1 ], the representation () AI F of A x induced by the representation of defined by the product of the representations ()�,IF for all choices of ()', is irredu cible ; by Theorem 3.4 of [6], this representation () AI F has its L-function and .;-factor given as products of the L-functions and .;-factors of the representations ()� ' IF · From the properties of inductivity and the theorem, we have the formulas, with ' n the rank n fn0 of A' :
P
P
IT L(() � ,IF) = IT L(() �o!F) = L( 8�;E0) = L(()), c;((} AI F• cj;) = Il c;((} �,IF• cj;) = Il( - 1 ) Cn ' �l) ord ¢ .; ((} ', ¢Eo/F) - l ) no (n ' �l) ord ¢ .; (()�!Eo • ¢E I F) ).(E/Eo, ¢Eo/F) = ( - l ) no (n '�l)ord ¢+ Cno�l)ord ¢ .;(() , cj;E I F) = ( n� = ( - 1 ) ( l) ord ¢ .;( (} , cj;E I F) . L(() AI F)
=
The other properties of functoriality given in the theorem for a regular quasi character () are still true here. REFERENCES 1. I . N. Bernstein and A.V. Zelevinskii, Induced representations of the group GL(n) over a P-adic field, Functional Anal . Appl . 10 ( 1 976) . 2. P. Deligne and G. Lusztig, Representations of reductive groups o ver finite fields, Ann. of Math. (2) 103 (1 976), 1 03-1 61 . 3. P. Gerardin, On cuspidal representations of P-adic reductive groups, Bull. Amer. Math. Soc. 81 (1 975), 756-758. 4. --, Construction de series discretes P-adiques, Lecture Notes in Math., vol. 462, Springer, Berlin, 1 975. 5. -- , Wei! representations associated to finite fields, J. Algebra 46 ( 1 977), 54-1 01 . 6. R. Godement and H. Jacquet, Zeta functions of simple algebras, Lecture Notes in Math., vol . 260, Springer, Berlin, 1 970. 7. R. Howe, Tamely ramified supercuspidal representations of GL. , preprint. 8. -- , Tamely ramified base change in local Langlands reciprocity, preprint. 9. G. Lusztig, Some remarks on the supercuspidal representations of p-adic semisimple groups, these PROCEEDINGS, part 1 , pp. 1 7 1 - 1 7 5 . 1 0 . T. Shintani, On certain square integrable irreducible unitary representations of some P-adic linear groups, J. Math. Soc. Japan 20 (1 968), 522-505. 11. T. Springer, The zeta function of a cuspidal representation of a finite group GL .(k), Lie Groups and Their Representations (Gel ' fand, Ed.), Halsted, New York, 1 975. UNJVERSITE PARIS VII
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part I , pp. 1 7 1 - 1 7 5
SOME REMARKS ON THE SUPERCUSPIDAL REPRESENTATIONS OF p-ADIC SEMISIMPLE GROUPS
G. LUSZTIG 1. Let G be a semisimple group over a local nonarchimedean field K with ring of integers (!) and finite residue field We define = G(K), = G(K ) where k is a maximal unramified extension of K (with ring of integers 19 and residue field Tc, an algebraic closure of Let T c G be a maximal torus, defined and anisotropic over K. We shall assume that T satisfies the following condition. There exists a Borel subgroup B of G containing T and defined over k. (This condition is certainly satisfied if T is split over k.) Let f be the set of homomorphisms of T = T(K) into Qf (I a prime #: char which factor through a finite quotient of T. One expects that to any (} e f satisfying some regularity condition, there corresponds an irreducible admissible supercuspidal representation of (over Q1). Such a correspondence has been established by Gerardin (for T split over k, with certain restrictions, see [2]) using methods of Shintani, Howe and Corwin. (For SL 2 the correspondence was estab lished by Gelfand, Graev, Shalika and others.) I would like to suggest another possible approach to the question of constructing this correspondence. This would use /-adic cohomology (or homology) of a certain infinite dimensional variety X over Tc. (The fact that /-adic cohomology might be used to construct representations of is made plausible by the work of Drinfeld and by that of Deligne and myself, in the case of finite fields [1] . Note that, even in Gerardin's approach, one has to appeal, in the case where the conductor is very small, to the representation theory of a reductive group over a finite field, which is, itself, based on /-adic cohomology.) Let U be the unipotent radical of B and let 0 = U(K ). Consider the Frobenius map defined by the Frobenius element
k).
k.
k)
G
G
G
G
G GF . G,
F: G --+ G I F G
G
A MS (MOS) subject classifications ( 1 970) . Primary 22E5 0 ; Secondary 1 4F20, 20H30. © 1 9 79, American Mathematical Society
171
G. LUSZTIG
1 72
by any nontrivial element in the small Weyl group of T, H;(X)0 should be nonzero for exactly one index i = i0 (depending on ()) and the resulting representation of G on HiX)8 should be an irreducible admissible supercuspidal representation. This should establish the required correspondence between characters of T and repre sentations of G. 2. Consider, for example, the case where G homomorphism defined by
1
0 (2. 1 )
F(A)
�
[
SLn (K ) and let F: G
0
0
o ··..
7C
=
0
·--
L
0
A¢
0
1
o
7C
0
__.
G be the
-1
1
0
where 7C is a uniformizing element of (I} and, for any matrix A over K, A¢ is obtained by applying ¢ to each entry of A . The fixed point set (}F is the group of elements of reduced norm one in a central division algebra of dimension n2 over K. Let f be the group of diagonal matrices in SLn( K) ; it is invariant under F. Let D c SLn( K ) (resp. D-) be the subgroup consisting of all upper (lower) triangular matrices with I 's on the diagonal. If A is a matrix in SLn {K ) satisfying A-1F(A) E ·D, we can find a unique B E D n F- 1 D such that (AB)-1F(AB) E D n FD-. Thus,
I
{A E SLn( K) A-1F(A) E D}l D n F- 1( D) can be identified with
I
X = {A E SLn ( K ) A-1F(A) E D n FD-} on which (}F acts by left multiplication and fF acts by right multiplication. X is just the set of all n x n matrices of the form
ttf. - 1 -
------------ a'f.-(n-l)
·-. .
(2.2)
l d!1 --.
- - ----- --- -- - '!C Cn-2) a'f -
-
--
-----
--
- -
2
-.
-�-(n-l) - -- - df-(n-l) 1
with a; E K and determinant equal to 1 . For such a matrix, we have automatically E (!j (I � i � n) and a1 !/= 7Ci9. It follows that X may be regarded as the projective limit proj limhXh of the algebraic varieties Xh over k, where Xh � is the set of all n x n matrices of the form (2.2) with a1 E ({!j I '!Ch {!j)*, a,- E (!j I '!C h -t {!j (2 � i � n), with determinant equal to (We regard '!CO; (2 � i � n) as elements of 'JC(!j I '!C h{!j ; the determinant of such a matrix has an obvious meaning as an element of (!j I '!C h {!j.) Let Gh be the set of all n x n matrices (a.-j ) with aij E 7C (!j I '!C h {!j (Vi > a,-i E i917Ch - t {!j (Vi < j), a,-; E ({!j 1 '!C h{!j)* (Vi), with determinant equal to I , as an element of (!j I '!C h {!j. Define F: Gh __. Gh by the formula Gh is an algebraic group over k and F: Gh __. Gh is the Frobenius map for a k-rational structure on Gh . Let Th be the group of diagonal matrices in Gh and let Uh (resp. uh-) be the group of upper a.-
I.
(h I)
(2. I ).
j),
173
REMARKS ON SUPERCUSPICAL REPRESENTATIONS
(lower) triangular matrices in Gh with 1 's on the diagonal. We may identify Xh with the variety {A E Gh 1 A- 1 F(A) E uh n FU;;} . Gf acts o n Xh by left multiplication and T[ acts o n Xh b y right multiplication. (Note that GF = proj limhGf, fF = proj limhT[). The following lemma is crucial.
ofaffinethespnataceuralTheofdimapactimension oofn (n 1and) overeachk. fibre of(h d, h QJ(d)), LEMMA .
on preserves is isomorphieachc tofibrethe k
� 2), Xh Kh = ker(Th -+ Th_1) Xh/Kh -+ Xh-l Xh -+ Xh - h -
Let us define H;( Y) for any smooth algebraic variety Y over of pure dimension to be H'f"-; ( Y, where I is a fixed prime =1- char k . The previous lemma shows that, for � 2, H;(Xh - l ) is canonically isomorphic to H;(Xh) Kh (fixed points of Kh on H;(Xh)) . In particular, we have a well-defined embedding H;(Xh _1 ) -+ H;(Xh). Using these embeddings, we form the direct limit inj limhH;(Xh). We define H,-(X) to be this direct limit. H;(X) decomposes naturally in a direct sum EBH;(X)8, where () runs through the set ( fF) V . It is clear that, for fixed (), H;(X)8 is of finite dimension and is zero for large On the other hand, it is in a natural way a GF-module (GF acting via a finite quotient). Let R8 .E ,{ - 1)iH;(X)8 • THEOREM. F If () =1E ( f V , ± R8 ± R8, ± R0
i. is an irreducible GF-module. =
areor dieach() stinct. F)
It suffices to prove that, if I = Gf\(Xh xh X Xh) we have
x
Xh) (with Gf acting diagonally on
IX - 1) i dim H;(I )v 1 = 1 i
-
()', then
if ()'
=
()
,
= 0 if () ' =1- ().
(Here Tf x Tf acts on 2 by right multiplication and H;(I)o, o•- 1 denotes the ((), ()'- i )-eigenspace of Tf x Tf.) To compute the (equivariant) Euler characteristic of I we use the principle that the Euler characteristic of a space can be computed from the zeros of a nice vector field. The map (g, g') -+ (x, x ' , y), x = g-lF(g), x ' = g'-lF(g'), y = g-lg' defines an isomorphism I � {(x, x ' , y) E ( Uh n FU-,;) X ( Uh n FU-,;) X Gh xF(y) yx '}
I
=
(compare [1, 6.6]). Now, any y E Gh can be written uniquely in the form Y = Yi y; y; Yz ,
Yi E U-,; n p-l Uh, y; E Th( U-,; n p- l U-,;) , Y� E uh n p-l u-,;, Y2 E uh n p-l uh.
We now make the substitution xF(yi ) = x E Uh, so the equation of I becomes xF(y;)F(y�)F(yz) = Yi y; Y� Yz x ' . Any element z E Uh can be written uniquely in the form z Yz E uh n p-l uh, x ' E uh n FU-,;, so the equation of I becomes
=
y 2x ' F(y2) - l
with
G. LUSZTIG
1 74
xF(y;)F(y�) E Yi y; Y� Uh = Yi y; Uh . Thus, we may identify I = { (x , yi. Y2 Yi) e ( Uh n FUJ;) X ( UJ; n F- 1 Uh) X Th( UJ; n F- 1 UJ;) X
The action of Tf
x
(Uh
n F- 1 UJ;) I Y;-1 yi_-1 xF(y2) F(y� ) E
Uh } ·
Tf is given by
(t, t') : (x, yi. yz, y�) -+ (txt -1, tyi ,-1, ty2 t'-1, t 'i{ t '-1). This action extends to an action of a larger group H, consisting of all pairs (t, t ') ofn x n-diagonal matrices with diagonal entries in (@{1&h@) * such that det t = det t ' and t - 1 F(t ) = t ' - 1 F(t') e centralizer o f Th( UJ; n F- 1 U J;) ; the action o f H i s given by (t, t') : (x, yi, yz, y�) -+ (F(t)xF(t)-1, F(t )yiF(t)-1, ty;t '-1, t 'y�t '-1). A diagonal matrix of the form 0 0
1 a
certainly centralizes Th ( U J; n F- 1 U J;) . Thus, any pair ( t, t) with of the form
(:;; :J
t
(with e e (@{1&h@)* a root of 1 of order prime to char is in H. The set of all such pairs ( t, t) is a one dimensional subtorus f7 of H (over k). This torus acts on I commuting with Tf x Tf. Its fixed point set on I is the finite set given by x = yi_ = y� = e , y; e Tf, hence it is isomorphic to Tf with Tf x Tf acting by left and right multiplication. It follows that
k)
I;( - 1) i dim H.-(I)0, 0, _ , = I; ( - 1 )i dim H,-(I�)v- 1 = dim H0( Tf>o. o' - 1 = 1
if O =
= 0 if O =1=
0', 0',
a s required. 3. Let be a 2-dimensional vector space over K and let V ® k k. Let F : V -+ V be defined by F = 1 ® ¢. Assume that we are given a nonzero element we The set X = { x e V I x 1\ Fx = w } is invariant under the obvious action of SL2( V) and under the action of the group T = {A e K* I il · il.P = 1 } which acts by scalar multiplication. (This set X can be identified with the set X defined in § 1 for G = SL2 and T a maximal torus associated to the unramified quadratic extension of K.) Objects similar to this X appear in the work of Drinfeld. We now
Jf2V.
V
=
V
REMARKS ON SUPERCUSPIDAL REPRESENTATIONS
1 75
show what is the meaning of the homology groups H;(X) in the present case. It can be easily checked that if x e X, the iii- module spanned by x and Fx is invariant under F, hence it comes from a lattice L in V. Thus X is a disjoint union X = I1 LXL over all lattices c: V with determinant (!J*w, where XL = {x e L ®e iii l x
1\
Fx = w}.
L
Each XL can be regarded in a natural way as proj limhXL.h where XL.h is the h-dimensional variety over k defined by XL . h = {x e L ®e> (i!ij7l:Li!i) i x
1\
Fx
=
w}.
The fibres of XL.h --+ XL,h-1 have a property similar to that in the Lemma in §2. This allows us to define H,{XL) = inj limhH;(XL,h) as in §2. We also define H;(X) = EBL H;(XL). Similar arguments should apply in the general case.
4. Let G, T, B, U be as in § 1 . Assume that G comes from a Chevalley group over Z by extension of scalars so that G((!J), G(i!i) are well defined. Let Gh = G(iii/7l: hi!i).
Assume that T(K ) c: G(i!i). Let th, Oh be the images of T(K ), U(K ) n G(i!i) under --+ Gh. F: G( K ) --+ G(K ) induces F: G h --+ Gh. This gives a k -rational struc ture on the k-algebraic group Gh. Define
G(i!i)
xh = {g e Gh 1 g-1 F(g) e Oh}l oh n F-1 oh.
The finite group G f x Tf acts on Xh, as before, by left and right multiplication. For each character () : Tf --+ Q/1 we form R0 = I; ( - l )•H;(Xh)o. i
± R8
It is independentIf()ofisthsueffichoicientcelyofregular, the virtual Gf-module is irreducible. h THEOREM.
In the case
B.
= 1 , this follows from [1].
REFERENCES
1. P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 ( 1 976), 1 03-1 6 1 . 2 . P . Gerardin, Cuspidal unramified series for central simple algebras over local fields, these PROCEEDINGS, part 1 , pp. 1 57-169. UNIVERSITY OF WARWICK, COVENTRY, GREAT BRITAIN
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part 1 , p p . 1 79- 1 8 3
DECOMPOSITION OF REPRESENTATIONS INTO TENSOR PRODUCTS
D. FLATH I. In this paper some generalizations of the classical theorem which classifies the irreducible representations of the direct product of two finite groups in terms of those of the factors will be discussed. The first generalization consists in expanding the class of groups considered.
1. Let be locally compact total y disconnected groups and let If ssibisleanirreduci admisbsilbelrepresentati e irreducibleonrepresentati on of 2, then is an(1.1.1)admi of If ibsleanrepresent admissibalteioinsrreduciofble represent ation of then tThehereclexiassesst admi s sibarele 2ir)detreduci such that o f the ermined by that of e ke algebra K K). K Irreduci bilitbyleCriterioK)-modul n. smoote forh all modul e i s i r reduci b l e if and onl y if is an i r reduci compact open subgroups K of G
THEOREM = G 1 x G2 . ( 11:1
Gb G2
G;,
G.
71:;
11:.
=
I,
11: 1 ® 11:2
G, 11: � 11: 1 ® 11:2 •
11:
71:;
i
G;
We recall some notation. For a locally compact totally disconnected group G, the H e H(G) of G is the convolution algebra of locally constant complex valued functions on G with compact support. For a compact open subgroup K of G, let ex be the function (meas K)- l . chx, where chx is the characteristic function of and meas is the Haar measure on G which has been used to define convolution in H(G). Then ex is an idempotent of H(G). The subalgebra exH(G)ex of H(G) will be denoted H(G, A smooth G-module W is in a natural way an H(G) module, and for every compact open subgroup c G the space wx = ex W is an H(G, K)-module. Before proving Theorem I we state an A G W wx H(G, G. PROOF. This follows from the fact that if U is an H(G, K)-submodule of w x , then ( H(G) · U)X = U. 0 Remark that in applying the irreducibility criterion it is sufficient to check that wx is an irreducible H(G, K)-module for a set of which forms a neighborhood base of the identity in G.
K of suchThenthat K) is com mutative, and letLet Kbebeanaadmicompact ssible open irreducisubgroup ble G-module. CoROLLARY.
G
W
H(G, dim wx � I .
A MS (MOS) subject classifications ( 1 970). Primary 22E55 .
© 1 9 79, American Mathematical Society
1 79
D . FLATH
1 80
PROOF OF THEOREM 1 . It is straightforward that (i) H(Gt x G2) � H(Gt) ® H(G2), (ii) H(G1 x G2, K1 x K2) � H(G�o K1 ) ® H(Gz, K2) and (iii) ( W1 ® W2) K 1 x K2 � W1Kt ® Wj<2 for every pair of compact open subgroups K; of G; and every pair of smooth G;-modules W;. Assertion ( 1 . 1) follows from (iii) and the irreducibility criterion. Conversely, let W be an admissible irreducible G-module. Let K = K1 x K2, where K; is a compact open subgroup of G;, = 1 , 2, be such that W K :f. 0. The space WK is finite dimensional, so by the corollary on p. 94 of [2] there exist irre from WK to ducible H(G;, K;)-modules Wl; and an H(G, K) isomorphism Wft ® Wf2. Similar remarks apply to every open subgroup K' = K{ x K� of K') : Wf; -+ Wf: such that the follow K. There exist H(G;, K;)-maps = ing diagram is commutative.
i
ax
b; b;(K, __3f__.
Wf t ® Wfz
WK ' __!!£
, Wft ® Wf 2
WK
find.
1
bt®hz ,
Moreover, the maps b;(K, K') can be chosen for every pair of compact open sub groups K, K' of this type in such a way as to form an inductive system. Then W � ® W2 , where W.· = ind limx; Wfi, and W.· is an admissible irreducible repre sentation of G;, = 1 , 2. The class of W; is determined by that of W, for the restriction of W to G; is W;-isotypic. D An analysis of the proof of Theorem 1 reveals that the groups G and G; enter only through their Heeke algebras. This leads one to define an (A, E) to be an algebra A with a directed family of idempotents E such that UeEE An W for (A , E) is an A-module W which is A in the sense that A W = W and is such that dim W is finite for all E E. The tensor product of two idempotented algebras is naturally idempotented . The proof of Theorem 1 is readily adapted to establish a similar theorem about the admissible irreducible modules of the tensor product of two idempotented algebras.
W1
i
idempotented
algebra eAe. admissible module nondegenerat e e =
e
2. The study of the representations of adelic groups, which are infinite re stricted products of groups, requires the notion of restricted tensor product of vector spaces which was introduced in [4] . Let { W. l v E V} be a family of vector spaces. Let V0 be a finite subset of V. For each v E V\ V0, let x. be a nonzero vector in w• . For each finite subset of V con taining Vo, let V5 = ®vES w. ; and if c let fs : Ws -+ W5, be defined by ®vES w. ..... ®vES w. ®vES'\S x •. Then W = ®x. w., the of the w. with respect to the x., is defined by W = ind lim s W5. The space W is spanned by elements written in the form w = ® w., where w. = x. for almost all v e V. The ordinary constructions with finite tensor products extend easily to restricted tensor products.
S S',
S restricted tensor product
DECOMPOSITION OF REPRESENTATIONS
181
Given linear maps Bv : Wv ---> W" such that Bvxv = Xv for almost all E V, then one can define B = 0Bv : W ---> W by B(0wv) = 0Bvwv. Given a family of algebras { Av v E V} and given nonzero idempotents v E A v for almost all v, then A = 0e,Av is an algebra in the obvious way. If W" is an Av-module for each v E V such that · xv xv for almost all v, then 0x, Wv is an A-module. The isomorphism class of W depends on {xv} · How ever, if {x�} is another collection of nonzero vectors such that xv and x� lie on the same line in Wv for almost all v, then the A-modules 0x, Wv and 0..; W" are iso morphic. ExAMPLE The polynomial ring in an infinite number of variables C[Xb X2 , . . . ] is isomorphic to 0e, C[X;], where is the identity element of C[X;]. EXAMPLE Let G n �Pv be the restricted product of locally compact totally disconnected groups G"' restricted with respect to the compact open subgroups K". Then G itself is locally compact and totally disconnected, and H(G) is isomor phic to @eK, H(Gv) . For each v E V let Wv be an admissible G"-module. Assume that dim W�v = 1 for almost all v. Choosing for almost all v a nonzero vector xv E W�', we may form the G-module W 0.., W". The isomorphism class of W is in fact independent of the choice of xv E W�v and will be called the tensor product of the representations Wv. On e sees that W is admissible, and that it is irreducible if and only if each Wv is. The admissible irreducible representations of G isomorphic to ones constructed in this way are said to
(I) (2) (3)
I
I. 2.
e"
v e
=
e;
=
=
befactorizable. 2.reduciSuppose that ation isofcommutis factativoeriforzable,almost all ThenTheeveryiso admi s si b l e i r b le represent morphism cla1.sses of the factors are determined by that of For almost all H(Gv, Kv) W W"
THEOREM
dim W�v =
v.
W � 0 Wv. W.
G
v,
PROOF. One first factorizes the finite dimensional spaces WK' for compact open subgroups K' TI K� of G, then continues as in the proof of Theorem 1 . D =
3. Let G be a connected reductive algebraic group over a global field F. Let A be the adele ring of and let V be the set of places of The adelic group G(A) is isomorphic to a restricted product TI �, G(Fv), where the subgroups Kv are defined for all finite v and are certain maximal compact subgroups of G(Fv). For almost all finite v E V, is a quasi-split group over and K" is a special maximal com pact subgroup. For these places H( G v) , K.) is commutative. See [5]. So the function field case of the following theorem, whose meaning has yet to be explained in the number field case, is a special case of Theorem 2.
F,
F.
G(Fv)
v, (F Fv, rreducible representation of is factorizable. Thefactors are3. Every uniqueadmi up tossiequible vialence. F v G(Fv), G(Fv ) . "). F G( F. v G(A)
THEOREM
Let be a number field. Then the class of admissible representations of G(A) has yet to be defined. For each archimedean place E V, let Kv be a maximal com pact subgroup of and let Bv be the real Lie algebra of Let Koo algebra of Lie Let real G the and be Boo 00 = TI arch TI an v Kv, TI arch v Kv, K be the ring of finite adeles of • Let A G00 1 =
=
D. FLATH
1 82
admissible
DEFINITION. An G(A)-module W is a vector space W which i s both a ( g "'' K"')-module and a smooth G(A1)-module such that ( 1 ) the action of G(A1) commutes with the action of g"' and K"', and (2) for each isomorphism class r of continuous irreducible representations of K, the r-isotypic component of w has finite dimension. In Theorem 3, the factors at the archimedean places are to be admissible ( g v, Kv)-modules. The proof when F is a number field is the same as that when it is a function field once an idempotented algebra is found for each archimedean place whose admissible modules are the same as admissible (g " ' K")-modules. Let G be a Lie group, and let K be a compact subgroup. Let g and t be the real Lie algebras of G and K. Let U( gc) and U(tc) be the universal enveloping algebras of the complexified Lie algebras. Define the (g , K) of ( g , K) to be the algebra of left and right K-finite distributions on G with support in K. It con tains the algebra of K-finite measures on viewed as distributions on G. The map f-1) ---> f-1 from U(gc) x A x to the space of distributions on G induces a vector space isomorphism of U(gc) ®uctcl with ( g , K). With the set E of central idempotents of (g , K) is an idempotented algebra. Let (n, W) be a ( g , K)-module. By means of the formula X ® f-1 w = n(X)n(fi.)W for U(gc), f-1 X• and W, the space W becomes a nondegenerate (g , K) module. Moreover, it is not difficult to verify that this construction establishes an isomorphism between the categories of admissible (g , K)-modules and of admissible ( (g , K), E ) -modules.
v
v
Heeke algebra H K Ax H
(X, X *Ax
Ax, H EA wE
XE
•
H
H
4. In practice, a more analytic theory than that described above is needed as well. Let { v be a family of Hilbert spaces. For almost all let be a unit vector in Hv. The = ®x. Hv is a Hilbert space which can be conveniently described by giving an orthonormal basis. Let be an orthonormal basis for for each V which extends for almost all The set of symbols ®hv such that hv Pv for all and hv = xv for almost all is an orthonormal basis for H. Constructions analogous to those described above with reference to the ordinary restricted tensor product are available in the Hilbert space context. For the following theorem see [3] and [1].
H I v E V} Hilbert restricted product H v E V, H. E v E v {x.}
x" P"v. v
Let abetionasofin Theorem Let be a continuous irreducible unitary Hi(I)lbertThere spaceexirepresent A Then slt ofcontiwhincuoush areirreduci bleed,unisuch tary Hithatlbertn space representati ons areof G(uniFqvue), almost al unramifi The factors up toeachisomorphi sm.sm class r of continuous irreducible representations of (2) For i s omorphi the r-iThesotypispacec component ofe vectors hasfiniofte diismiensi onatn. ural way an admissible irreducible o f Kfi ni t n a orizateioton tofhe nXspacegivenof Kvby-:fiTheorem G(A)-modul esomorphiLetc as an admissibebletG(heFfact.)-modul Then i s i nite vectors ofn THEOREM 4.
G(A)
3.
G( ) .
n
�
nv nv
K,
n
(3)
n
nX.
n{f
•.
®nv·
nX
�
@n{f
3.
DECOMPOSITION OF REPRESENTATIONS
1 83
REFERENCES 1. I. N. Bernshtein, All reductive p-adic groups are tame, Functional Anal. Appl. 8 (1 974), 9 1 -93. 2. N. Bourbaki, Modules et anneaux semi-simples, Hermann, Paris, 1 958.
3. I. M. Gelfand, M. I. Graev and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co. , Philadelphia, 1 969. 4. H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math . , vol . 1 1 4, Springer-Verlag, Berlin and New York, 1 970. 5. J. Tits, Reductive groups o ver local fields, these PROCEEDINGS, part 1 , pp. 29-69. DUKE UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 1 85-1 8 8
CLASSICAL AND ADELIC AUTOMORPHIC FORMS. AN INTRODUCTION
I. PIATETSKI-SHAPIRO * 1. Classical Heeke theory. Let H be the upper half-plane E Cjlm > a congruence subgroup of SL(2, i.e., :::> for some integer N � (where
{z 0 z 0}, F Z), F FN {(� �) Z) }) (� �) (� �) f: modularform of weightz k-+ (az b)j(cz F)d). f ((az b)j(cz d))(cz d)-k k F. f(z) n�OI: an e2rrinJ-z . a0 0. Q(f) cuspform ) { ( �: : � (cz d)-k j (� �) } I Q(f). i( s J (2F(s)n: ) 0oo f(iy)y <s-u 12 dy n=l£ ann-s f). s)oinals a "niequatce"ioentn). irefunction when fis a cuspform ("nice" means that s) s)hashasafuncti an Euleunder r productthe actifiQ(f) in variant linear subspaces on of is algebraically irreducible, i.e. , has no rN
E SL(2,
=
(mod N) .
=
such that
The group SL(2, R) acts on H by + + We say that a function H -+ C is a (with respect to iff (a)fis holomorphic on H. (b) + + + = f(z), for all (� �) E r, and some strictly positive integer k. (c)fis ho lomorphic also at the cusps of H with respect to For example, at co , this means thatfhas the Fourier expansion =
We say thatfis a if in the Fourier expansion at each cusp, Let be the C-linear span of the set
1
=
E GL+(2, Q) .
+
Here G = GL+(2, Q) = {g E GL(2, Q) det(g) > 0} . Note that GL+(2, R) acts on H. There is an obvious representation of G on Heeke defined the L-function attached to the modular form/ by the formula : L( f, s) =
(the Dirichlet series corresponding to
=
Heeke [1] proved the following theorem :
THEOREM. (1) L(f, L(f, (2) L(f,
G.
The second statement is the more interesting ; it is equivalent to Heeke ' s actual
A MS (MOS) subject classifications (1 970) . Primary 1 0D20 ; Secondary 1 0 D 1 0, 22E5 5. *The author was supported by an NSF grant while preparing this paper. © 1 9 7 9 , Amercian Mathematical Society
185
I . PIATETSKI-SHAPIRO
1 86
statement. He proved that L(f, s) has an Euler product if/is an eigenfunction of the "Heeke operators". This is equivalent to Q(f) being irreducible. Already (2) sug gests that it might be better to study modular forms from the point of view of representation theory. This leads us to the main purpose of this paper, which is to motivate the transi tion to the adelic setting and the systematic use of representation theory in the study of modular forms. First let us explain the notion of Euler product. L(s) = L(f, has an Euler product means that L(s) =
p
a
s)
fi
prime
Lp(s),
where ap , �P E C.
In Heeke's theorem we have a functional equation of the sort : L(s) = .s(s) i ( I s), where L(s) (resp . .s(s)) can be written as an infinite product IT Lp (s) (resp. IT .sp(s)). Now Tate's theory of L-functions associated to Grossencharakters and Artin L-functions suggests the problem of finding objects such that the local factors Lp(s) and ep (s) are .s- and L-functions for these objects. It was the beautiful idea of Jacquet-Langlands [2] to take as such objects irreduci�le representations of GL(2,Qp). More precisely, let us look at Q(f). Any element h E O(f) is fixed by a congruence subgroup £ Now we define a topology on by taking a basis of neighborhoods of the identity to be the set of congruence subgroups of and let G be the completion of in this topology. Then G = {g E IT nnite primes GL(2, Qp ) l det gp = r > 0, r E Q}, where IT means restricted direct product, acts on Q(f), via the action of and it can be shown that this repre sentation 1r is an infinite tensor product (interpreted in a suitable sense) 1r = ® finite primes 1rp , where 1rp is a representation of GL(2, Q p). In the author's interpretation, it is the idea of Jacquet-Langlands to attach an L-function to the irreducible representation 1r = O(f), rather than to f, and then to interpret Lp (s) and ep (s) as the L- and .s-factors for the representations 1rp of GL(2, Qp) . Finally, let us give the following additional motivation for introducing the adelic framework into the study of modular forms. We wish to study modular forms with respect to different congruence subgroups simultaneously, and every such form is stabilized by some congruence subgroup. To each inclusion relation F c cor responds a projection H/ F -+ H/F1 • A small computation shows that the projective limit of this system, -
F G.
G,
G,
G
G
F1
r
a
proj lim
congruence subgroup
H/F
= K""\SL(2 , A)JS L(2 , Q).
Hence the study of modular forms has been transformed into the study of a cer tain space of functions on this double coset space. 2. Automorphic forms for adelic groups. Assume G is a connected reductive algebraic group over a global field For such a we can define the group GA c A as a discrete subgroup. (cf. Springer's paper for the construction), and
k.
Gk G G
CLASSICAL AND ADELIC AUTOMORPHIC FORMS
1 87
We need some basic facts from reduction theory. First let n {ker x . X a rational character of G} . Then is compact iff there are no additive unipotent elements. (This is due to Borel-Harish-Chandra-Mostow-Tamagawa-Harder.) Let A = be the global lwasawa decomposition (which follows directly from the local lwasawa decompositions), where K is a standard maximal compact subgroup of A • the set of adelic points of a maximal split torus, and is a maximal unipotent subgroup. EXAMPLE. GL(2 , Then
G NAK GA G k). Kp
G�
Gk\G�
ll
=
N
=
=
=
=
0(2) U(2) GL(2 ,
if p is real, if p is complex, for a finite place ;
Q p)
N {(� �)} , A {(a� �J} . A CA0 0A AA0 C Z(G0 N0 N, G sN0s-A1 A0 A 0 j j ap l p Cp cp 1 N0SK, N0 N, mental domainThere for Gexik contsts aainSiedegelin ®C.set s.t. G Gk®C. Hence there exists afunda f: Gk\GA --> (1) C: f(cg) w(c)f(g). w (3) I dz 0, Z P. =
=
Note that =::1 = A ) , and that we can write = for some subgroup s.t. c n is finite. We say that a set S c is semibounded iff for any relatively compact subset c the set UsES is relatively compact. � for all places {(g ?)} , S {(3 �) E EXAMPLE. GL(2). Then = for almost all p} . Then S is semibounded. p, and We now give the important definition of a Siegel set @3. 1 @3 = is an open compact subset of S is semibounded, and when K as above. Then the main result of reduction theory says : =
THEOREM.
=
=
@3
A
=
We now come to the definition of a cusp form (the definition of an automorphic form is given elsewhere in these PROCEEDINGS) : A function C is said to be a cusp form iff fis an eigenfunction with respect to (2) .f is smooth (i .e., f is coo at archimedean places and locally constant at the finite places). ScckiG A f(g) i 2 dg < oo (here we suppose that is unitary). = where is the unipotent radical of any parabolic sub (4) J zk1zJ(zg) group ExAMPLE. For GL2 it is sufficient to consider the standard unipotent group ( 6 i). For GL one has the following conjugacy classes of unipotent radicals : =
( 0010101 **), ( 00110 *) ( 0011 **) Oh ,
Oh .
In general, for GLm one associates to each partition jugacy class of unipotent radicals
n1
+ ... +
n" n, =
a con
' For simplicity we give the definition only for split groups. There is a similar definition in general.
I . PIATETSKI-SHAPIRO
I 88
[n1 0
*
[n2 In ,
We shall write L5(w) for the space of cusp forms with respect to w. The first main result is the following. THEOREM ( GELFAND, PIATETSKI-SHAPIRO, HARDER) . L5(w) is
irreducible admissible representations of G each occurring with .fianicount te multiableplisum city. of G s) A•
We now describe a naive form of Langlands' philosophy. Let n be a representa tion of A occurring in L5(w ). Then one can attach to n a function L(n, which is a product 11 L(np , s), and furthermore L(n, s ) is "nice" (i.e., it is meromorphic, with a finite set of poles, and satisfies a functional equation), and the t-factor c(n, s) = Il t(n p , s) . A precise exposition of Langlands' philosophy will be given by Borel in his article [5].
3. The case of GL2 • Now we shall explain what follows from Langlands' conjec ture for GL2 . Let n be an irreducible cuspidal automorphic representation of GL2 (A). Then n = 0.11 pr imes n P• and for almost all primes np � lnd8(,u 1 ® ,u2), where ,Ub ,u2 are unramified characters of Such !11 > ,u2 are described by ,u 1 (p), ,u2(p), where p is the generator of the prime ideal Let be an integer � I . We can define local factors of the form
kj;. p. n
n
Ln (np , s) = ( I - ,u�/P/ s) -1(1 - .ur-l.uz /P/') -1 . . . ( 1 - ,u�/P/')-1 . Here we are writing !1 1 > .uz for ,U l (p), ,uz(p) . For n, as above we can attach to prime p a local factor Ln (np , s ) , agreeing with the definition above at the unramified primes, such that the function Ln (n, s) = IT Ln (np , s) exists, and is "nice". For = I this conjecture is proved in the book by Jacquet-Langlands. For = 2, it has been proved by Gelbart-Jacquet and Shimura. For = 4 it can be proved that L n (n, exists and has a meromorphic continuation. Beyond these cases, the situation is unresolved. Notes of this talk were made by Lawrence Morris (I. H. E. S.) and Ben Seifert (U. C. Berkeley).
Conjecture. n
n
every
n 3,
s)
n
REFERENCES I. E. Heeke, Mathematische werke, Vandenhoek and Ruprecht, Gottingen , 1 959. 2. H . Jacquet and R.P. Langlands, Automorphic forms on GL(2) , Lectu re No tes i n Math . , vol. 1 1 4, Springer, 1 970. 3. A. Borel, Introduction aux groupes arithemetiques, Hermann, Paris, 1 969 . 4. G. Harder, Minkowskische Reduktionstheorie uber Funktionen korpern, Inven t . Math. 7 ( 1 969), 33-54. 5. A. Borel, Automorphic L-functions, these PROCEEDINGS, part 2, pp. 27-61 .
UNIVERSITY OF TEL AVIV
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part I, pp. 1 89-202
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
A. BOREL AND H. JACQUET Originally, the theory of automorphic forms was concerned only with holomor phic automorphic forms on the upper half-plane or certain bounded symmetric domains. In the fifties, it was noticed (first by Gelfand and Fomin) that these au tomorphic forms could be viewed as smooth vectors in certain representations of the ambient group G, on spaces of functions on G invariant under the given discrete group r. This led to the more general notion of automorphic forms on real semisimple groups, with respect to arithmetic subgroups, on adelic groups, and finally to the direct consideration of the underlying representations. The main purpose of this paper is to discuss the notions of automorphic forms on real or adelic reductive groups, of automorphic representations of adelic groups, and the relations between the two. We leave out completely the passage from automorphic forms on bounded symmetric domains to automorphic forms on groups, which has been discussed in several places (see, e.g., [2], or also [5], [6], [15] for modular forms) . 1. Automorphic forms on a real reductive group. 1 . 1 . Let G be a connected reductive group over Q, the greatest Q-split torus of the center of G and K a maximal compact subgroup of G(R). Let g be the Lie algebra of G(R), U(g) its universal enveloping algebra over C and (g) the center of U(g) . We let .Yf or .Yf(G(R), K) be the convolution algebra of distributions on G(R) with support in K [4] and A K the algebra of finite measures on K. We recall that .Yf is isomorphic to U(g) ®u(!) A K · An idempotent in .Yf is, by definition, one of A K , i.e., a finite sum of measures of the form d(0")- 1 . Xu · where O" is an irreducible finite dimensional representation of K, d(O") its degree, Xu its character, and dk the normalized Haar measure on K. The algebra .Yf is called the Heeke algebra of G(R) and K. 1 .2. A norm 11 11 on G(R) is a function of the form llgll = (tr O"(g)* · O"(g)) 1 12, where O" : G(R) ---4 GL(E) is a finite dimensional complex representation with finite kernel and image closed in the space End(£) of endomorphisms of E and * denotes the adjoint with respect to a Hilbert space structure on E invariant under K. It is easily seen that if -r is another such representation, then there exist a constant > 0 and a positive integer n such that
Z
Z
dk,
C
A MS (MOS) subject classifications ( I 970). Pri mary J OD20 ; Secondary 200 3 5 , 22E5 5 . © 1 9 79, American Mathematical Society
1 89
A . BOREL AND H. JACQUET
1 90
ll xll a
(1)
�
C
·
l l x l l � , for all x
slowlny increasing
A function / on G(R) is said to be and a positive integer such that
G(R) , a constant (2)
C
I J(x) I
�
C
E G(R) .
· ll x l l n ,
if there exist a norm I I on
for all x E G(R) .
n
In view of ( l ) this condition does not depend on the norm (but does). REMARK. If (J : G --+ GL(E) has finite kernel, but does not have a closed image in End E, then we can either add one coordinate and det (J (g)- 1 as a new entry, or consider the sum of (J and (J * : g ,__. (J(g- 1 ) * . The associated square norms are then l det (J(g)- 1 1 z + tr((J(g) * (J (g)) and tr((J (g) * (J (g)) + tr((J(g- 1) (J (g- 1 ) * ) . 1 . 3. Let F be a n arithmetic subgroup of G(Q). A smooth complex valued func tion f on G(R) is an automorphic form for (F, K), or simply for F, if it satisfies the following conditions : (a) f(r · x) = f(x) (x E G(R) ; r E F). (b) There exists an idempotent � E Jft'(cf. 1 . 1) such that f * � = .f (c) There exists an ideal of finite codimension of Z(g) which annihilates f: f * X = 0 (X E (d) f is slowly increasing ( 1 .2). An automorphic form satisfying those conditions is said to be of type (�, We let d (F, �. K) be the space of all automorphic forms for (F, K) of type (�, 1 .4. ExAMPLE. Let G = GLz , F = GLz(Z), K = 0(2, R), � = 1 , the ideal of Z(g) generated by ( C - A), where is the Casimir operator and A E C. Then f is an eigenfunction of the Casimir operator which is left invariant under right inva riant under K, and invariant under the center Z of G(R). The quotient G(R)/Z · K may be identified with the Poincare upper half-plane H. The function f may then be viewed as a F-invariant eigenfunction of the Laplace-Beltrami operator on It is a "wave-form," in the sense of Maass. See [2], [ 5] , [6] for a similar interpreta tion of modular forms. 1 . 5. REMARKS. (1) Condition 1 . 3(b) is equivalent to : fis K-finite on the right, i.e., the right translates off by elements of K span a finite dimensional space of func tions. These functions clearly satisfy 1 . 3(a), (c), (d) iff does. (2) Let r : K --+ GL( V) be a finite dimensional unitary representation of K. One can similarly define the notion of a V-valued automorphic form : a smooth func tion ({! : G --+ V satisfying (a), (c), (d), where I I now refers to the norm in V, and
J).
J). J,
J
J).
J F,
C
X.
(b')
f(x · k) =
r(k-1) • f(x)
(x E G(R) ; k E K) .
For semisimple groups, this is Harish-Chandra's definition (cf. [2], [11]). For E V, the functions x ,__. (({!(x), v) are then scalar automorphic forms. Conversely, a finite dimensional space E of scalar automorphic forms stable under K yields an £-valued automorphic form. (3) A similar definition can be given if G(R) is replaced by a finite covering H of an open subgroup of G(R) and r by a discrete subgroup of H whose image in G(R) is arithmetic. For instance, modular forms of rational weight can be viewed as automorphic forms on finite coverings of SL2(R). v
191
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
GrowtG,h condition.
1 . 6. ( 1 ) Let A be the identity component of a maximal Q-split torus S of and gi/J the system of Q-roots of with respect to S. Fix an ordering on gi/J and let L1 be the set of simple roots. Given > 0 let =
At
(1)
E A:
{a G
i a(a) i
Gt t, (a
;;:::
E J)} .
Let f be a function satisfying 1 .3(a), (b), (c). Then the growth condition (d) i s equivalent to : ( d') Given a compact set R c (R), and > 0, there exist a constant > 0 and a positive integer such that
m i · a) C :$;
if(x
(2)
t
· i a(a)!m, for all
a
E At,
a E J,
x
E R.
C
G' G'(R).
This follows from reduction theory [11, §2]. More precisely, let be the derived group of Then A is the direct product of Z(R)" and A' A n For a function satisfying 1 .3(a), (b), the growth condition (d') is equivalent to (d) for a E A ; ; but says nothing for E Z(R)". However condition (c) implies thatf depends polynomially on E Z(R), and this takes care of the growth condition on Z(R). (2) Assume f satisfies 1 . 3(a), (b), (c) and
G.
=
z
(3)
a
f(z .
=
x) x(z)f(x) (z
E
Z(R),
X E
G(R))
· F\G(R).
where X is a character of Z(R)/(Z(R) n r) . Then /// is a function on Z(R) If /// E LP(Z(R)F\G(R)) for some p � 1 , then fis slowly increasing, hence is an auto morphic form. In view of the fact that Z(R)F\G(R) has finite invariant volume, it suffices to prove this for = 1 . In that case, it follows from the corollary to Lemma 9 in [11], and from the existence of a K-invariant function a E ";' ( (R) ) such that f = f * a (a well-known property of K-finite and Z(g)-finite elements in a dif ferentiable representation of G(R), which follows from 2. 1 below) .
p
1 . 7. THEOREM [11, THEOREM 1 ] .
The space
.sd(F, �,
CG J, is finite dimensional. K)
This theorem i s due to Barish-Chandra. Actually the proof given in [11] is for semisimple groups, but the extension to reductive groups si easy. In fact, it is im plicitly done in the induction argument of [11] to prove the theorem. For another proof, see [13, Lemma 3.5]. At any rate, it is customary to fix a quasi-character K ) x of elements in X of Z(R)/(F n Z(R)) and consider the space .sd(F, �, .sd(F, �, K) which satisfy 1 .6(3). For those, the reduction to the semisimple case is immediate. Note that since the identity component Z(Rt of Z(R) (sometimes called the split component of has finite index in Z(R) and Z(Rt n F = { 1 } , i t is substantially equivalent t o require 1 . 6(3) for a n arbitrary quasi-character of Z(R)". The space .sd(F, �, K) is acted upon by the center of by left or right translations. Since it is finite dimensional, we see that C(G(R))-finite. 1 .8. A continuous (resp. measurable) function on is cuspidal if
J,
J,
is Cusp forms.
G(R))
J,
(1)
for all (resp. almost all)
S
(r n N (R) ) \ N (R)
x G(R), in
f(n·x)dn
=
C(G(R))any automor G(R), phic form G(R)
0,
where N is the unipotent radical of any proper
A. BOREL AND H. JACQUET
1 92
parabolic Q-subgroup of G. It suffices in fact to require this for any proper maxi mal parabolic Q-subgroup [11, Lemma 3]. A is a cuspidal automorphic form. We let o d(F, �. J, K) be the space of cusp forms in d(F, �. J, K). Let f be a smooth function on G(R) satisfying the conditions (a), (b), (c) of 1 . 3 . Assume that/is cuspidal and that there exists a character X of Z(R) such that 1 .6(3) is satisfied. Then the following conditions are equivalent : (i) / is slowly increasing, i.e. , / is a cusp form ; (ii) f is bounded ; (iii) I f I i s square-integrable modulo Z(R) (cf. [11, §4]). In fact, one has much more : Ill decreases very fast to zero at infinity on Z(R)F\G(R), so that if g is any automorphic form satisfying 1 .6(3), then 1 / · g l is integrable on Z(R) F\G(R) (loc. cit.). The space o d(F, �. J, K ) x of the functions in °d(F, �. J, K) satisfying 1 .6(3) may then be viewed as a closed subspace of bounded functions in the space V(F\G(R))x of functions on F\G(R) satisfying 1 . 6(3), whose absolute value is square-integrable on Z(R)F\G(R) . Since Z(R)F\G(R) has finite measure, this space is finite dimensional by a well-known lemma of Godement [11, Lemma 1 7]. This proves 1 .7 for d(F, �. J, K) x when Z(R)F\G(R) is compact, and is the first step of the proof of 1 . 7 in general. is an arithmetic subgroup of G(Q), 1 .9. Let E G(Q). Then ar = and the left translation a by induces an isomorphism of d(F, �. J, K) onto d(ar, �. J, K) . Let .E be a family of arithmetic subgroups of G(Q), closed under finite intersection, whose intersection is reduced to { I } . The union d(.E, �. J, K) of the spaces d(F, �. J, K) (F E .2) may be identified to the inductive limit of those spaces :
cusp form
·r
·
1 a·F·al a
a
d (-2, � . J, K ) = ind limr "'I d (F, � . J, K),
(1)
where the inductive limit is taken with respect to the inclusions ( 2)
j
rnr ' : d (F', �. J, K)
-+
s:Y(F", �. J, K)
(F" c F')
associated to the projections F \ G(R) -+ F '\G(R). Assume .2 to be stable under conjugation by G(Q) . Then G(Q) operates on d(.E, �. J, K) by left translations. Let us topologize G(Q) by taking the elements of .2 as a basis of open neighborhoods of I . Then this representation is admissible (every element is fixed under an open subgroup, and the fixed point set of every open subgroup is finite dimensional). By continuity, it extends to a continuous ad missible representation of the completion G(Qh of G(Q) for the topology just de fined. For suitable .2, the passage to d(-2, �. J, K) amounts essentially to considering all adelic automorphic forms whose type at infinity is prescribed by �. J, K; the group G(Q)z may be identified to the closure of G(Q) in G(A1) and its action comes from one of G(A1). See 4.7. 1 . 10. Finally, we may let � and J vary and consider the space d(.E, J, K) spanned by the d(.E, �. J, K) and the space d(4, K) spanned by the s-1'(4, J, K). They are G(Q)z-modules and (g, K)-modules, and these actions commute. Again, this has a natural adelic interpretation (4. 8) . Let Je"(G(Q), F ) b e the Heeke algebra, over C , o f G(Q) 1.11.
Heeke operators.
"
1 93
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
G(Q)
mod F. It is the space of complex valued functions on which are hi-invariant under F and have support in a finite union of double cosets mod F. The product may be defined directly in terms of double cosets (see, e.g., [17]) or of convolution (see below). This algebra operates on d(F, t;, J, K). The effect of FaF e is given by ...... I: bE (rn•rJ 1r More generally, let Z) be the Heeke algebra spanned by the characteristic functions of the double cosets F'aF" (F', F" e Z, e [17, Chapter 3]. It may be identified with the Heeke algebra of locally constant compactly supported functions on This identification carries F) onto f), where f is the closure of F in [12] . The product here is ordinary convolution (which amounts to finite sums in this case). Since d(Z, t;, J, K) is an admissible module for the action of extends in the standard way to one of The space d(F, t;, J, K) is the fixed point set of f, and the previous operation of £'( F) on this space may be viewed as that of f). For an adelic interpretation, see 4.8.
.n"(G(Q), G(Q)z. Jf'(G(Q)z, G(Q)z, .n"(G(Q)z). G(Q), .n"(G(Q)z, G(R).
f a G(Q)) Jf"(G(Q),
ld
(a G(Q)) Jf'(G(Q)z) G(Q)z G(Q)z
2. Automorphic forms and representations of The notion of automorphic form has a simple interpretation in terms of representations (which in fact sug gested its present form). To give it, we need the following known lemma (cf. [18] for the terminology).
(n, V)
be s finisible.te and )-fiLetnite Thenbethae diff smalerenti lest ablK)-e represent submodulateioofn of G(R). containLeting is admi G(Rt G(R) Jf'oJf'o G(Rt, 2. 1 . LEMMA. Z(g
.
(g ,
V
ve V v
K
Indeed, £' · v is a finite sum of spaces · w, where £'0 is the Heeke algebra of the identity component of and K0 = K n and w is K0-finite and · v is an admissible (g , K0)-module. Z(g)-finite. It suffices therefore to show that By assumption, there exist an ideal R of finite codimension of the enveloping algebra U(t) of the Lie algebra t of K and an ideal J of finite codimension of Z(g) which annihilate v and moreover U(t)/R is a semisimple t-module. Then £'0 v may be identified with U(g)/ U(g) · R · J. By a theorem of Harish-Chandra (see [19, 2.2. 1 . 1]), U(g)/ U( g) · R is t-semisimple and its t-isotypic submodules are finitely generated Z(g)-modules. Hence their quotients by J are finite dimensional. 2.2. We apply this to acted upon by via right translations. Therefore, if f is automorphic form, then f * £' is an admissible £'- or (g , K) module. This module consists of automorphic forms. In fact, 1 .3(a) is clear, and 1 . 3(b) follows from 2. 1 ; its elements are annihilated by the same ideal of Z(g) as v, whence (d). Finally, there exists a e such that / * a = f so tha t / * satis fies 1 .2(c) (with the same exponent as f) for all X e U(g) [11, Lemma 1 4]. Thus the spaces •
coo(F\G(R)),
C�(G)
G(R)
X
d (F, J, K) = Ze d(F, t;, J, K), are (g , K)-modules and unions of admissible (g , K)-modules. If f is a cusp form, then f * £' consists of cusp forms. Thus the subspace d(F, K) of cusp forms is also an £'-module. If X is a quasi-character of Z, then the space d(F, K)x of eigenfunctions for Z with character X is a direct sum of irreducible admissible (g , K)-modules, with finite multiplicities. In fact, after a twist by l x l - 1 , we may assume X to be unitary, and we are reduced to the Gelfand Piatetski-Shapiro theorem ([7], see also [11, Theorem 2], [13, pp. 41-42]) once
o
o
1 94
A.
BOREL AND H. JACQUET
we identify o d(F, K) to the space of K-finite and Z(g)-finite elements in the space
x o L 2 (F\G(R)) x of cuspidal functions in L2 (F\G(R)) x (see 1 . 8 for the latter).
3. Some notation. We fix some notation and conventions for the rest of this paper. 3 . 1 . F is a global field, OF the ring of integers of F, V or VF (resp. Voo, resp. V1) the set of places (resp. archimedean places, resp. nonarchimedean places) of F, F. the completion of F at v E V, 0 v the ring of integers of F. if v E V 1. As usual, A or AF (resp. A1) is the ring of adeles (resp. finite adeles) of F. 3.2. G is a connected reductive group over F, Z the greatest F-split torus of the center of G, £'. the Heeke algebra of G. = G(F.) (v E V) [4]. Thus £'. is of the type considered in § I if v E Voo and is the convolution algebra of locally constant com pactly supported functions on G(F.) if v E V 1. We set
£' = £-'oo ® £'1, £'1 = ® £' v , v eV1 where the second tensor product is the restricted tensor product with respect to a suitable family of idempotents [4]. Thus £' is the global Heeke algebra of G(A) [4]. If F is a function field, then V00 is empty and £' = £' 1. If L is a compact open subgroup of G(A1), we denote by eL the associated idem potent, i.e., the characteristic function of L divided by the volume of L (relative to the Haar measure underlying the definition of Yf1). Thus f eL = f if and only iff is right invariant under L. The right translation by x E G(A) on G(A), or on functions on G(A), is denoted r" or r(x). 3.3. A continuous (resp. measurable) function on G(A) is cuspidal if (1)
Jf"oo = @V £'., vE
oa
*
J
N (F) \N (A)
f(nx) dn = 0,
for all (resp. almost all) x E G(A), where is the unipotent radical of any proper parabolic F-subgroup of G. It suffices to cheek this condition when runs through a set of representatives of the conjugacy classes of proper maximal par abolic F-subgroups.
P
N
4. Groups over number fields. 4. 1 . In this section, F is a number field. An element e if it is of the form
P
E
£' is said to be
� � E £'1, �oo idempotent in £'oo•
(1)
simple
We let Goo = Ti v E V oo G. and goo be the Lie algebra of Goo, viewed as a real Lie group. We recall that Goo may be viewed canonically as the group of real points H(R) of a connected reductive group H, namely the group H = RFIQ G obtained from G by restriction of scalars from F to Q. This identification is understood when we apply the results and definitions of §§ 1 , 2 to GooThe group G(A) is the direct product of Goo by G(A1). A complex valued function on G(A) is if it is continuous and, if viewed as a function of two arguments x E Goo, y E G(A1), it is c oo in x (resp. locally constant in y) for fixed y (resp. x). 4.2. Fix a maximal compact subgroup Koo of G00• A smooth function / on G(A) is a K00-automorphic form on G(A) if it satisfies the following conditions :
smooth Automorphic forms.
195
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
x)
(a) /(r /(x) ( r E G(F), x E G(A) ). (b) There is a simple element � E .Y"e such that/ * .; = f (c) There is an ideal J of finite codimension of Z(goo) which annihilates [ (d) For each y E G(A1), the function x >-+ f(x · y) on Goo i s slowly increasing. We shall sometimes say that f is then of type (.;, J, K00). We let d(.;, J, Koo) be the space of automorphic forms of type (.; , J, K00). There exists a compact open subgroup L of G(A1) such that .;1 * .;L = .;1. Then d(.;, J, Koo) c d (.;oo * .;L, J, K00). We could therefore assume .;1 = .;L for some L without any real loss of generality. 4.3. We want now to relate these automorphic forms to automorphic forms on G=. For this we may (and do) assume .; = .; (8) �L for some compact open sub group L of G( A 1) . There exists a finite set C c G(A) such that G( A) = G(F) · C · G • [1]. We assume that C is a set of representatives of such cosets and is contained in G(A1). Then the sets G(F) · c · G= · form a partition of G(A) into open sets. For c E C, let ·
=
=
00
L
L
(1)
I t is a n arithmetic subgroup o f G00 • Given a function f on G(A) and c E C, let fc be the function x >-+ f(c · on Goo . Suppose f is right invariant under Then it is immediately checked that f is left invariant under G(F) if and only if/c is left invariant under Fe for every c E C. More precisely, the map f >-+ (/Jc E C yields a bijection between the spaces of functions on G(F)\G(A)/L and on llc(Fc\Goo). It then follows from the definitions that it also induces an isomorphism
x)
L.
d(.; oo (8) .;L , J, Koo) ----=::.... cEB d(Fc , � oo ' J, Koo) , EC
(2)
so that the results mentioned in § I transcribe immediately to properties of adelic automorphic forms. In particular, 1 . 6 , 1 .7 imply : (i) The space cc1"(�, J, K=) is finite dimensional. (ii) A smooth function / on G(A) satisfying 4.2(a), (b), (c), and f(z · x)
(3)
=
x(z) ·f(x)
(z E Z(A), x E G(A)),
for some character X of Z(A)/Z(F), such that 1/1 E LP(Z(A)G(F)\G(A)) for some ;;;; is slowly increasing. (iii) For a smooth function satisfying 4.2(a), (b), (c), the growth condition 4. 2(d) is equivalent to 1 .6(d'), where R is now a compact subset of G(A). (iv) Any automorphic form is C(G(A))-finite, where C(G(A)) is the center of
p
I,
G(A) .
We note also that one can also define directly slowly increasing functions on --+ GLn with finite
G(A), as in 1 .2, using adelic norms : given an F-morphism G kernel define, for x E G(A),
(4)
( or simpl y max lo- (g0) 1 v if o-(G) is closed as a subset of the space of n
x n matrices) . For continuous functions satisfying 4.2(a), (b), this is equivalent to 4.2(d).
196
A. BOREL AND H. JACQUET
4.4. A is a cuspidal automorphic form . We let o d(�. J, Koo) be the space of cusp forms of type (�, J, Koo). The group N is unipotent, hence satisfies strong approximation, i.e., we have for compact open subgroup Q of N(A1)
any
(2)
cuspform
N(A)
=
)
N(F) · Noo · Q ; hence N(F)\N(A � (N(F) n (Noo
G(A)
X
Q))\Noo
[1]. Let now f be a continuous function on which is left invariant under G(F) and right invariant under From (2) it follows by elementary computations that f is cuspidal if and only if the fc's (notation of 4.3) are cuspidal on Goo- Hence, the isomorphism of 4.3(2) induces an isomorphism
L.
(3) so that the results of 1 . 8 extend to adelic cusp forms. In particular, assume that / satisfies 4.2(a), (b), (c), and also 4.3(3) for a character X of Z(A)/Z(F). Then the following conditions are equivalent : (i) fis slowly increasing, i.e. , / is a cusp form ; (ii) is bounded ; (iii) is square-integrable modulo Z(A) · G(F). REMARK. In 4.3, 4.4, we have reduced statements on adelic automorphic forms to the corresponding ones for automorphic forms on G(R), chiefly for the conven ience of references. However, it is also possible to prove them directly in the adelic framework, and then deduce the results at infinity as corollaries via 4.3(2), 4.4(3). In particular, as in 1 . 8, one proves using (ii) above and Godement's lemma that o d(�, J, Koo) is finite dimensional.
fI f I
P ROPOSITION.
Let ftiobens aaresmootequihvfunct ion on satisfying Then(I)fithes folan lautomor owing condi al e nt: pe hiplacec form.v, the space ! * £. is an admissible £.-module. For each i n fini t ForTheeach e space! £. is an admissible £.-module. / * v is anthadmi spaceplace ssible* £-module. 4.5. (2) (3) (4)
G(A)
4.2(a), (b), (d).
E V,
.Yt'
PROOF. The implications (4) => (3) => (2) => ( 1 ) are obvious ; (1) => (4) follows from 4. 3(i). 4.6. An irreducible representation of is (resp. if it is isomorphic to a subquotient of a representation of .Yt' in the space of automorphic (resp. cusp) forms on G(A). It follows from 4.5 that such a representation is always admissible. It will also be called an automorphic representation of G or G(A), although, strictly speaking, it is not a G(A)-module. However, it is always a G(A1)-module. More generally, a topological G(A)-module E will be said to be if the subspace of admissible vectors in E is an automorphic representation of .Yt'. In particular, if X is a character of Z(A)/Z(F), any G-invariant irreducible closed subspace of
phic Automor cuspipdhial)c representations.
.Yt' automor
automorphic
V(G(F)\G(A)) x = {I E V (G(F) ·
Z(A)\G(A)) , ,f(z x(z) z · X)
=
-f(x), E Z(A), x E G(A))}
is automorphic in this sense, in view of [4, Theorem 4]. By a theorem of Gelfand and
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
1 97
Piatetski-Shapiro [7] (see also [8]), the subspace o V(G(F)\G(A)) x of cuspidal func tions of £2( G(F )\G(A)) x is a discrete sum with finite multiplicities of closed irreduci ble invariant subspaces. Those give then, up to isomorphisms, all cuspidal auto morphic representations in which Z(A) has character X · The admissible vectors in those subspaces are all the cusp forms satisfying 4.3(3). 4.7. Let L ::J L' be compact open subgroups of G(A1). Then �L �L ' = �L ; hence d(�oo ® �L ; J, Koo) c d(�oo ® �L' • J, Koo). The space ..91(�00, J, Koo) spanned by all automorphic forms of types (,;:00 ® ,;=1, J, Koo), with �� arbitrary in Jlt'1, may then be identified to the inductive limit d(,;=oo , J, Koo) = ind l i m d(�oo ® �L• J, Koo) , (I)
*
L
where L runs through the compact open subgroups of G(A1), the inductive limit being taken with respect to the above inclusions. The group G(A1) operates on Jlt'1 by inner automorphisms. Let us denote by x,;= the transform of � E Jlt'1 by Int x. We have in particular (2)
(x e G(A1), ,;=L as in 4. 1 ) ;
if/ is a continuous (or measurable) function on G(A), then (3) Therefore, G(A1) operates on ..91(�00' J, Koo) by right translations. It follows from 4.3(i) that this representation is admissible. In view of 4.3(3), ..91(�00, J, Koo) is the adelic analogue of the space d(Z, ,;=00, J, Koo) of 1 .9, where Z is the family of arithmetic subgroups of G(F) of the form rL = G(F) n (Goo x L), where L is a compact open subgroup of G(A1). These are the of G(F), i.e., those subgroups which, for an embed ding G 4 GLn over F, contain a congruence subgroup of G n GLn(OF). This analogy can be made more precise when G satisfies strong approximation, which is the case in particular when G is semisimple, simply connected, almost simple over F, and Goo is noncompact [16]. In that case, as recalled in 4.4(2), we have G(A) = G(F) · Goo · L for any compact open subgroup of G(A1), so that we may take C = { I } i n 4.3. Then 4.3(2) provides an isomorphism
congruence arithmetic subgroups
(4) for any L, whence (5) where Z is the set of congruence arithmetic subgroups of G(F). Moreover, the projection of G(F) in G(A1) is dense in G(A1) and G(A1) may be identified to the completion G(Fh of G(F) with respect to the topology defined by the subgroups n. It is easily seen that the isomorphism (5) commutes with G(Q), where, on the left-hand side x E G(Q) acts as in 1 .9, via left translations, and, on the right-hand side, x acts as an element of G(A1) by right translations. It follows that the isomor phism (5) commutes with the actions of G(F)z = G(A1) defined here and in 1 .9. Also, the isomorphism G(F)z ...:::. G(A1) induces one of the Heeke algebra Jlt'( G(F)z) (see 1 . 1 0) onto Jlt'1 and, again, (5) is compatible with the actions defined here and in 1 . 1 0. Note also that FL is dense in L, by strong approximation, so that this
198
A.
BOREL AND
H.
JACQUET
isomorphism of Heeke algebras ind uces one of .Yt' G(F) x , fL) , which is equal to .Yt' ( G(F), FL), onto ( G(A1) , [Strictly speaking, this applies at first for F = Q, but we can reduce the general case to that one, if we replace G by RF1Q G (4. 1 ) .] In the general case, the isomorphisms 4.3(2) for various �L are compatible with the action of G(F) defined here and in 1 .9 respectively, and this extends by continu ity to the closure in G(A 1) of the projection of G(F). 4. 8. Let d(J, K=) be the span of the spaces d(�=• J, K=) and d(K=) the span of the d(J, K=). These spaces are £-modules, and union of admissible .Yt'-submo dules. When 4.6(5) holds, they are isomorphic to the spaces d(Z, J, K=) and d(Z, K=) of automorphic forms on G(R) defined in 1 . 1 0. Otherwise, the relation ship is more complex, and would have to be expressed by means of the isomor phisms 4. 3(2).
.Yt'
(
L).
5. Groups over function fields. In this section, F is a function field of one variable over a finite field. A function on G(A) is said to be smooth if it is locally constant. 5. 1 . Let X be a quasi-character of Z(A)/Z(F) and K an open subgroup of G(A). We let 0i""( x, K) be the space of complex valued functions on G(A) which are right invariant under K, left invariant under G(F), satisfy
f(z · X) = x(z) ·
(1)
f(x)
(z E Z(A),
x
E G(A)),
and are cuspidal (3 .3). [These functions are cusp forms, in a sense to be defined below (5.7), but the latter notion is slightly more general.] We need the following : X
Let andof be given.hasThensupportthereinexists a compact subset o f such that every element In particular, isfinite dimensional. 5.2. PROPOSITION (G. HARDER). C G(A) 0i""( x, K)
K
0i""(X , K)
Z(A) · G(F) · C.
This follows from Corollary 1 .2.3 in [10] when G is split over F and semisimple (the latter restriction because ( I ) is not the condition imposed in [10] with respect to the center). However, since G(A) can be covered by finitely many Siegel sets, the argument is general (see [9, p. 1 42] for GLn).
X
Letthe space be a finite setm,of quasiof-cuspi charactdalefunct rs of ions which areandrimghta posi t i v e i n t e ger. Then invariant under and satisfy the condition is finite dimensional. m, m, s, s s , q: (j 5.3. CoROLLARY.
0i""(X,
Z(A)/Z(F)
K)
K
IT (r (z) - x
(1)
'/. E X
PROOF. The space 0i""(X, K) is the direct sum of the spaces 0i"" ( {x}, K) ; hence we may assume that X consists of one quasi-character X · By 5.3, 0i"" (X, 1 , K) is finite dimensional. We then proceed by induction and assume that 0i"" (X, K) is finite dimensional for some � 1 . The group Z(A)/Z(F) · K , where K ' = Z(A) n K, i s finitely generated. Let (zi) t &.i&. q be a generating set. Set A s+t, o = 0i"" (X, + 1 , K) and, for t = I , . . . '
As + l.t = {I E A s+l.o i (r (zi) - x(zi)) • ·f = 0
Then
= 1 , . . . , t)} .
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS As+l . O
=>
A s+l , I
::>
::>
• • •
A s+I.t
) (r(a1+1 -
=> . . .
=>
A s+I. q
=
0.,V(X,
s,
1 99
K).
Fix (0 � < Then / �-+ x (at+I ) ·/ maps A s+l . t into A s+ I . t+ I and its kernel is contained in A s + l , t+ l · It follows, by descending induction on t, that As+I.O is finite dimensional. 5.4. REMARK. If H is a commutative group, let C[H] be its group algebra over C. Every complex representation W) of H extends canonically to one of C[H] . An element W is H-finite if and only if it i s annihilated by some ideal I of finite codimension of C[H]. If I is such an ideal, there exist a finite set of quasi-charac ters of H and a positive integer such that
t t q). wE
IT
(I)
(1t'(h) - x(h))m w ·
XEX
=
(1t', m
)
X
0, for all
hE
H and all
w w
annihilated by I.
If H is finitely generated, then, conversely, all elements E W satisfying (1) for all H are H-finite, and annihilated by some ideal of finite codimension of C[H]. Therefore, 5 . 3 implies that the space 0.,V(I, of cuspidal functions on G(F)\G A)/ which are annihilated by some ideal I of finite codimension of C[Z(A)/Z(F)] is finite dimensional. In fact, since Z A)/Z F) n Z(A)) is finitely generated, these two statements are equivalent. 5 . 5 . Let E be a local field, R a connected reductive group over E, .YI'R the Heeke algebra of R(E). A left ideal J of £'R is said to be if the natural repre sentation of £'R on £'RfJ is admissible.
hE
K)
( K
( ( (K admissible Letbe abeparaboli an admic £-ssisbubgroup le ideal ofof£' andtheKunia pcompact opencal ofsubgroup of Let P otent radi P and M aofLevi £-swiubgroup ofloP.winThere is any :admiif 1t'ssiisblae smooth ideal represent in the Heeke algebra on£' t h t h e fol g propert a t i o n of space(cf andis wanniE hilatiseK-jid byxed, annihilated by then the image of w in the space Ev v0 1t' w w. � E £' £' 1t'(�) ww., £' v0 Let KThen be a tcompact open subgroup of G(A). Let v E and an admi s si b l e i d eal of£ h e s p ace v K) of compl e x valued functi o ns onby isfiniwhitecdih mareensilefotnal.invariant under G(F), rig, ht invariant under K and annihilatefd LEMMA. R(E).
J
·
R H, N
JM
M(E) W, [3]) WN
J,
W
JM ·
M
w
a
R(E)
PROOF. Let q;0 .YI'R be the characteristic function of K, its image in .Yt'RfJ and v0 the image of 0 in (£'RfJ)N. The representation of R(E) on £'RfJ is admissible by assumption ; therefore the representation of M(E) on (.Yt'RfJ)N is admissible [3], and the annihilator JM of v0 in £'M is admissible. We claim that it has the required properties. In fact, if and are as in the lemma, then there is a unique R(E) It maps R onto a scalar multiple of morphism R --t W taking q;0 to Therefore, it factors through an R(E)-morphism RfJ --t W, mapping onto whence an M(E)-morphism (.Yt'RfJ)N --t WN mapping v0 onto w. It follows that w is annihilated by J M · ·
5.6. THEOREM. G(A) J
V
•.
.,V(G,
J
J,
PRooF. Since the representation of G. on .Yt'vfJ is admissible (and finitely gen erated), there exists an ideal I. of finite codimension of C[Z(F.)] which annihilates ./J.
£
200
A. BOREL AND H. JACQUET
Let J, Then fz ..... fz being the canonical involution of gives a G0-morphism Y'fvfJ --+ £0• Therefore J, is annihilated by · · Let = n and regard as a subgroup of Then in As a consequence, there exists an ideal of finite codi mension of which annihilates J, K). The space of cuspidal elements in J, is then contained in °-r(I, (notation of 5.4) , hence is finite dimensional. We now prove the theorem by induction on the F-rank rkFG' of the derived group of G. If rkFG' = 0, then J, = 0-r, and our assertion is already proved. So assume rkFG' and the theorem proved for groups of strictly smaller semisimple F-rank. Let now = · vary through a set fllJ of representatives of the conjugacy classes of proper parabolic F-subgroups of where is a Levi F-sub group and the unipotent radical of For each such let Cp be a set of repre sentatives of It is finite. The intersection of the kernels of the maps / �-> p, c , where fp,c(m) = fN (F) \ N (AJ i · · Cp, fllJ) J, i s then °-r, hence is finite dimensional. It suffices therefore to show that, for given P fllJ, Cp, the functions fP. c vary in a fixed finite dimensional space. After having replaced J and by conjugates, we may assume that = 1 . We write for fP. l · Let now Uc (resp. UM) be the space of functions on (resp. which are right invariant under some compact open subgroup (de pending on the function). The representation of by right translations on Uc is smooth. Ifx thenfp = J p ; hence pp : ..... p factors through ( Uc)N
f E -r(G, v, K). £0) /0•Z' hasfiniZ' te Z(A) iZ(A)/Z(F) ndex KZ(A). -r(G, v, K) G' �P1 N P(A)\G(A)JK. f E cE K M(F)\M(A)) E N(A) K' K -r(M,
h ....f. f* * (h -r(G, v, K) Z(F0) Z(A). I Z(F) Z(Fv) o-r -r(G, v, K) -r(G, v, K) MN G,P, M P. (n m c) dn(cE PE (fe -r(G, v, K)) c G(F)\G(A) fp G(A) (r ) (fer -r{G, f fv, K)) M(F0). -r(G, v, K) M(A),
Letfbesomeafunct ion onopenG(A)subgroup which oifs G(A). left invariThenantthunder G(F) and ri g ht i n vari a nt under compact e fol l o wing conditiThere ons areis equi v alent: a pledacebyvr(G0ofF) -such that tshsiebrepresent ation of Gv on the G(F0)-invariant subspace generat f i s admi l e . (2) Forssibeachpl acevofF, the representation of Gv on the space generated by r(G0) f is admi l e . The representation ofG(A) on the space spanned by r(G(A)) fis admissible. (2) f * v G(A). K Kv Kv, K. KGv Kv G• v G(A) G•.v Kv K• ) K. E E g K, g * 5.7. COROLLARY.
(1)
•
(3)
·
PROOF. Clearly (3) => => ( 1 ) . Assume (1). Let be as in ( 1) . Then / is annihi lated by an admissible ideal J of £0• Let U = £. We have to prove that UK is finite dimensional for any compact open subgroup K of There is no harm in replacing by a smaller group, so we may assume that fixes f We may also assume that K = x is compact where is compact open in and open in the subgroup of elements in with v-component equal to 1 . We also have £ = £0 ® £ where Jfv is the Heeke algebra of Let �v (resp. �·) be the idempotents associated to (resp. (3.3). Then �v ® � = �K is the idempotent associated to Any element in U is a finite linear combination of elements of the form / * a * (3 (a Jfv , (3 £0). If such an element is fixed under then �K =
AUTOMORPHIC FORMS AND AUTOMORPHIC REPRESENTATIONS
20 1
hence we may assume that each summand is fixed under K, and that a * �· = a, = (3. Since / is fixed under K, it follows that f * a is fixed under K. The ele ments * a then belong to the space "f/" ( G, v, J, K), which is finite dimensional by the theorem. For each such element * a * f3 is contained in the space of K.-fixed vectors in the admissible .no.-module * a * .no. = * .no. * a, whence our asser tion. 5 . 8 . DEFINITIONS. An on G(A) is a function which is left in variant under G(F), right invariant under some compact open subgroup, and satis fies the equivalent conditions of 5. 7. A is a cuspidal automorphic form. Any automorphic form is Z(A)-finite (as follows from 5.7(3)) . An G(A) if it is isomorphic to a sub quotient of the G(A)-module .91 of all automorphic forms on G(F)\G(A). It follows from 5. 7 that it is always admissible. More generally, a topologically irreducible continuous representation of G(A) in a topological vector space is automorphic if the submodule of smooth vectors is automorphic. As in 4.6, it follows from [4, Theorem 4] that if X is a character of Z(A)/Z(F), then any G-invariant closed irreducible subspace of V(G(F)\G(A)) x is automorphic.
g;
f3 * �.
f
ff f automorphic form cusp form irreducible representation of is automorphic
Letfbe a function on Then thefollowing conditions are(1)equival e nt: f iiss aZ(A)-fini cuspform.te, cuspidal and right invariant under some compact open f subgroup of I 5.9. PROPOSITION.
G(F)\G(A).
(2)
(3 . 3),
G(A).
PROOF. That ( 1 ) � (2) is clear. Assume (2). Then /is annihilated by an ideal of finite codimension of C[Z(A)/Z(F)]. Let U be the space of functions spanned by r(G(A)) · f Every element of U is cuspidal, annihilated by /and right invariant under some compact open subgroup. If L is any compact open subgroup, then UL is con tained in the space 0"f/"{/, L) (notation of 5.4), hence is finite dimensional. Therefore U is an admissible G(A)-module and ( 1 ) holds. 5. 1 0. It also follows in the same way that the space o .91(1) (resp. o .91(X, of all cusp forms which are annihilated by an ideal of finite codimension of (resp. which satisfy 5.3( 1 )) is an admissible G(A)-module. Moreover, if X consists of one element X• and if = in which case we put .91(X, m) = o .91 x• then this space is a direct sum of irreducible admissible G(A)-modules, with finite multiplicities. To see this we may, after twisting with lxl- 1 , assume that X is unitary. Then, since o .91 x consists of elements with compact support modulo Z(A),
CfZ(A)/Z(F)]
I
m 1,
(f,
g)
=
J
Z (A) G (F) \G (A)
f(x) g(x) dx
m))
·
de fines a nondegenerate positive invariant hermitian form on o .91 x · Our assertion follows from this and admissibility. This is the counterpart over function fields of the Gelfand-Piatetski-Shapiro theorem (4.6). We note that, by [14], every automorphic representation transforming according to X is a constituent of a representation induced from a cuspidal automorphic representation of a Levi subgroup of some parabolic F-subgroup, for any global field F.
A. BOREL AND H. JACQUET
202
REFERENCES 1. A. Borel, Some finiteness theorems for adele groups over number fields, lnst. Hautes Etudes Sci . Pub! . Math. 16 (1 963), 1 0 1 - 1 26. 2. , Introduction to automorphic forms, Proc. Sympos. Pure Math . , vol. 9, Amer. Math . Soc . , Providence, R . I . , 1 966, pp. 1 99-210. 3. P. Cartier, Representations of reductive P-adic groups : A survey, these PROCEEDINGS, part 1 , pp. 1 1 1 - 1 5 5 . 4 . D . Flath, Decomposition of representations into tensor products , these PROCEFDINGS, part 1 , pp. 1 79- 1 8 3 . 5 . S . Gel bart, Automorphic forms o n adele groups, Ann . o f Math. Studies, n o . 8 3 , Princeton Univ. Press, Princeton, N.J., 1 975. 6. I . M . Gelfand, M . I . Graev and I . I . Piatetski-Shapiro, Representation theory and automorphic functions, Saunders, 1 969. 7. I . M . Gelfand and I. Piatetsk i-Shapiro, A utomorphic functions and representation theory, Trudy Moscov. Mat. Obsc. 12 ( 1 963), 3 89-4 1 2 . 8. R . Godement, The spectral decomposition of cusp forms, Proc. Sympos. Pure Math . , vol . 9 , Amer. Math. Soc . , Providence, R . 1 . , 1 966, p p . 225-234. 9. R. Godement and H . Jacquet, Zeta functions of simple algebras, Lecture Notes i n Math . , vol. 260, Springer-Verlag, New York, 1 972. 1 0. G . Harder, Cheval/ey groups over function fields and automorphic forms, Ann . of Math. (2) 100 ( 1 974), 249-306. 1 1 . Harish-Chandra, A utomorphic forms on semi-simple Lie groups, Notes by J. G. M . Mars, Lecture Notes i n Math . , vol. 68, Springer-Verlag, New York, 1 968. 12. M . Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties, Mimeo graphed Notes, Univ. of Chicago, 1 963-64. 13. R . P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol . 544, Springer, New York, 1 976. 14. On the notion of an automorphic representation, these PROCEEDINGS, part 1, pp. 203207. 15. I . P iatetski-Shapiro, Classical and adelic automorphic forms . An introduction, these PROCEED INGS, part 1 , pp. 1 8 5-1 8 8 . 16. V . I . Platonov, The problem of s trong approximation and the Kneser- Tits conjecture, lzv. Akad. Nauk SSSR Ser. Mat. 3 ( 1 969), 1 1 39-1 1 47 ; Addendum, ibid. 4 ( 1 970), 784-786. 17. G . Shimura, Introduction to the arithmetic theory of automorphic forms, Pub!. Math. Soc. Japan 11, I . Shoten and Princeton Univ. Press, Princeton , N.J., 1 9 7 1 . 18. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1 , pp. 7 1 -86. 19. G . Warner, Harmonic analysis on semi-simple Lie groups. I , Grundlehren der Math. Wis senschaften, Band 1 88, Springer, Berlin, 1 972. --
-- ,
INSTITUTE FOR ADV ANCED STUDY
COLUMBIA UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 1 , pp. 203-207
ON THE NOTION OF AN AUTOMORPHIC REPRESENTATION. A SUPPLEMENT TO THE PRECEDING PAPER
R. P. LANGLANDS The irreducible representations of a reductive group over a local field can be obtained from the square-integrable representations of Levi factors of parabolic subgroups by induction and formation of subquotients [2], [4]. Over a global field F the same process leads from the cuspidal representations, which are analogues of square-integrable representations, to all automorphic representations. Suppose is a parabolic subgroup of G with Levi factor and fJ = @fJv a cuspidal representation of Then Ind fJ = ®v Ind fJv is a representation of which may not be irreducible, and may not even have a finite composition series. As usual an irreducible subquotient of this representation is said to be a constituent of it. For almost all v, Ind fJv has exactly one constituent 11:� containing the trivial representation of G(O.). If lnd fJv acts on x. then 11:� can be obtained by taking the smallest G(F.)-invariant subspace v. of x. containing nonzero vectors fixed by G(O.) together with the largest G(F.)-invariant subspace u. of v. containing no such vectors and then letting G(F.) act on v.; u• .
G(A)
P
M
M(A).
is a constituentTheof constitanduents of forarealmostthe alrepresent l ations LEMMA 1.
lnd fJv
Ind fJ 1l:v = 11:�
v.
11:
= ®1rv
where
1l:v
That any such representation is a constituent is clear. Conversely let the con stituent 11: act on V/ U with 0 £ U £ V £ X = ®X• . Recall that to construct the tensor product one chooses a finite set of places S0 and for each v not in S0 a vector x� which is not zero and is fixed by G(O.). We can find a finite set of places S, containing S0, and a vector Xs in Xs = ®vES x. which are such that x = Xs ® (®voes x�) lies in V but not in U. Let Vs be the smallest subspace of Xs containing xs and invariant under Gs = 11 vES G(F.). There is clearly a surjective map Vs ® (®voes V.) ----> Vf U,
and if v0 ¢ S the kernel contains Vs ® U.0 ® (®voesu rvol V.). We obtain a surjec tion Vs ® (®vES v.; u.) -+ V/ U with a kernel of the form Us ® (®vES V./ U.), Us lying in V5. The representation of Gs on Vs/ Us is irreducible and, since Ind fJv has a finite composition series, of the form ®vES 1l:v, 1l:v being a constituent of lnd fJv· Thus 11: = ®1rv with 1l:v = 11:� when v ¢ S. AMS (MOS) subject classifications ( 1 970). Primary 30A58, 20E35, 1 0D20. © 1979, American Mathematical Society
203
R. P. LANGLANDS
204
The purpose of this supplement is to establish the following proposition.
an automorphic representation if and only if is a constituentrepresent of atiforon someof G(A)andis some PROPOSITION 2. A n
i nd (J
n
(J .
P
The proof that every constituent of Ind (J is an automorphic representation will invoke the theory of Eisenstein series, which has been fully developed only when the global field F has characteristic zero [3]. One can expect however that the analogous theory for global fields of positive characteristic will appear shortly, and so there is little point in making the restriction to characteristic zero explicit in the proposition. Besides, the proof that every automorphic representation is a constituent of some Ind (J does not involve the theory of Eisenstein series in any serious way. We begin by remarking some simple lemmas.
Let Z be the centre ofG. Then an automorphicform is Z(A)-.finite. Suppose isLeta maxibemaalparaboli compactcsubsubggrouproupofofG(A)G. andChooseangautomor pandhic form wi t h res p ect t o G(A) letP(A)K'onbeM(A). a maxiThen mal compact subgroup of M(A) containing the projection ofgKg-1 (m; g) J qJ(nmg) is an automorphicform on M(A) with respect to g) M qJ(g)cj;(g) dg G(A). M(A) g. g) for all then is zero. LEMMA 3.
This is verified in [1].
LEMMA 4.
qJ
K
K.
P
=
([Jp
N (F) \ N (A)
E
n
dn
K'.
It is clear that the growth conditions are hereditary and that ([Jp( ; is smooth and K'-finite. That it transforms under admissible representations of the local Heeke algebras of is a consequence of theorems in [2] and [4]. = 0 whenever We say that qJ is orthogonal to cusp forms if JoG , ,. CAl ¢ is a cusp form and [) is a compact set in If P is a parabolic subgroup we write ¢ j_ P if ([Jp( ; is orthogonal to cusp forms on for all We recall a simple lemma [3]. ·
·
LEMMA 5. If qJ
j_
P
P
qJ
We now set about proving that any automorphic representation n is a con stituent of some Ind (J. We may realize n on Vf U, where U, V are subspaces of the space .91 of automorphic forms and V is generated by a single automorphic form (/) · Totally order the conjugacy classes of parabolic subgroups in such a way that {P} < {P'} implies rank P � rank P' and rank P < rank P ' implies {P} < {P ' } . Given qJ let {PI' } be the minimum {P} for which { P } < {P '} implies qJ j_ P ' . Amongst all the qJ serving t o generate n choose one for which { P } = {PI' } i s minimal. If ¢ E .91 let cpp(g) = cpp( I , g) and consider the map ¢ -> cpp o n V. I f U and V had the same image we could realize n as a constituent of the kernel of the map. But this is incompatible with our choice of ([J, and hence if Up and Vp are the images of U and V we can realize n in the quotient Vpf UP · Let .91� be the space of smooth functions ¢ on satisfying (a) ¢ is K-finite.
N(A)P(F)\G(A)
AN AUTOMORPHIC REPRESENTATION
205
(b) For each the function --+ = is automorphic and cuspidal. Then Vp � sl�. Since there is no point in dragging the subscript P about, we change notation, letting 1C be realized on Vf U with U � V � sl�. We suppose that V is generated by a single function
g
m cf;(m, g) cj;(mg) rp. We may(g)forso alchoose character ofLet A)besatithsfeyicentng rp(reaog)f M. x(a)q; l g E G(A)rp andand allthaatE thereA).is a P(A)\G(A)/Krp q; a. E C {l(a)rpla E A } a E A) ( a) ) ) ( a ) (( ) ) ( a ( (a) ) (/(a) - a)rp, /(a) q; rp {bE Bl l(b)rp {jrp, {3 E C}, rp M(A) rp l x a) , a eA(A) , M(F)\M(A), x) M(Q)\M(A) m E M(A) a dmA(A), cj;(ag) x(a)cf;(g). m cj;(m, g) grp( ·,g)G(A). g E G(A).cf;( ·,g) cf; ' ( ·,g) cf;'(g) cf;'(m, m1g). cf;'(g) cf;'(l, g). m1 E M(A),{cf;' lcf;cf;'(mmM(A)l > g) N(A) M(A) v-1 LEMMA 6. X
A
V
=
A(
A(
Since is finite, Lemma 3 implies that any in d� is A(A)-finite. Choose V and the generating it to be such that the dimension of the span Y of A( ) is minimal. Here /(a) is left translation by If this dimension is one the lemma is valid. Otherwise there is an A( and a such that 0 < dim I( - a Y < dim Y. There are two possibilities. Either / a - a U = /( ) - a V or /( ) - a U # / - a V. In the second case we may replace rp by contradicting our choice. In the first we can realize 1C as a subquotient of the kernel of - a in V. What we do then is choose a lattice B in A(A) such that BA(F) is closed and BA(F)\A(A) is compact. Amongst all those and V for which Y has the minimal possible dimension we choose one for which the subgroup of B, defined as = has maximal rank. What we conclude from the previous paragraph is that this rank must be that of B. Since is invariant under A(F) and BA(F)\A(A) is compact, we conclude that Y must now have dimension one. The lemma follows. Choosing such a and V we let v be that positive character of which satis fies
v(a)
=
( j
and introduce the Hilbert space L� = L� ( of all measurable functions cf; on satisfying : (i) For all and all e = < 00 . (ii) JA (A ) M(Q) \M ( A) v -Z(m)jcp(m)j 2 L� is a direct sum of irreducible invariant subspaces, and if cf; E V then --+ lies in L� for all e Choose some irreducible component H of L� on which the projection of is not zero for some For each cb in V define to be the projection of on H. It is easily Thus we may define by seen that, for all = = If V' = e V}, then we realize 1C as a quotient of V'. However if o2 is the modular function for on and (J the representation of (J. on H then V' is contained in the space of lnd To prove the converse, and thereby complete the proof of the proposition, we exploit the analytic continuation of the Eisenstein series associated to cusp forms. Suppose 1C is a representation of the global Heeke algebra .Yt', defined with respect to some maximal compact subgroup K of Choose an irreducible representa tion ,. of wh i ch is contained in 1e . If E. is the idempotent defined by let .Yt'k = E,.Yt'E. and let 1e. be the irreducible representation of .Yt'. on the IC-isotypical sub space of 1C. To show that 1C is an automorphic representation, it is sufficient to show that 11:. is a constituent of the representation of .Yt'. on the space of automorphic
K
G(A).
K
206
R. P.
LANGLANDS
forms of type " · To lighten the burden on the notation, we henceforth denote 1T: K by 7T: and £K by £. Suppose P and the cuspidal representation u of are given. Let L be the lattice of rational characters of M defined over F and let Lc = L Each ele ment p. of Lc defines a character �I' of Let be the IC-isotypical subspace of Ind �l'u and let = I0 . We want to show that if 1T: is a constituent of the representa tion on I then 1T: is a constituent of the representation of £> on the space of au tomorphic forms of type '" · If {g;} is a set of coset representations for then we may identify II' with by means of the map rp -> rpl' with
M(A). II'
I
I
M(A)
® C.
P(A)\G(A)/K
rpl'(nmg;
k)
= �l'(m) rp(nmg;k).
In other words we have a trivial ization of the bundle {II'} over Lc, and we may speak of a holomorphic section or of a section vanishing at p. = 0 to a certain order. These notions do not depend on the choice of the g;, although the trivialisa tion does. There is a neighbourhood U of p. = 0 and a finite set of hyperplanes passing through U so that for p. in the complement of these hyperplanes in U the Eisenstein series E(rp) is defined for rp in To make things simpler we may even multiply E by a product of linear functions and assume that it is defined on all of U. Since it is only the modified function that we shall use, we may denote it by E, although it is no longer the true Eisenstein series. It takes values on the space of automorphic forms and thus E(rp) is a function g -> E(g, rp) on It satisfies
Iw
G(A).
E(pl'(h)rp) = r(h)E(rp)
if h E £> and pi' is Ind �l'u. In addition, if rpl' is a holomorphic section of {II'} in a neighbourhood of 0 then E(g, rpl') is holomorphic in p. for each g, and the deriva tives of E(rpl'), taken pointwise, continue to be in d. Let I, be the space of germs of degree r at p. = 0 of holomorphic sections of I. Then rpl' -> pl'(h)rpl' defines an action of £> on I,. If s � r the natural map -> I5 i s an £-homomorphism. Denote its kernel by I:. Certainly I0 = I. Choosing a basis for the linear forms on Lc we may consider power series with values in the ,._ isotypical subspace of d, I: lal � r p.a c/Ja · £> acts by right translation in this space and the representation so obtained is, of course, a direct sum of that on the K-isotypical automorphic forms. Moreover rpl' -> E(rpl') defines an £-homomorphism ). from to this space. To complete the proof of the proposition all one needs is the Jordan-Holder theorem and the following lemma.
I,
I,
LEMMA 7. For r sufficiently large the kernel of ). is contained in I�.
Since we are dealing with Eisenstein series associated to a fixed P and u we may replace E by the sum of its constant terms for the parabolic associated to P, modify ing ). accordingly. All of these constant terms vanish identically if and only if E itself does. If Qr. . . . , Qm is a set of representatives for the classes of parabolics as sociated to P let E;(rp) be the constant term along Q;. We may suppose that is a Levi factor of each Q;. Define v(m) for m E by �l'(m) = e . Thus v(m) lies in the dual of LR. If rp E II' , the function E;(rp) has the form
M(A)
M
AN AUTOMORPHIC REPRESENTATION
E,-(nmgik, cp) =
b
a
I; I; Pa( v(m) )f;v�(p.) (m) c/Jafl(m,
207
k).
a=l {3=1 Here c/Ja{J lies in a finite-dimensional space independent of p. and v13 , l � � is a holomorphic function of p. ; and { Pa } is a basis for the polynomials of some degree. This representation may not be unique. The next lemma implies that we may shrink the open set U and then find a finite set h in such that (cpp.) is 0 for p. E U, (/)p. E /" if and only if the numbers 1 � � l � � are all 0.
gi; {j b h ...E,, -(h,hi, cp"G(A) Ej n ), i m, Let abebasia neis forghbourhood ofOainls of some givenholdegree. omorphicfunct ionsison t h e pol y nomi Then t h ere and neithenghbourhood ofO contained in and a finite set . in such that if ,I:p;(y) = for all y if and only if it is for y = y E E" Suppose Eis awhere holomorandphic functi ontei-ndimensia neionalghbourhood of Osuppose on with val u es in are fini s p aces, and tishatanalyt E" isicinnearjectifl.ve=on anandopenthe subset ger r such Tayloroserif esThen of E"tcph"erevaniissanhesintoteorder r thenthatcp0 if= = = s. E" r s = E" E s. 0, vh . . . , vk,
LEMMA 8 . U U, PI- . . . , Pa V
Yh . . , Yb
U
e"i Cp.) ·y
(*)
0
Yr.
. . .
0
p.
E
a
V
0
, Yb·
To prove this lemma one has only to observe that the analytic subset of U defined by the equations (*), E 0, is defined in a neighbourhood of O by a finite number of them. We may therefore regard as a function on U with values in the space of linear transformations from the space /, which is finite-dimensional, to the space Cm n. One knows from the theory of Eisenstein series that is injective for p. in an open subset of U. Then to complete the proof of the proposition, we need only verify the following lemma. LEMMA 9.
Hom(/, J),
I
U,
J U.
0
0,
(/)p.
0.
Projecting to a quotient of J, we may assume that dim I dim J and even that have I J. Let the first nonzero term of the power-series expansion of det degree Then we will show that can be taken to be + l . It is enough to verify this for I 1 , for we can restrict to a line on which the leading term of still has degree s. But then multiplying fore and aft by nonsingular matrices we may sup pose it is diagonal with entries z a , 0 � a � Then the assertion is obvious. In conclusion I would like to thank P. Deligne, who drew my attention to a blun der in the first version of the paper. REFERENCES
1. A. Borel and H. Jacquet, Automorphic forms and automorphic representations, these PRo CEEDINGS, part 1, pp. 1 89-202. 2. P. Cartier, Representations of p-adic groups : A survey, these PROCEEDINGS, part 1 , pp. 1 1 1 - 1 5 5 . 3 . R . P. Langlands, O n the functional equations satisfied by Eisenstein series, Lecture Notes i n Math . , vol . 544, Springer, New York, 1 976. 4. N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1, pp. 7 1 -86. THE INSTITUTE FOR ADVANCED STUDY
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part 1 , pp. 209-21 2
MULTIPLICITY ONE THEOREMS
I. PIATETSKI-SHAPIRO *
G k. Leticity ofbenanin itrhreduci bleofsmoot h admiisssiequal ble reto present a t i o n o f G(A). Then t h e mult i p l e space cus p forms one or zero. n n @pnp, Gp· np n and n nl. Let n @ ('i9 2, pisbea 1 2 p p p n , . for every p ¢: S, where S i r reduci b l e represent a t i o ns; suppose n two 1 2 p p finin2. pteforsetal, whil c(Hence h in casenn1 n2i.s) assumed to contain only finite places. Then n1 .p k (n, k G k H G k. x�(: 1 ;) G. W(n, (n, . -l 1. Let
=
GLn over a global field
( 1 ) MULTIPLICITY O N E THEOREM.
We shall discuss the following two results. 7C
(2) Recall that any irreducible admissible smooth representation can be written = where each is an irreducible admissible smooth representation of the local group STRONG MULTIPLICITY ONE THEOREM. p.
>
�
2
=
=
�
=
We begin by sketching the proof of the first Theorem ( 1 ) . The basic tool is the Whittaker model. We introduce this first in the case a local field, and V) an irreducible smooth representation. In the case of archimedean, we mean by "smooth representation" the representation of on the space V of Coo-vectors in some Hilbert space on which acts unitarily ; for nonarchimedean, this notion was introduced in Cartier's lectures. Let ¢ be an additive character of Let
be the standard maximal unipotent subgroup of Then a Whittaker model ¢) for V) is the image of V under an element of Homc ( V, lnd�(¢)) where ¢ (x) = ¢ (x l + . · + Xn ) if x2
X =
,
X* n- l . . !
0
More explicitly, it is given by a set of smooth functions { w. : G which (i) wv (xg) = s}(x) w.(g), for all X g E G.
E X,
--> C, E v
V} for
AMS (MOS) subject classifications ( 1 970) . Primary 22E5 5 ; Secondary 1 0020. *The author was s upported by an NSF grant while preparing this paper. © 1 9 7 9 , American Mathematical Society
209
I . PIA TETSKI-SHAPIRO
210
(ii) W:n: (h) v(g) = w.(gh), for all g, h E We have the following important result due to Gelfand-Kazhdan for the case k nonarchimedean, and Shalika for general local fields ([1], [2]).
G. ble admissible smooth representation V), there exists at most oneFor each irreduci (forfixed V V H I (E�x, .@x) 9 (:n:, V)(x, x). H V (.!Zlx, 9x), 9 UNIQUENESS THEOREM.
W(:n:, cp )
( :n:,
cp).
For k archimedean we assume that is a unitarizable representation and = {x E < oo 't;f E enveloping algebra} . Here means the comple tion of with respect to the inner product We assume also that W. ( l ) is a continuous linear functional on with respect to the topology defined by semi norms E enveloping algebra. Returning to the global case, we point out that the preceding discussion easily implies uniqueness of global Whittaker models (defined in the obvious way). 2. Global Fourier analysis. Let (:n:, Then we can define rp E
V.
J g
W10( ) =
V)
be admissible irreducible cuspidal as before,
Xk\XA
rp
(xg)cjJ-l(x) dx.
Global Fourier analysis says that this " Fourier transform" defines a cusp-form uniquely. In the classical setting this is due to Heeke ; for = 2 it is proved in Jacquet-Langlands [3] ; for > 2 it is due independently to Piatetski-Shapiro [4] and Shalika [2]. The proof is motivated by a corresponding result over a finite field due to S . I. Gelfand [5] It is now easy to see that these results imply Theorem ( 1), since
n
n
dim Homc( V, W(:n:, cjJ)) = I � dim Homc( V, La). We now turn to the proof of the strong multiplicity one theorem. First we dis cuss the case = 2 ; we need the following
n locals,. Then (:n:l > Vth1ere), (:n:exi2, sVt 2vl) twovhirreduciv2bsuch le admithatssible representations with WhiAssume t aker kmodel (� ?) W02(� ?) we assumeations.thatDenot (:n:1 , eHby1) andV1 (V( 22), tHhe2)setareofiralreduci bleh invectfiniotrse diinmHlensi(Honal2). Then unitaorrythCererepresent l smoot exist vl vl> v2 v2 such that k k C. k) v1 VI > vP.2 V2 SMALL LEMMA. ( I )
E
W. 1
v2
E
( W., E W(:n:;, .1)) .
=
(2) If k = R
:n:
E
E
Wv ,
E
W(:n:;, cp).
PROOF. For a local nonarchimedean field it is known that V contains al l Schwartz-Bruhat functions with compact support in k*. Hence we have what we want. Now let = R or The Kirillov theorem (see [8, p. 22 1 ]) says that each ir reducible infinite-dimensional unitary representation of GL(2, remains irreduci ble after restriction on the subgroup {(3 n} = p and hence as a representation of P is isomorphic to the standard representation of Hence, if rp (x) is a Coo-function e with compact support then there exist E such that
21 1
MULTIPLICITY ONE THEOREMS
(� �)
wv,
=
�(x).
REMARK. Assume that for a unitary representation with a Whittaker model the inner product can be written as an integral similar to the case for n = Using this result we can prove the "small lemma" for any n as we did for n = This implies the strong multiplicity one theorem for any n . Next we give the formula for recovering � from its Whittaker model due to Jac quet-Langlands, for
2.2.
GL(2, A):
n-1 > n-2
satisfy the hypotheses of the theorem. To prove the assertion, Now suppose it is enough to produce a nonzero � E V n V2 , since then the irreducibility of V,.) implies equality. Further, since A is dense in A • it is enough to pro duce two functions (nonzero) � .. E V,. which are equal on BA (as usual is the group of upper triangular matrices). From the properties of Whittaker models and ( * ), it is enough to produce Whit = taker functions wb w2 such that X E A * . One can suppose such are of the form IT P and then it suffices to construct the appropriate W,. at a finite number of places (by assumption). But then one can use the small lemma. This type of argument was found independently here by Shalika and in Moscow. For we need a similar small lemma (Gelfand-Kazhdan) : Suppose k local, nonarchimedean, V,.), i = I , 2, irreducible admissible representations with Whittaker models. There exist E V,. such that
(n-,.,
Bk\BI
Gk\G
B
W,. Wf WI ((� m W2((� �)), n � 3, (n-,., v,.
One can then employ induction using arguments similar to the case n = in order to prove the general case. It should be possible to prove this lemma also for k archimedean ; then the restriction we made that contains no infinite places could be removed . Now suppose G is quasi-split and satisfies the T( ) acts transitively on TI a a simple root Here T is a maximal k-torus in a Borel group, x: = Xa {I} where Xa is the root group associated to the simple root a. Define an automorphic cuspidal irreducible representation V) to be hyp (degenerate cuspidal) if
-
A
cuspidal
2,
S Xd'(A). transitivity condition: (n-,
er
for all � E V. Holomorphic cusp forms lifted from symmetric spaces which contain no copies of = Im z > 0} are of this type. A cuspida! automorphic form will be called if it is orthogonal to all hyper cuspidal automorphic forms (under the usual scalar product Jcck\ GA �¢ Counterexamples to the Ramanujan conjecture given during this conference by Howe and the author are hypercuspidal forms [6]. The author does not wish to kill
H {
generic
dg).
212
I . PIATETSKI-SHAPIRO
all belief in the Ramanujan conjecture ; he conjectures it to be true for the generic cuspidal automorphic irreducible representation. Now I shall sketch the proof of the multiplicity one theorem for generic cuspidal automorphic forms. First, the uniqueness theorem for local Whittaker models is true [2]. But of course now the Whittaker function does not define an arbitrary cusp form uniquely. Now it follows immediately from the definition that a generic cusp form is uniquely defined by its Whittaker function. This implies multiplicity 'one just as before. It can be proved that for each quasi-split reductive group there exist generic cusp forms. It also can be proved that for such groups there exists a unipotent subgroup U such that Juk1uA can be expressed in terms of the Whittaker function. Since for any group except GL(n) there exist hypercuspidal forms, we cannot of course expect to be able to recover itself from its Whittaker function. Details concerning this will be given in a forthcoming publication of Novodvorskii and the author. (Notes prepared by B. Seifert and L. Morris.)
rp(ug) du
rp
R EFERENCES 1. I. M. Gelfand and D. I. Kazhdan, Representation of the group GL(n, K), where K is a local field, Lie Groups and their Representations (Proc. Summer School of the Bolya-Janos Math. Soc., Budapest, 1 97 1 ) Halsted, New York, 1 975. 2, J . A . Shalika, The multiplicity one theorem /or GL(n), Ann. of Math. (2) 100 ( 1 974) . 3. H. Jacquet and R. P. Langlands, A utomorphic forms on GL(2), Lecture Notes in Math . , vol. 1 1 4, Springer-Verlag, New York, 1 970. 4. I . I. Piatetski-Shapiro, Euler subgroups, Lie Groups and their Representations (Proc. Sum mer School of the Bolya-Janos Math. Soc., B udapest, 1 97 1 ) Halsted, New York, 1 975. 5. S . I . Gelfand, Representations of the general linear group o ver a finite field, (Proc. Summer School of the Bolya-Janos Math. Soc. , Budapest, 1 97 1 ) Halsted, New York, 1 975. 6. R . Howe and I . Piatetski-Shapiro, A counterexample to the "generalized Ramanujan con jecture" for (quasi-) split groups, these PROCEEDINGS, part 1 , pp. 3 1 5-322. 7. R. P. Langlands, On the notion of an automorphic representation, these PROCEEDINGS, part 1 , pp. 203-207. 8. I. M. Gelfand, M. I . Graev and I . Piatetski· Shapi ro, Representation theory and automorphic functions, W. B. Saunders Company, 1 969. UNIVERSITY OF TEL AviV
Proceedings of Symposia in Pure Mathematics Vol . 33 ( 1 979), part I , pp. 21 3-251
FORMS OF
GL(2)
FROM THE ANALYTIC POINT OF VIEW
STEPHEN GELBART * AND HERV E JACQUET Introduction. Suppose is a reductive group defined over an A-field Pcusp is the representation of G(A) in the space of cusp forms, and rp is a well-behaved function on G(A). Then rp defines a trace-class operator Pcusp( rp , and the goal of the trace formula is to give an explicit formula for this trace. In most cases, one takes rp TI v iP v and wants to compute tr Pcusp( rp ) in terms of Although it is desirable to have invariant distributions, this is not necessary for all applications. In any case, contrary to common belief, the goal is to express the trace in terms of characters of irreducible representations. Indeed the distribution rp(e) always appears, and one is quite content to leave it as is. In the anisotropic case, Pcusp is actually an induced representation. In this case a general technique-valid for all induced representations and all cocompact groups gives the trace of Pcus/ rp) in terms of local orbital integrals. In general, Pcusp is a subrepresentation of an induced one p. Our purpose here is to describe the analytic theory of the trace formula for forms of GL(2). Since the trace of Pcusp(rp) is always computed as the integral of the kernel which defines it, we need an explicit formula for this kernel. For GL(2) this turns out to be the kernel of p(rp) minus the sum and integral of the product of two Eisenstein series. But since only the difference of these kernels is integrable not one or the other-we need to use a truncation process. For this we closely follow J. Arthur's exposition ([Ar 1] and [Ar 2]). Any improvement in clarity over earlier references (such as [DL], [JL], or [Ge]) should be credited to him. A large part of these notes-§§3 through 5-is devoted to the analytic continua tion of Eisenstein series. In particular, nearly complete proofs are given for the analytic continuation, functional equation, and location of poles of the of the Eisenstein series. This analysis enters precisely because we have to subtract off the continuous spectrum of p(rp) before we can compute its trace. § I is included to show how easy things are when there is no continuous spec trum . § 8 is included to give some idea of the power and range of applicability of the trace formula ; here discussions with D. Flath on his thesis [FI] were invaluable.
G
F,
butions. = not
only
) local invariant distri
constant
term
1. Division algebras. First we derive a statement of the trace formula for a division AMS (MOS) subject classifications ( 1 970) . Pri mary 22E5 5 ; Secondary I OD99. *Alfred P. Sloan Fellow. © 1979. American M athematical Society
213
STEPHEN GELBART AND HERVE JACQUET
214
algebra whose rank is the square of a prime. Then we give a simple, but in a sense typical, application. A. Let denote a number field and H a division algebra of degree over The multiplicative group G of H may be regarded as an F-group whose center Z is isomorphic to Set = GjZ. If is a character of G) the space of (classes of) functions on G(A) such that F<\Ax denote by
Some pZ generali H. ties. F Fx. G fw L2(w, f(rzg) w(z)f(g), r (F) z S - f(g) 2 dg L2(w, p, (z (g) z i p2, F. R v Fv Kv rpv(gv) v, !f! v !f!v(gv) w;1(zv) gv kvzv, kv Kv, Zv Zv, =
(1 . 1)
E
G
,
E
Z(A)
and
(1 . 2)
l
l
-
G (F) \G (A)
< + 00 .
Let denote the natural representation of G(A) in lation. Suppose now that rp is a function on G(A) satisfying
G) given by right trans
rp(zg) = w �I )rp
( 1 . 3)
for in Z(A). Suppose also that rp is c= and of compact support mod Z(A). More precisely, let C;, I � � be a basis of H over Then, for almost all v, the module generated over the ring of integers of In is a maximal order particular, = Ot is a maximal compact subgroup of H� for almost all v. We assume that rp(g) = TI where, for each is a c=-function of compact sup port satisfying the analogue of ( 1 . 3) ; moreover, for almost all v,
=
=
if
E
Ov .
E
otherwise.
0
Then the operator
is an integral operator in
When
G (A)
with kernel
= I:
(I . 5)
K(x, y)x
p ( ) s rp(g)p,(g) dg L2(G(F)\G(A)) K(x, y) rp(x-1ry). w rp =
( 1 .4)
r"'G (F)
and y lie in fixed compact sets the sum in (I . 5) is actually finite. Therefore is continuous.
The operator is of trace class. G(F)\G(A) K(x, y) pw(rp) m products ationngp,widecomposes irreducible representaThetionsrepresent each occurri th finite mulastipali(cdiitsycret. e) direct sum of p,(rp) THEOREM ( 1 .6).
p,(rp)
PROOF. Since is compact, and is continuous, is at least of Hilbert-Schmidt class. But rp can be written as 1: rpff.· with /,·, rp; satisfying the same conditions as rp (except at infinity where /, may only be of class Cm with large ; cf. [DL, p. 1 99]). Thus p,(rp) is a sum of of Hilbert-Schmidt opera tors, and hence is of trace class. CoROLLARY ( l . 7) .
PROOF. According to ( 1 .6) the operator
is compact for well-behaved rp.
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
215
Thus the corollary follows from a basic result o f functional analysis (cf. [GGPS] or [La 1]. If we denote by m (1e) the multiplicity (possibly zero) with which the irreducible representation of occurs in then we have
1C G(A)
p.,
( ) tr
p.,(
m 1e
=
tr
Note that each component of perforce admits as central character, i.e. , = Thus the sum in ( 1 . 8) need only be extended over repre sentations of with central character On the other hand, we can also compute the trace of in terms of the kernel viz.,
1e
=
tr
(1 .9)
_
G (F) \G (A)
B. From now on assume that p is a prime. Then any element of G(F) - Z(F) is regular. On the other hand, if = (in G (F) ) then = Thus we may rewrite the right-hand side of (1 .9) as
( 1 . 1 0)
I;
+
=
T *e
But every element � of G(F) - Z(F) generates an extension of F of degree p in M oreo er if e is conjugate to � in G(F), then the extension generated by e is F-isomorphic to and there is an F-isomorphism of onto taking to Thus, if we let X be a set of representatives for the isomorphism classes of ex tensions of degree p of F which imbed in any element #- of G (F) can be expressed in the form r =
where � is in for
f px -
for some
in X and
belongs to a set of representatives
�
Note that the extension ber of F-automorphisms of
is uniquely determined by r and, if is the num admits decompositions like ( * ) . Thus
T
and
S K(x, x) dx vol(G(F)\ G (A))
1
J
+
I;
I; €
UZ (A) \G (A)
In the last integral, the integrand depends only on the class Thus we also have +
=
(1 . 1 1)
I;
�E (U-P) /P
mod
I;
_
U (A) \G (A)
= tr
STEPHEN GELBART AND HERVE JACQUET
216
In this formula the measures on G (A) and P(A)\U(A) are chosen arbitrarily and U(A)\ G (A) is given the quotient measure. Suppose now that the measure on U(A)\ G (A) is a product measure ; given �. for almost all � is in Kv and rpv{x-;; 1 .;x.) = 0 unless x. E K.L�. Thus we have
v,
S rp(x- 1�x) dx
=
IT V
J.
c
L v\ v
rpv(x-;; 1 .;x.) dx.
almost all factors being equal to vol(L�\L�K.). Thus we have expressed tr Pw (rp) in terms of the local invariant distributions v 'P ...... rp .(e) and 'P v .__, fL�'Gv rp .(x-;; 1 .;x.) dx• . C. Let denote another division algebra of degree p 2 and sup pose H' and H both fail to split at the same set of places For S there is an isomorphism --> H� which is uniquely determined up to inner automorphism. For almost all we may suppose that this isomorphism takes a. to a similarly defined maximal order 0 � (and hence K. = 0� to K� = (O�Y) . The resulting iso morphisms G. --> G� then give rise to an isomorphism of the restricted product groups G S = IT v'¢. S Gv and G' S = IT v$ S G� which is again determined up to inner automorphism. Let V denote the space of functions on L2( w, G) which are invariant under Gs = IT ""' s • . Since G5 and Gs commute, V is invariant for the action of as. Thus we have a representation a of G5 on V and (similarly) a representation a' of G'5 on V'. Via the isomorphism a s --> G' s we may transport a' to a representation of as which we again denote by a'. (Its class does not depend on the choice of isomor phism.)
An application. H' H.v,
S. v f/:
G
equivalent.(1. 1 2). Suppose
THEOREM
are
mv
=
1
for all v S. Then the representations and a
E
a'
PROOF. Let (} = IT v fi- S (}" be any function on G5 satisfying conditions analogous to those satisfied by rp in § l .A. To prove our theorem it will suffice to show that
(1.13)
=
tr a((})
tr a'((}) .
Indeed a basic result in harmonic analysis asserts that a is a subrepresentation of ' if and only if
a
(1.14) (1.13) 0(1.13),
for sufficiently many "nice" f Cf. Lemma of [JL] ; it will suffice to apply with = O r * or and Ot (g) = O r (g- 1 ) . To prove extend (} to a function rp on G(A) by setting rp(g) = IT v fl-s (} (g.) and extend (} similarly to rp ' by identifying G' S with a s . Now take the volume of G . (resp. G �) to be one for each E S and the measure on G (A) (resp. G '(A)) to be a product measure. Furthermore, assume that the isomorphism G. --> G� takes the Haar measure of Gv to the Haar measure of a;,. Then we find that
(1. 1 5) (1.13) (1. 1 6) Thus
tr Pw (rp)
16.1.1
v
=
tr a(O)
(resp. tr p�(rp')
holds if and only if tr Pw (rp)
=
tr p�(rp ' ) .
=
tr a'(O)) .
FORMS OF
GL(2)
FROM THE ANALYTIC OINT OF VIEW
P
217
To prove ( 1 . 1 6) we apply the trace formula ( 1 . 1 1 ) as follows. Assume that the measure on P(A)\Lx(A) is a product measure so that
S
_
U (A) \G (A)
-
�( x 1 .;x) dx =
( IT J v<".S
_
L�\Gv
)
(),, (x- 1 .;x) dxv IT vol (L�\Gv) . veS
A similar identity holds for G ', and the extensions of F which imbed in are the same. Thus ( 1 . 1 1 ) implies that
H H' or
tr Pw (�) - tr Pw(�') = ( c - c') () (e) . Here c = vol( G ( F)\ G (A)) and c' is defined similarly. To prove the theorem it re mains to show c = c'. Suppose c > c' . If () = () 1 * () f with () f (g) = iMg- 1 ), then the right side of ( l . 1 7) is strictly positive. So by ( 1 . 1 5)-and the result described by ( 1 . 1 4)- 0'1 must be a subrepresentation of 0', and the quotient representation n = 0"/0'' must satisfy the identity ( 1 . 1 7)
for all nice 0 1 . This implies that the regular representation of GS is quasi-equivalent to n and hence that the regular representation of G5 decomposes discretely. Since the same is then also true of the regular representation of each G "' v rj: S, we obtain an obvi ous contradiction . The assumption c < yields a similar contradiction and the theorem is proved. REMARK ( 1 . 1 8). Suitably modified, the proof above shows that G and G ' have the same Tamagawa number. The restriction on w is not really necessary.
c'
2. Cusp forms on GL(2). Henceforth G will denote the group GL(2). The center Z of G is z
Again Z
� p:
=
{(g �)}.
and we set G = G/Z. We also introduce the subgroups
P
=
{(� :)}.
A =
{(� �)}.
N
=
{(b i )} .
If � is a function on G(F)\G(A) we say that � is cuspidal if
S
N (F) IN (A)
�(ng) dn
=
0,
gE
G(A) .
As in § 1 , we introduce a space V(w, G) and a representation Pw · We denote by La(w , G) the subspace of cuspidal elements in V(w, G) . THEOREM (2. 1 ) . Let Pw . 0 denote the restriction of Pw to the invariant subspace La(w, G). Let � denote a Coo-function which satisfies ( 1 . 3) an d is of compact support mod Z( A). Then the operator Pw . o(�) is Hilbert-Schmidt. SKETCH OF PROOF. For each c > 0, recall that a Siegel domain '1J in G(A) is a set of points of the form g
_- (0I I )( 0 O)k x
tab
a
STEPHEN GELBART AND HERVE JACQUET
218
where x is in a compact subset of A, t is an idele whose finite components are trivial and whose infinite components equal some fixed number u > c > 0, a is arbitrary in Ax, lies in a compact subset of Ax, and is in the standard maximal compact subgroup K of G(A). If '!J is sufficiently large then G(A) = G(F)'!J . For a proof of this fact see [Go 3]. By abuse of language, if '!J is a Siegel domain in G(A), we call its image in N(F)\G(A) a Siegel domain in N(F)\G(A). If '!J is such a domain, we denote by V(w, '!J) the space of functions on '!J such that f(zg) = w(z)f(g) and dg < + oo . Clearly V(w, G) can be identified with a closed SN (F) Z (A) \(§' 1 subspace V of V(w, '!J) if'!J is large enough. In this case there is a bounded operator with bounded inverse A from V(w, G) onto V. Iff is in V(w, G) and
b
k
f(g) 1 2
Pw(
with
J
_
N (F) \G (A)
H(x, y)f(y) dy
H(x, y) = I;
However, ifjis actually cuspidal, we also have Pw(
(2.2) with
JH(x, y)f(y) dy
H(x, y) = I;
(2. 3)
r;cN (F)
f
.;y).
-J
N (A)
Indeed cuspidal implies
S
_
N (F) \G (A)
f(y)
J
N (A)
To prove the theorem we need to estimate H(x, y) . If x is in a fixed Siegel domain '!J ( of N(F)\G(A)) it is easy to see that H(x, y) = 0 unless y is in another such domain '!J ' . Then using the fact that the term subtracted in (2.3) is precisely the Fourier transform of
Ss
(N (F) Z (A) \'!J) (N (F) Z (A) \'!1')
I H(x, y) dx dy < + oo ,
12
i.e., H defines a Hilbert-Schmidt operator B from V(w , 0'') to V(w, 0') . For a detailed discussion of this type of argument see [Go 1] or [La 1]. Now enlarge 0'' (if necessary) so that G(A) = G(F)'!J ' . Then Pw. 0 ( V(w, '!J'), the operator B, the projection of V(w, 0') onto V, and the operator A- 1 . This proves Pw. o (
is trace class. of
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
219
PROOF. The conclusion o f (2. 1 ) i s valid i f rp is highly (as opposed t o infinitely) differentiable at infinity. Thus one argues as in § LA (cf. [DL, p . l 99]).
The representation decomposes discretely with finite f f(nrzg) )f(g r P(F), f(rg) F(G)
CoROLLARY (2. 5).
multiplicities. 1)
Pw. o
3. P-series. Suppose is a function on G(A) satisfying
(3 .
with
n E
N(A),
=
P-H(serige)s.
a Let
(
and z E Z(A). Then we call the series
E
=
(3.2)
)
(JJ z
I;
P (F) \G (F)
To study the convergence of such a series we need a well-known lemma. be the function on G(A) defined by
H(g) I � I g (6 �)(� �)k. Fix a Siegc'elfordomai tive number Then the set ofr in G(F[JL,) suchP.197]).that H(rg) someng andisfinia posite modul o P(F). f {(g :H(g) c'}) F L5((J), The scalar product of two P-seri(1.1)es. : F (cf;, F) S- cj;(g) (g) dg/ (rg) dg J cf;(g) f (r dg f d ( r ( g) g) cf; g) (g) g J J cf; dg J cj;(ng)f (g) dg, J f d F) s g (g) g ¢ constant term ¢N(g) J cj;(ng) dn. (¢, F. =
=
if
CS
LEMMA (3. 3) (CF.
E
�
' c .
CS
� From this lemma we see that if the support of is contained in a set -in particular if it is compact mod N(A)Z(A)P(F)-then the series (3 . 2 is finite. Moreover, in the latter case, the function has compact support mod G(F)Z(A). Our goal in this section is to analyze the orthocomplement of G) in terms of these P-series. A. First we compute the scalar product of a P-series with a function ¢ satisfying =
G (F) \G (A)
=
G (F) \G (A)
=
=
-
I:
P (F) \G (F)
=
I:
P (F) N (A) Z (A) \G (A)
P (F) Z (A) IG (A)
N (F) \N (A)
t .e . ,
(cj; ,
(3 .4)
where (3 . 5)
N
is the
=
P (F) N (A) Z (A) \G (A) cj;
N(
)
of ¢ :
=
N (F) \N (A)
From (3.4) it follows that if ¢ is cuspidal then
F)
=
0 for any P-series
STEPHEN GELBART AND HERVE JACQUET
220
Conversely, if cp E 2 ( , is orthogonal to all P-series (with! of compact support mod then ¢ N = 0 and cp is cuspidal. Thus the P-series (with f of compact support mod . . . ) span a dense subspace of the orthocomplement of
Lij(w, F
N(A)P(F)Z(A)),L w G) G).
Now we compute the scalar product of two P-series. According to (3.4) we need first to compute the constant term of But by Bruhat ' s decomposition for GL(2), (g) = f(g) + L: rE N (F) f( w T g) with w = (_q A). Thus
F.
N(g =
F )
JN (F)\N (A)f(ng) dn + JN (F) \N (A) dn I; f(w r ng) .
Normalizing the Haar measure on we see that
by the condition vo i ( ( )
N F \N A))
N(g) = f(g) + f' (g)
where
F
(3.7)
f' (g)
(3.6)
Thus (3. 8)
N(A)
(Fh F
z) =
=
J.rl
(
=
1
JN (A)f(wng) dn.
· fz(g) dg +
Jf{
· fz(g) dg
P(F)N(A)Z(A)\G(A).
where both integrals extend over the space To further analyze formula (3 . 8) we need to carefully investigate the map .f ,..... f'. Note that.f' still satisfies (3. 1). Note also that all the computations above are merely formal unless certain assumptions onf1 and f2 are made to i nsure convergence. For instance, if bothf1 and f2 have compact support (mod . . . ) then these computations valid. Under these same conditions, however, the support off� need no longer (in general) be compact. Another formal relation which is easy to prove is this :
are
(3.9)
J f�(g)Jz(g) dg = Jft (g)J;(g) dg.
Indeed the left- (resp. right-) hand side of (3.9) may be written as
SP
(F)
( w Z (A) \G (A) ft( g)fz g) dg
(resp. S P ( F) Z (A) \G (A dt(g) fz( wg ) dg) and these integrals are equal since
P(F)Z(A). Induced representations and intertwining operators. Fd; u FD(A) 1 FD(A) F�). we assume that w is trivial on F�.
w
normalizes
To investigate the map B. f ,..... f ' (and the scalar product (3. 8)) we need to perform a Mellin transform on the group Z(A)A(F)\A(A). Right now it will suffice to deal with a subgroup of this group. Thus we let denote the group of ideles whose finite components all equal 1 and whose infinite components all equal some positive number (independent of the infinite place). By we denote the ideles of norm and by AO (resp. A�) the group of diagonal matrices with entries in (resp. For convenience We also normalize Haar measures as follows.
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
221
The Haar measure on Ft, is obtained by transporting the usual measure on Rt through the map t �---+ I t 1. Thus
2 1 . Jxx+•-zco:oo ds JF!J(t)j t j�s dxt � l
=
f(e).
Recall that the measure on N(A) is chosen so that vol(N(F)\N(A)) = On Ax we select any Haar measure and give Fx\A x the quotient measure. Then we use the isomorphism Fx\A x � Ft, x Fx\FO(A) to get a Haar measure on Fx\FO (A) (which in general will not have volume We also use the isomorphism (g �) ._. to get a measure on Z(A)\A(A). Finally we select measures on G(A) and K such that
1.
a
1).
1
(3 . 0)
1.
Note that vol (K) is not necessarily Now for each complex number s we introduce a Hilbert space H(s) of(classes of) functions on G(A) such that
rp
(3
. 1 1)
a, (3
a
) rp [(aau 0 (3av gJ x
u, v
=
l w(a) I u s 2 rp(g) v
+I
/
E Ft,, and x E A. Such functions are completely determined for E P, E A x , by their restriction to the set of matrices of the form
a E FO(A), k
(3. 1 2)
E
K.
rp to belong to H(s) we require that (3. 1 3) JK JPIFO(Al j rp j 2 [(g ?)k] da dk + oo . The restriction of rp to matrices of the form (3. 1 2) satisfies For
<
(3. 1 4)
a
each time E px and (3 �) E A(A) n K. Moreover, the group G(A) operates on H(s) by right translation and the resulting representation is denoted .,.,. This re presentation is continuous but not always unitary. It is, however, unitary when is purely imaginary. We may think of the collection as a holomorphic fibre bundle of base C. The sections over an open set U of C are the functions on G(A) x U satisfying
rp[(aau0
H(s)
,8av) sJ x
g,
=
w(a) ] i
rp
u
,s+ I I 2
v!
H(s) s)
rp( g, s)
s
rp [g, s] .
Of course this bundle i s trivial since every in is uniquely determined by its restriction to the set of matrices of the form (3. 1 2). Thus we may think of as the Hilbert space of functions (in dependent of satisfying (3. 1 3) and (3 . 1 4) .
H(s)
STEPHEN GELBART AND HERVE JACQUET
222
)s.
Put another way, we may set H = H(O). Then every element rp of H defines a sec tion of our bundle over namely (g, >-+ rp(g ) H(g With this understanding let f be a function satisfying (3 . l), say of compact sup port mod N(A)Z(A)P(F). Then define the ofj by
C,
s)
Mellin transform
Since the integrand has compact support, j clearly defines a section of our fibre bundle. The Haar measure is chosen so that
(g = f )
-2-. sxx+i-ioooo - s) ds 1
f(g,
TC l
with x any real number. Now it is easy to see that the first term of the inner product formula (3.8) is
i J J (P\FD(A)) J F;!: f1 z [(� �)k] \ t \-1 dxt da dk -= 2 � i s:::·: sP\/ FJK/ i{(� �)k, ]lz [(� �)k, -s] dadkds 1 . sxx+i-£oooo ( l(s), z( - ds. =_ 2 j/1 j1 s) (f1(s), rp21( rp1 rp2 F). N(A)Z(A)At,A( (3.15) f{Jz) JF<\FD(A) s f/J I rpz[(oaQJ)k] da dk H(s) s) (3. 1 6) 1.24- 1.26].) (2rci)-1 J:::.: (/�C - s), j2 ) ds. x j -s) j ·
K
s
s))
1l: l
(s) denotes the value of the function (g, at s and the scalar product - s) ) is taken by identifying all fibres with H. Alternatively, observe that if is in H(s) and rp2 is in H( - s) then transforms on the left according to the modular function of the group Thus if we set
Here
(rp l ,
=
K
·
we obtain a nondegenerate sesquilinear pairing between
and H( -
such that
This is the pairing which appears in our formula. (For more details see [Go 2, pp. Similarly we find that the second term of (3.8) is
(s )
Here we must integrate on a line where j{( - s) is given by a convergent integral, i.e., we must have > l It remains then to compute '( in terms of (s). For Re(s) > t we find that
j'( - s)
= Jr[(� ?)g] \ tl s-l!Z dxt = f \ t j s-1 1 2 dx t J![w(� �)(� t � x)g] d l
x
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW =
=
=
223
J j t j s+ l 1 2 dx t J ![(b ?)w(b I)g] dx I t j -s-1 /2 dxt J1[(� w I) dx ?) (b gJ S
J ![ w(b I) g, s] dx .
(Recall that w was assumed to be trivial on Fd;, ; therefore
to H(
M(s) H(s) -s) M(s)\l}(g) J \l}[wng] dn /'( -s) M(s)/(s). Summisupport) ng up, weisfindgiventhatby tthhee formul scalar product compact a of two P-series FI > F2 (with fi > of F2) s (!1. (s), . ds . s (M(s)f.i (s), f2(s)) ds. s) s) s)
for t E F�. ) If we define an operator
=
(3 . 1 7)
when Re(s)
from
>
by
N (A)
-!, we conclude that
=
(3. 1 8)
f2
(F�>
(3. 1 9)
=
/2( - s) )
1 2 71;{
1 + 2 1d
The integrals are taken over any vertical line with Re(s) > -! and the pairings we have written are on H( x H( - and H( - x H(s) (or simply H x H). It is worth noting that (3.20) for in and in H(s). (This is analogous to the identity (3.9) and proved just the same way.) If we think of as an operator from H to H then (3.20) simply asserts that
rp1 H(s) \1)2
(3. 2 1 )
M(s) M(s)* M(s). / (s)) Pw(g)F n,h(g)/(s)) . M(s)n.(g) n_,(g)M(s). M(s). with s purely imaginary. M(s).
Note finally that if/ is replaced b y (resp. right translate, i.e., by that (3.22)
=
f-+
/(hg) then F (resp. is replaced b y its On the other hand, it is also clear
=
These facts play a key role in the next section .
4. Analytic continuation of Our goal is to use the inner product formula (3 . 1 9) to construct an intertwining operator between a subrepresentation of Pw and a continuous sum of the representations 11: 5 To this end we need to analytically continue the operator Indeed the integration in (3. 1 9) is
STEPHEN GELBART AND HERVE JACQUET
224
over a line Re( ) = x with x > t. So if we want to rewrite this formula using purely imaginary we need to know something about the poles of M( ) . A. Now it will be more convenient to replace ns and M(s) with the operators we get by performing a Mellin transform on the whole group Fx \ " rather than just Fd;,. Let r; = (fl., denote a pair of quasi-characters of with = cv . Let H(r;) be the space of functions on G(A) such that
ss Prelimminary rearks. F"\A"A fJ.V (4. 1)
q;
and
s
[(g �) J = g
v) fJ.(a)v(b) I � 1
1 12
q;(g)
(4.2) Once more we can think of the collection H(r;) as a fibre-bundle over the space of pairs r;. These pairs make up a complex manifold of dimension with infinitely many connected components and our fibre-bundle is trivial over any such com ponent because the character (3 g) ,_. of n is fixed there. In par ticular, H(r;) may be regarded as the subspace of functions in such that
1
fJ.(a)v(b) A(A) K L2(K)
when (3 �) E K. In any case, we denote by n� the natural representation of G(A) on H(r;). The fiber decomposes as the sum of the fibres H(r;), with r; = (fl., and fl. - 1 ( ) I Is for a E Fd;, ; the representation ns decomposes correspondingly as the sum of the "principal series representations n�" ; see [Ge, p. 67] . As far as pairings are concerned, we have a natural one between H(r;) and H(7j- 1 ) defined by a
v a = H(s)a
v),
(q;l > q;z) = J q;1(k)ipz(k) dk. K
(n�(g)q;l >
As before, q;2) = (q; 1 , nii -t(g-1 ) q;2 ) . Moreover, if we set 7j = formula (3 . 1 7) defines an operator M(r;) from H(r;) to H(7j) satisfying (4. 3)
(v,
fl.), then
M(r;)n� (g) = n�(g) M(r;)
and (4.4) Here q; 1 i s in H(r;) and q;2 is in H(�- 1 ) ; cf. formulas (3.20) and (3 . 22) . In particular, if we identitfy H(r;) with and H(ij) with H (�- 1 ) (keeping in mind that r; and belong to the same connected component) then (4.4) reads
7]-1
(4. 5)
H(7j-1)
M(r;)
*= -
M(� 1 ).
Note that (3 .21) is essentially a direct sum of identities like (4. 5). To continue analytically M(r;) we are going to decompose it as a tensor product of local intertwining operators and thereby (essentially) reduce the problem to a local one. B. Let v be a place of F and F. the corresponding
Local intertwining operators.
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
225
local field . If 7Jv = (f.J.v• lJv) is a pair of quasi-characters of F� such that fl.v lJv = Wv we can form the space H(r;.) analogous to H(r;), the representation 1r�.· and the operator M(r;.) : H(r;.) --+ H(ij.) defined by (M(r;.) rp) (g)
(4.6)
=
JF. rp[w(6 �)gJ dx.
Since the group G. operates by right matrix multiplication on F. x F., and the stabilizer of the line F.(O, 1 ) is the group P., we may define global sections of the bundle H(r;.) by the formula (4.7)
rp(g, r;.)
I (det ) 'i'.{t• fl.vlJv � JF. cP [(O, t)g]f1.vlJ;1(t) t l d"t.
=
.
Here L(s, fl.vlJ;1) is the usual Euler factor attached to the character fl.vlJ;l, a . (x) = x I . , and cP is a Schwartz-Bruhat function on F. x F•. Recall that if X is a quasi character of F�. andf is Schwartz-Bruhat on F., the integral sF� f(t) I t l s x( t ) d"t = Z(J, x, s) converges for Re(s) � 0 ( > 0 if x is unitary) and the ratio Z(f, X• s)/L(s, x) extends to an entire function of s. (This is Tate ' s theory of the local zeta-function.) Thus the formula (4. 7) actually makes sense for all r; and indeed defines a section of our fibre-bundle. Now we · want to apply the operator M(r;.) defined by (4.6) to an element of H(r;.) given by the section (4.7). After a change of variables we get fl.v I (det f) w.( (4.8) cP[(t, x)g]f1.vlJ;1 ( t ) dxt dx . - 1) a � Z L(l , fl.vlJv ) Fv F�
I
JJ
Next recall the functional equation for the quotient Z(f, X• s)/L(s, x) :
Z(/, x- 1 , 1 - s)
____,. __, -'-'--..----,-� - s,
X
L(1
1)
_ -
(
e s, X• ,, 'f' v)
Z( f, X• s) . L(s, x)
Here f denotes the usual Fourier transform taken with respect to the fixed additive character cf;., and e(s, X• cf;.) is an exponential function which also depends· on cf; •. If we define the Fourier transform of cP to be
iJ (x , y)
=
J JcP(u, v) cf;. (yu
-
xv) du dv
we see that (4. 8) can be written as the product of L(O, fl.vlJ; 1) L( C fl.vlJv 1 ) e (0, fl.vlJv 1 ' cf;.)
(4.9) and (4. 1 0)
Note that we pass from (4.7) to (4. 1 0) by the substitutions r; --+ ij, cP --+ iJ , and multiplication by w . ( I ) Next we write 7Jv = (f.J.v• JJ. ) , fl.v = x 1 a �2 , lJv = x 2a;s12, where x 1 and x 2 are characters. For Re s > 0, we can write M(r;.) as the product of the scalar (4.9) with an operator R(r;.) which (for each r;.) takes the element (4.7) of H(r;.) to the -
.
STEPHEN GELBART AND HERVE JACQUET
226
element (4. 10) of H(7j.). But for Re(s) > - !, every element of H(7;.) can be re presented by an integral of the form (4.7) ([JL, pp. 97- 98]). An easy argument then shows that R(r;.) extends to a holomorphic operator valued function of r;. in the domain Re (s) > - ! which again takes (4.7) to (4. 1 0). Moreover, from the obvious relations (7])- = r;, (C/JA)f' = C/J, and w�( - 1) = 1 , we find that R(7j.)R(r;.) = Id.
(4. 1 1)
Since (4.9) is clearly meromorphic, this also gives the analytic continuation of the operator M(r;.) in the domain Re(s) > - t . Although we do not need to, we note that R(r;.) (and hence M(r;.)) extends to a meromorphic function of all r;. (or s E It is important to note that the operator R(r;v) is "normalized" in the following sense. If r;v is unramified then H(r;v) (resp. H(7j.)) contains a unique function
C).
(4. 1 2) Moreover, R(r;.) is unitary whenever r;. is unitary. Indeed if ii v = r;-;; 1 , the opera tors M(r;v) : H(r;.) --> H(7j.), w. ( - 1 )M(7Jv) : H(7Jv) --> H(r;v) are adjoint to one another. Therefore, since the scalar (4.9) changes to its imaginary conjugate times wv( - 1 ) when r;v is replaced by 7Jv, we conclude from (4. 1 1) that R(r;.)* = R(r;.) - 1 . REMARK (4. 1 3) (oN THE RANGE OF R(r;v)) . If fi.v'.,J-;; 1 = av then fi. v = Xva;12 and )..I V = xva;;- 1 12 with X� = Wv· In this case the space H(r;.) consists of functions
and the kernel of M(r;v) (or R(r;v)) has codimension one. The space H(ij.) consists of functions
so the range of M(r;.) (or R(r;.)) must be the one-dimensional space spanned by the function g ,_. x.(det g) . Moreover, for the sesquilinear pairing between H(r;) and H(7j), (4. 1 4)
(R(r;)
where c ' is a known constant. C. Our task is to piece together the local intertwining operators M(r;.) in order to analytically continue M(s ) . First we define an operator R(r;) as the "infinite tensor product" of the local operators R(r;.). If
Global theory.
(4. 1 5)
M(r;)
=
- -- -- L(O,1
fi.V-::2__
- R(r;) .
L(l , fl.)) ) c (O, fl.)..l - 1 )
This gives the analytic continuation of M(r;) since the scalar factor in (4. 1 5) has a
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
227
R(r;)
known meromorphic behavior and the operator is a meromorphic function of To get the functional equation of we use the functional equation of the L-function This allows us to write the scalar in (4. 1 5) as
r;.
(4. 1 6) Therefore
M(r;) L(1, ,u-1v)1)
L(s, ,uv-1) .
M(r;)
L(l , W
satisfies the functional equation
M(ij)M(r;) M(r;) * M(r;-1) r;-1 . M(r;) unitary R(r;) M(r;) t l ,uv-1 1 M(r;) ,u R(r;xa)1 12, v xa-1 12, M(s) n of siotnalhe operat complex plane(4.and19). satiAs safifunct es theiofunct equatioonr M(s) is meromorphic in the whole ) M( -s)M(s) Itthsatonly pole in the ha(fplane Re(s) is at s 1 and the residue there is such (4. 1 7)
= ld.
(Cf. (4. 1 1 ).) By (4.5) we also have (4. 1 8)
=
if ?]
=
Thus is when r; is unitary. Recall that also satisfies (4. 1 7) and (4. 1 8). 1 We sum up the analytic behavior of as follows. Write a with t real. Then is meromorphic in the half-plane � 0 and its only poles there are simple ones which occur for = xz w . The residues are scalar multiples of the operator and the ranges are the one-dimensional spaces spanned by the functions g ,_. x(det g). Going back to we get (from (4. 14)) : =
=
=
THEOREM
(4.20
=
� 0
Id. =
where c is a known constant. r um. F o F 1 2 .2 ysis of the contiF1nuousFspect f1 ,fAnal 2 / / / / (FJ > Fz) in: J:oo {/( 1(iy), z(iy)) / (M(iy) 1(iy), 2( - iy))} dy c ( 1(1), x ( 2(1), x2 F:':;, x2
The last assertion follows from (4. 1 4). D. Suppose are two P-series belonging to We know that and lie in the orthocomplement of L5(w, G) and their scalar product is given by the formula (3. 1 9). If we use the residue theorem, Theorem (4. 1 9), and some simple estimates, we can shift the integration in (3 . 1 9) to the imaginary axis and write =
(4.2 1 )
+
+
I;
X2=w
o
det)
X o det) .
Here the sum is extended over all characters X of P\Ax such that = w and the scalar product is the pairing between H(!) and H( - 1). (Note that since w is assumed to be trivial on so is X if = w ; thus X o det indeed belongs to H( - 1) .) But (cf. (3. 1 5) )
STEPHEN GELBART AND HERVE JACQUET
228
( /1 ( t) , x o det)
= =
J J
P (F) N (A) Z (A) \G (A) G (F) Z (A) \G (A)
/1 (g) x (det g) dg
F1 (g) x (det g) dg
since F1 is the P-series attached to /1 . Therefore the be written as
c
(4.22)
second
term in (4. 2 1 ) can also
_E (Fl> X o det) (F2 , X o det),
XZ=w
the scalar product now being taken in LZ((JJ , G). Now note that (4.20) and (3 . 2 1 ) together imply that, for y E R, M(iy)*M(iy) Id. In particular, M(s) is unitary on the imaginary axis and, if we set
=
a(iy) = t {/ (iy) + M( - iy) / ( - iy)} ,
(4.23)
then
M( - iy) a ( - iy)
(4.24)
=
a(iy),
and the first term in (4. 2 1 ) can be written as (4.25)
This is significant for the decomposition of Pw because if we replace /1 by h ,__. f1 (hg) then F1 is replaced by Pw(g)Fh y ...... J(iy) by y ,__. ns(g)j(iy), and y ,__. a(iy) by y ....... ns(h)a(iy). To sum up, let .P denote the Hilbert space of square-integrable functions a on with values in (i.e., square-integrable sections of our bundle over iR) satisfying (4.24). Equip .P with the inner product (4.25) and let n denote the representation of G(A) on .P given by n(g)a(iy) = n5(g)a(iy) . Because of (4.24), we may also regard as the space of square-integrable functions a(y) from R+ to H, the scalar product being given by
iR
H
.P
Thus n is a continuous sum of the representations n;y· For each X such that x z = (JJ we let .Px denote the space of the one-dimensional representation g ,__. x(det g) . Then, since (FI > F2) is the sum of (4.22) and (4.25), it follows that there is an isometric map (with dense domain) from (Lfi)-L to .P E£) This map (given by F ,__. (a(iy), /(1-))) extends by continuity to a map from (Lfi) _l_ to a dense subspace of .PE£) ( EB x .Px) and, by the remarks above, it is also an intertwining operator. To conclude, we see that L2((JJ , G) decomposes as a direct sum
CEB ) . T x.Px
Lfi ((JJ , G) EB L�on t ( (JJ , G) EB L;p((JJ , G)
where L;P is the space spanned by the functions x(det g), i.e., L;P is the space EB.Px • and L�ont is isomorphic to .P via the intertwining operator S : L�on tC(JJ , G) --+ Moreover, (4.22) is the scalar product of the orthogonal projections of F1 and F2 on L;P . Therefore = [vol(G (F)\ G (A)))- 1 and we have computed this volume.
c
.P.
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
229
5. Eisenstein series and the truncation process. Let Pcusp denote the orthogonal projection onto Lij(w, G). Since we need an explicit formula for the kernel of the Hilbert-Schmidt operator PcuspPw(f/J) Pcusp we need an explicit description of the operator of §4. This description is given in terms of Eisenstein series. A. Eisenstein series. If qJ is a section of the fibre-bundle H(s) we set
S
(5. 1 )
E(qJ(s), g)
=
I;
T E P (F) \G (F)
f/J (r g, s).
This series converges only for Re(s) > t and in general has to be defined by analytic continuation. Recall that if/satisfies (3. 1) and has compact support mod N(A)Z(A)P(F) then f(g) = (2n-i)- 1 J��}: /(g, s) ds. Interchanging summation and integration we see that the P-series F defined by f is given by ( 5 . 2)
1 F(g) = _ . 2 Jt" l
Jxx-+iioooo E( /(s), g) ds
provided x > t. Thus the functions of a dense subspace of (Lij)l. are "continuous sums of Eisenstein series". However, as before, to obtain a useful formula we must analytically continue E and shift the integration in (5.2) to the imaginary axis. Since E is a P-series, its constant term is easily computed (cf. (3.6) and (3 .7)) : EN( f/J (s), g) or (5.3)
=
qJ(g, s) +
J qJ[wng, s] dn
EN(f/J(S) , g) = qJ(s)(g) + [M(s)qJ(s)](g).
In both equations, Re(s) > f. To analytically continue E we shall primarily deal with sections obtained from identifying H(s) with H. In other words, if h E H = H(O), we define a section h(s) by the formula
h(g, s) = h(g)H(g) s .
(5.4)
The corresponding Eisenstein series will be denoted E(h(s), g) . B. Properties of the truncation operator. We shall use the truncation operator to obtain the analytic continuation of E. If c > I we let X c denote the characteristic function of [c, + oo). For any function f/J on G(F)\G(A) set (5. 5)
The second term here is a P-series attached to a function with support in the set {g : H(g) > c}. Thus if g is in a Siegel domain the series has only finitely many terms ( cf. Lemma (3.3)). If rp is a cuspidal function, i.e., f/JN = 0, then clearly .Ac f/J = 'P · In general, we need to appeal to the following lemma :
P.
LEMMA (5.6) (cF. [DL, 1 97]). If there is a g in G(A) such that H(g) > I then r belongs to P(F).
H(rg)
>
1 and
STEPHEN GELBART AND HERVE JACQUET
230
This lemma shows that, for a given g, the series in (5. 5) has at most one term for
c > 1 . In particular,
A c �(g) = �(g) - � N (g) X c (H(g)) if H(g) > 1, c > 1 ,
(5.7)
= �(g) - � N ( g )
if H(g) > c > 1 .
Now let 'll be a Siegel domain and Q the set of g in 'll such that H ( g ) ::;;; c. Then Q is compact (mod Z(A)) and, if c > 1 , A<� is given on 'lJ - Q by the second for mula in (5.7). Thus A c �(g) is bounded on 'll - Q under very mild assumptions on � · For instance, this is the case if � is the convolution of a "slowly increasing" or square-integrable function on G(F)\G(A) with a c�-function of compact support. Actually (5.7) will then be "rapidly decreasing" . On the other hand, if Q is any compact set, then A c �(g) = �(g) for g E Q and c large.
(5. 8)
Indeed suppose g E Q and Ac �(g) # �(g) with c > 1 . Then there is a finite non empty set of elements r of P(F)\G(F) such that H(rg) > c > (cf. Lemma (3. 3) again ; this finite set depends on Q but not on c). Since the resulting element rg must belong to a compact set mod P(F)N(A)Z(A), and since H is continuous, we must have H( rg) < c0• But if c > c0 we get a contradiction. Therefore (5. 8) must hold . In other words, Ac � ---> � uniformly on compact sets as c + oo . W e also want t o point out that A c i s a continuous hermitian operator o n V(w, G) :
1
--->
(5.9) Indeed the left-hand side is (�b �2) minus the scalar product of a P-series with �2. In particular, by (3.4) the left-hand side equals (� ! > �z)
-S
P (F) N (A) Z (A) \G (A)
Similarly the right-hand side is (� ! > �2) is real the desired equality follows. We also have, for c > 1 ,
� l.N (g) x c (H(g) ) �z .N (g) dg.
- f�1. N(g )952• N(g )xc (H(g)) dg ; since
Xc
(5. 1 0) i.e., Ac is an orthogonal projection in V(w, G). Indeed ( 1 - Ac)� 1 is also a P-series. So the left-hand side of (5. 1 0) is
S
N (A) P (F) Z (A) \G (A)
.
� l N ( g ) x c (H ( g ))
(J
N (F) \N (A)
)
A c �z(ng) dn dn.
But X c (H(g)) = 0 unless H(g) > c > 1 , in which case the inner integral is (by (5.7))
S
N (F)\N (A)
(�z(ng) - �z. N (ng)) dn = 0
and ( 5. 1 0) follows. (In fact (5. 1 0) is sometimes true even when �; does not belong to
V(w, G).)
C. Analytic continuation of Eisenstein series. Using the truncation operator (5. 5) we can write our Eisenstein series as
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
E(h(s), g )
(5. 1 1 )
=
AcE(h(s), g) +
I:
P (F) IG (F)
23 1
EN(h (s), rg) x c (H(rg)) .
The second term on the right side is the P-series attached to a function with support in the set {g : H(g) � c } . Thus, as noted before, for g in a Siegel domain the second term has only finitely many terms. In particular, it represents a meromorphic func tion of s whose singularities are at most those of M(s) (cf. (5.3)). On the other hand, the first term on the right side of (5. 1 1) is initially defined only for Re(s) > -!. However, since it is square-integrable (cf. (5.7) and the remarks immediately following it), it will suffice to continue it analytically as a V(w, G) valued function. Thus we need to examine the inner product (5. 1 2) By (5. 1 0) (which is true in this case) the inner product (5. 1 2) is just (E(h 1 (s 1 )), Ac £(h 2(s2) )) , which, since Ac E is a difference of P-series, we compute (using (3.4)) to be
Now recall that EN is given by (5.3) with cp(g, s)
for t E Ft,,
a
=
h(g, s) and
E FO(A), and k E K. So by Iwasawa's decomposition, we compute
(Ac E(hl > (sl)), A c E(h z (.i'z)))
(5. 1 3)
=
{(hl> h2)c• 1 +sz - (M(s 1 )h 1 , M (s2)h2)c- <s 1 +szl }
1
sl + Sz
Here Re(s1) > Re(s2) > -t. Note that the right side of (5. 1 3) is a meromorphic function of (sl> s 2) which seems to have a singularity on the line s 1 + s2 0. However, by (3.21) and (4.20), the expression in the first bracket vanishes precisely along this line. Similarly the ex pression in the second bracket vanishes when s 1 - s2 0. Thus we conclude that (5. 1 3) is meromorphic in s l> s 2 with singularities at most those of M(s 1 ) and M(s2) . In particular, for s s 1 - s 2 #- 0, =
=
=
=
(Ac£(h 1 (s)), Ac£(h2( - s)))
=
(5. 1 4)
2 (h l > h2) log c + ( M( - s)M' (s)hl> h 2)
+ { (hl> M ( - s)h 2)c 2c - (M( s )hl> h2) c- 2• }
is .
Now suppose that for h1 h2 h, the right-hand side of(5. l 3) is analytic in the polydisc I s 1 - so I < R, I Sz - s0 < R, while E(h(s)) is holomorphic in some small disc centered at s0• Then the double Taylor series of the left-hand side converges in this polydisc. Since =
=
I
STEPHEN GELBART AND HERVE JACQUET
232
I ;; AcE(h(s)) /t = ( ;;" AcE(h(s)), :s: AcE(h(s)) ) = os�;s� (AcE(h(s1)), AcE(h(sz))) 11.1=52=•
it is easy to conclude that the Taylor series of AcE(h(s)) converges in the disc I s - so l < R, i.e., AcE(h(s)) is analytic in thi s disc. Thus we conclude A cE(h(s)) (and hence E(h(s), is meromorphic in Re(s) � 0 with singularities at most those of M(s). Note that in (5. 1 3) and (5. 1 4) we needed to identify H(s) with H to define the scalar products and the derivative M'(s) of M(s). Summing up, we know that if cp(s) is any meromorphic section of our bundle H(s) then E(cp(s), is defined as a meromorphic function of s (at least for Re(s) � 0). Moreover,
g))
g)
(5.15)
E(M(s)cp(s),
g) =
E(cp(s),
g).
Indeed since cp(s) belongs to H(s) and M(s)cp(s) belongs to H( - s), the function EN(M(s) cp(s), is equal to
g --+
g) g) =
=
M(s)cp(s) + M( - s)M(s)cp(s) cp(s) + M(s)cp(s) = EN(cp(s), Thus EN(M(s)cp(s), i.e., the difference between the two sides of EN(cp(s), (5. 1 5) is a cuspidal function. Therefore, since any P-series (or its analytic continua tion) is orthogonal to all cuspidal functions, this difference vanishes as claimed . In general, say for a group whose derived group has F-rank one, the same facts can be proved. The only difference is that the operator M(s) may have a finite number of poles and these poles are not necessarily known. Nevertheless, the op erator M(s) still similarly controls the analytic behavior of the Eisenstein series (cf. [Ar 1] and [La 1]) . D. The kernel Pw(cp) L�ont· Suppose F1 (g) is a P-series attached to.fi (of com pact support mod Z(A)N(A)P(F)). Our immediate goal is to prove that, for all h e H,
g),
g) .
of in F E(h(iy), g) \(g) dg SF1 (iy)) = -2!_ s (5.16) SF1(iy) = a1(iy) (4.y.23). E(h(iy), g) F oo i ) g) dsF} 2(g) dg (Fh Fz = J - {� 2 7r: l. J"x-+ioo 12 7r: l J"+ioo ds S =� g) z(g) dg =x> F j (5.17) (Fh = 217r: s:oo dy s E( l (iy), g) z(g) dg X�w (Fh y} M(iy(5.15) )/1(iy) y - y.(5.17) a1(iy) (4. 23), /1(i(5.17) (h,
G (F) Z (A) \G (A)
for almost every Here is defined by analytic continuation and as in Note first that if F2 is the P-series attached to/2 we get from (5.2) that 1
_
E( jl (s),
G (F) \G (A)
.
Fz}
reads
_
G (F) \G (A)
E ( j l (s),
t. Shifting the integration to the imaginary axis we then get
for Re(s)
But
x- ioo
+ c
x) (x. Fz).
says that the integral in is unchanged if we replace as in and then change to Thus, with formula
by
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
On the other hand, computing the inner product as in
233
(4.21), we also have
� J:oo (a1(iy), az(iy)) dy + c x� (Fh x )(x. Fz) . Thus we conclude that oo (5 . 18) J oo (a(iy), SF(iy)) dy 1 J {J E(a(iy) , g) F(g) dg} dy (FI > Fz)
=
..
=
_
2
-
_
_
Gmww
for all a in 2 . This formula is then also true for any square-integrable function a(y) with values in H since both sides remain unchanged when a(iy) is replaced by t (a(iy) + M( - iy) a( - iy)) . Thus we may take a(iy) = c( y)h( iy) with c a scalar function and h a "constant section" to conclude from that indeed
(5.18)
(5.16)
holds. Now extend the map S: L�ont(w, G) -jo 2 to S: V(w, G) -jo 2 by setting it equal to on Lij and r;p(w, G). Then S * S is the orthogonal projection of L2(w , G) onto L�ont· Since S is an intertwining operator, S * Sp.,(cp) S * S = S * n:(cp) S for any function cp of compact support. Therefore, if F1 and F2 are in V(w, G) then
coo
0
( S * Sp "'(cp) S * SF1o F2)
(5 . 19)
= =
( S * n:(cp) SF1o F2)
=
(n:(cp) SF1 o SFz)
! s:oo (n:;y(cp)SF1(iy), SF2(iy)) dy.
Before applying the identity (5.16) we make the following observation. Although (5.16) was proved only for F1 in the orthocomplement of Lij(w, G) it is actually true all F1 V(w , G). Indeed if F is in Lij( w, G) then SF 0 by definition. On the
for in
=
other hand, since every Eisenstein series is orthogonal to any cuspidal function (in the domain of convergence at first but for all s by analytic continuation), the right is also identically zero. side of to get If { tlla } is an orthonormal basis for H we can finally apply to
(5 . 16)
(5.16) (5 . 19)
=
-111:- J:oo dy � SFl(g) E (n:� (cp)tlla(iy), g) dg J Fz(h) E(tlla(iy), h) dh ·
with g, h in G (F)\G (A ) . Interchanging the integrations and summations then yields ( S * Sp., (cp) S * SF1 o F2)
(5.20)
= JJ Fl(g)Fz(h) dg dh {� 4� J:oo E(tlla(iy), h) E (n,�(cp) tll2(iy), g) dy} .
So if we let Pcont denote the projection S * S onto L�ont we find that the kernel of
234
STEPHEN GELBART AND HERVE JACQUET
Pcon t p., (
(5.21)
-4-1_1C a,I;�
S""
-oo
brackets in (5.20). Alternately, since
(n;y(
On the other hand, if Psp is the projection onto L;P , then the operator P5pP., (
[vol( G (F)\ G (A) ) ] - 1 I:
X2=w
x (h)x(g)
S-
G (A)
The operator Pw(
where
S
=
I: c cFJ
Kcusp(h, g)F(g) dg,
Kcusp(h, g) = K(h, g) - Kconlh, g) - Ksp(h, g) .
6. The trace formula. Suppose
(6. 1 )
J-
_
G (F) \G (A)
Kcusp(x, x) dx.
What we are going to do now is give an explicit formula for the right-hand side of
(6. 1 ) . Note that for f in L5(w, G).L, J Kcusp(x, y)f(y) dy = 0. Thus for each x, the function y ...... K(x, y) is orthogonal to L5(w, G).L and hence in L5(w, G). In other
words, it is a cuspidal function of y. If we denote by A� the truncation operator with respect to the second variable then Kcusp(X, y) = A2Kcusp(x, y) = A 2K(x, y) - A2 Kconix, y) - A� Ksp(x, y) .
(6 . 2)
But one can show that each term in (6.2) is integrable over the diagonal. Thus (6. 3)
tr(PcuspPw (
S
A� K(x, x) dx -
J
A2Kcont(X, x) dx -
S
A2Ksp(x, x) dx.
We shall content ourselves with computing each of these integrals. Note that since the left-hand side of (6.3) does not depend on c, we can let c tend to + oo and-when computing-ignore all terms which tend to zero. A. Contribution from the kernel A2K. When x = y, A2 K(x, y) is by definition A�K(x, x) =
(6.4)
I:
rE G (F)
I: - < E P (F) \G (F) Let us recall the following lemma :
LEMMA
J
N (F) \N (A)
dn �
rE G (F)
(6.5) (cF. [Ge, P. 201 ]). If Q is a compact set in Z(A)\G(A) then there exists
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
235
a number d0 with the property that if r in G(F) and n in N(A) are such that x-1 rnx E 0 for some x in G(A) with H(x) > d0 then r E P(F). We shall apply this to the support Q of IP · After changing r to �-1r we may write the second term in (6.4) as (6.6) Here we need only sum over those r for which there is x, �. n such that g = � x and rirng is in Q. Thus if c is sufficiently Iarge (lfJ being given) it follows from Lemma (6.5) that r must belong to P(F), i.e., if c is large enough, we need only sum over those r which lie in the image of P(F) in G (F). But for such r we can write
r
(6.7)
= p.))
with p. =
(g ?) and ))
E
N(F).
Thus in (6.6) we can combine the integral over N(F)\N(A) with a sum over N(F) and rewrite (6.4) as (6.8) with p. as in (6.7). Now we break up the sums in (6.8) and write
A�K(x, x) = (6.9)
ip(e)
(6. 1 0)
+ l:; ip(x-1rx)
(6. 1 I )
+ L; ip(x-1rx) -
(6. 1 2)
T
+
l:; ip(x-1rx) T
(r £-elliptic)
I;
I;E P (F) \G (F)
I;
I;E P (F) \G (F)
J J
N (A)
N (A )
ip(x-1� - In�x)x c (H(�x)) dn
(
(r nilpotent regular)
I; lfJ x- I �- I
a'/"1
(a 01)n�x)xc (H(�x)) dn o
(r £-hyperbolic regular).
Here r F-elliptic means r is not G(F)-conjugate to anything in P(F) ; r nilpotent regular means r is conjugate to a nontrivial element of N(F) and r £-hyperbolic regular means r is conjugate to some (3 �) with a of. 1 in P. One can see that each of the terms (6.9)-(6. 1 2) is integrable over G (F)\ G (A) ; our task is to evaluate the resulting integrals. The integral of (6.9) is clearly (6. I 3)
vol ( G(F)\G(A ))IfJ(e).
As for (6. 1 0), it has compact support mod G (F). Indeed if ip(x-1 rx) of. 0 then x-1 rx belongs to Q = support(ip). So since r is elliptic, Lemma (6.5) implies H(x) < d0. The integral of (6. 1 0) over G (F)\ G (A) is
STEPHEN GELBART AND HERVE JACQUET
236
J-
(6. 14)
-
G (F) \G (A.)
I; gJ{x-1 rx) dx (r elliptic) r
and this can be transformed further just as in the division algebra case. In (6. 1 1) we can write r in the form with Thus (6. 1 1) takes the form
r = �-1(}, i)�
� E P(F)\G(F), 7J #- 0.
I; { I; gJ[x-1�-1(b i)�xJ - J gJ(x-1�-1n�x) Xc (H(�x)) dn}
eE P (F) \G (F)
�;<0
N (A.)
which we now have to integrate over over of
P(F)Z(A)\G(A)
?to [_x-1 (b i)xJ
{6. 1 5)
gl
-J
G(F)\G(A).
A gl
This is the same as the integral
[ x-1(b nx]xc(H(x)) du.
To evaluate this integral we shall use Poisson's summation formula. Set (6. 1 6)
F(x)
A. P(F)Z(A)\G(A)
so that is a Schwartz-Bruhat function on we get that the integral of (6. 1 5) over (6. 1 7)
Using lwasawa's decomposition is
Using Poisson's summation formula we find this is
F 1 F J ( I; F(a r;)) 1 aj d"a J I; (a- r;) d"a (O) ! ! l 1 xc< a - )) d"a - F(o) JF Ia d"a. .J F (O) c c) (O)vol(F"\FO(A)). J F{a)jaj • d"a= J I; F(ar;) j a j • d"a l a l �l
+
�;<0
1 a 1 ;;;; 1
l a l ;;;; l
+
�;<0
(1 -
1 a 1 ;;;; 1
This shows that the integral converges. The term involving equals (log For Re(s) > 1 we also have A.•
depends ·on and
F•\A• n;
F
s= (O) F j (J F(a) aj d"a) c) (O)vol (F"\FO(A)) ...) = U F(a) a 1 • dxa -
Here each term is analytic at 1 except the term involving I . Thus we conclude that (6. 1 7) equals
( (O) vol(F"\FOA))fs f.p.
F
A•
where the symbol f.p. means f.p.(
which equals
+ (log
value at 1 of
A•
I
principal
FORMS OF
GL(2)
a a-1
part at I } . Changing to of (6. 1 1) is
FROM THE ANALYTIC POINT OF VIEW
237
in the "f. p. integral" we get (finally) that the integral
(6. 1 8) Now we deal with the term (6. 1 2). In this term we can write r in the form �) .; with .; in A(F)\G(F), in P, -:f. 1 . But when .; varies through a system of representatives for A(F)\G(F) we obtain each r twice since r =
.;-1(�
(g
w
?)
w- 1
ex ex (b �)
=
Thus the first term in (6. 1 2) is
=
(ex;1 ?)
in G.
[x-1�-1(ex O)�xJ rp 0 [ ( ) J rp x-1_;-1v-1 ex0 0 v.;x . [ ( rp x-[1�-1v-1( ex0 ?o))v.; x]] f rp x- 1 .;- 1 ex0 n.;x xc(H(.;x)). (ex [rp x-lv-l(ex 0)vx] [rp x-ln0-1(g ?)nx]x (H(x)) dn. c ? I; rp[[x-'" ln-l((gg ?))nx]] dn rp x 1n-1 nx xc(H(x)) dn
z:; z:; _I_ 2 f; E A (F) \G (F) a ;t l
J
- l_ I; I; I; 2 f;EP (F) \G (F) vE N (F) a ;t l
1
The integral of all of (6. 1 2) is therefore the integral over G (F)\ G (A) of l_ I; I; I; 2 f; E P (F) \G (F) vEN (F) a ;t l -
l:
f;EP (F) \G (F)
N (A)
1
I;
a ;t l
Making the change of variables on N(A) given by
-:f. I )
we see that the integral of (6. 1 2) over G(F)\G(A) equals the integral over P(F)Z(A)\G(A) of l_ I; 2 a ;tl
I;
1
vEN (F)
-f
N (A)
I;
a ;t l
To compute this last integral we have to first integrate over N(F)\N(A) and then over N(A)A(F)Z(A)\G(A). The integration over N(F)\N(A) gives
i
f -f I;
a ;t l
N (A)
N (A)
I;
a ;t l
and the subsequent integral over N(A)A(F)Z(A)\G(A) is the same as the integral of
STEPHEN GELBART AND HERVE JACQUET
238 (6. 1 9)
over A(F)Z(A)\G(A). Note now that the first factor in (6. 1 9) does not change when x is replaced by wx but the second factor does. In any case, w normalizes A(F). Thus we see that the integral of (6. 1 2) is also the integral over A(F)Z(A)\G(A) of
� �1 �[x-I (g ?)x]U - Xc (H(x) )
-
X c (H(wx)) ) .
Using Iwasawa's decomposition we get that this integral is (6. 20)
__!_ 2
JJ K
( ) ] · (J ( I - Xc( l al) - Xc( lai-1 H(wn))) dxa) dk du.
[
0 I; � k - In-I a0 1 nk
N (A) a;"l
FX\AX
But c > and H(wn) � 1 . Therefore in (6.20) the integrand in the inner integral vanishes unless c I H(wn) < lal < c in which case it equals 1 . Thus the integral of (6. 1 2) over G (F)\ G (A) is
I
-
s s ?t1 {k-l (g ?)nk] dk dn I; �[k I n I (g ?)nk] log H(wnk) dn - __!_ vol(P\FO(A)) J J 2
(Jog c)voJ(Fx\FO(A) ) (6. 2 1 )
K
N (A)
K
N (A) a;"l
- -
dk.
Summing up (6. 1 3), (6. 1 4), (8. 1 6), and (6.2 1 ), we get PROPOSITION (6.22).
JA�K(x, x) dx =
(6.23) (6.24) (6.25)
+J_ _ I; �(x-I rx) dx + f.p. J �[ - 1 (01 11 ) ] d I; �[k -I (g ?) nk] dn + (log c)voJ(Fx\FO(A ) ) J J L: �[h - In-I (g ?) nh] log H(wnk) dn - 1 vol (F x \FO(A) ) J J 2
vol( G (F)\ G (A) )� (e)
G (F) \G (A)
g
Z (A) N (A) \G (A)
g
K
K
r elliptic
g
N (A) aEFX
N (A) a;"l
dk
dk .
REMARK. The term (6.24) depends on c but tr Pw( �) does not. Thus we can expect (6.24) t o cancel with another term later on. B. Contribution from the continuous spectrum. Recall that
It is not hard to see that this also equals
239
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
So taking it for granted that we can interchange orders of integration, we find
sA�Kcont(x, x)
dx
£ oo 1 J = -4 {J -oo ("";y(rp)f/JfJ, f/Ja) dy J - E(f/Ja(iy), x)Ac (f/Jrliy), x) dx. §5 (5.14) h1(s) = f/Ja(s), h2(s) = f/Jf3(s)). (5.14) SA�Kcont(X, x) dx (6. 26) - -2""c) a, {J J-oooo (""iy(rp)f/J{J, f/Ja) (f/Ja, f/J{J) d (6.27) (6.28) J (6.26) c/2"") J<:><>oo "";y(rp) dy. "";y (J rp(k- 1(toa 01)nk) i t i iy+l l 2dxt) dn dk. ss (6. 26) (6. 24), (6.27)(6. 24) iy)M'(iy)"";y(rp)) dy. (6.29) - 1"" s:oo (6.28), c22i. y - ("";y(rp)f/Jf3, M( -1.y)f/JfJ) c2-2ir.yy } dy -41"" f3 J-oo {("";y{rp)f/JfJ, M(1.y)f/Jf3) 1"" f3 J-oo (M( - l.y)"";y(rp) f/JfJ, f/Jf3) c2iy 21.yc-2iy dy 4 (6. 30) -1 oo Joo { ("";y(rp)f/Jf3, M(iy)f/JfJ) - ("",)rp)f/Jf3, M(-iy)f/Jf3) } c-2i. y dy. 4"" 21y l:; (c"";y(rp)f/JfJ, M(iy)f/Jf3) ... ) cf"" 9. 1 4 I;
_
G (F) \G (.A)
"" a,
Note that the inner integral here has already been computed in Plugging in then gives
- (log
� I...J
(cf.
with
Y
After exchanging l:; and the term can also be written (log · is an induced representation whose trace is easily But tr computed to be vo1(Fx\FO(A))
K
I;
N ( A)
aE F'
F;!;,
So after using the Fourier inversion formula we find that i.e. , it cancels just as expected. Now rewrite as
precisely equals
tr(M( -
As for
it can be written as
I;
-
ry
or
I;
+
_
I; 13
-oo
-
But the last term here is the Fourier transform of an integrable function (namely at log (cf. Lemma of [Ge]). Thus it tends to zero as -> + oo . To evaluate the first term we need the following lemma :
STEPHEN GELBART AND HERVE JACQUET
240
(6.31). IfF is continuous on R, and F, F are integrable, then Ztcixy e -Ztcixy F(y) dy = 2n iF(O). lim S e = Y x �+ 0, and PROOF. Set G(x) = J�= ((eZtcixy - e -Ztcixyjy))F(y) dy. Then G(O) G'(x) 2ni J (eZtcixy + e-Ztcixy) F(y) dy 2n i(F(x) + i ( - x)). LEMMA
_
=
=
=
Therefore
G(x) = 2n i J: (F (t) + F( - t)) dt 2ni S�x i {t) dt for x > 0 . So as x + oo, G(x) tends to 2n iF(O). Applying Lemma (6.31) to (6.30) we conclude that (6.28) tends to ( h ) (6.32) t � M(O o( rp) r/J� , r/J� = t tr M(O)no ( rp) =
---+
as c
---+
+ oo .
�
C . Trace formula. We leave it to the reader to check that
J A�Ksp(x, x) dx S Ksp(x, x) dx [vol( G (F)/ G (A)) ]- 1 x;;;w J rp (x) x (x) dx ---+
=
as c and
+ oo . Therefore, by combining
->
(6.32) we obtain THEOREM (6 . 33) . tr Pcusp(rp) + tr Psp(rp)
s - ( + f.p. S +
(6.34)
G (F) \G (A)
r
�
Z (A) N (A) \G (A)
Proposition
(6.22), (6 . 26), (6.27),
vol( G F) \ G A
( ( ))rp(e) rp(x-1 rx)) dx rp[g-1 (10 I1)gJ dg
=
elliptic
(6.3),
- 21 - vol (Fx\FO(A)) J J � rplk-1n-1 (g ?)nk] log H[wnk] dn dk (M ) + 4k s := t r ( - iy M' ( iy)7Ciy( rp )) dy - t tr (M(O)n0( rp)) .
(6.3 5)
K
(6 . 36) (6 . 37)
N (A) a 'l" l
L
Here he f. p. term is computed as the value at s
t
=
I
of
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
] I a I• dxa dk 1 � ) k �[ k{J (b
principal part at
s
=
1}.
24 1
Recall also that we have identified all fibres H( ) with H in order to define
dM/ds.
s
M'(s)
=
7. A second form of the trace formula. Our next goal is to express the right-hand side of (6.33) in terms of distributions, most of them invariant. This involves replacing the Mellin transform on Ft, by a Mellin transform on F"\Fx(A) , the fiber bundle H( ) by the bundle H(7]), and the operator by the operator M(7J). To express the right-hand side of (6.33) as a sum of invariant distributions entails applying a form of Poisson summation to some nonsmooth functions. This analysis is carried out in §7 of [La 2] but is not needed for the application we have in mind. A. We normalize the Haar measure on Ax as follows. Select (in any way) a nontrivial additive character ¢ = n ¢. of F\A. For each place v let be the self-dual Haar measure on F. with respect to ¢ • . Then the Haar measure we take on F� is = Note that on A the Haar measure = is self-dual. On Ax the (normalized) Tamagawa measure is
local
s
M(s)only
Normalization of Haar measures. dx. dx @dx. d"x. L(I, I.)dx.J i x.l .
(7. 1 ) where (7.2)
......
)._ 1 = lim s -1
(s - I )L(s, IF) . d"x/
The map x I allows us to identify F0(A)\A" with Rt. The Tamagawa measure fl. on FO(A) is such that the quotient measure fl. ( on FO(A)\A" or Rt) is and as shown in Tate's thesis-
xI
vol (Fx\F O(A) ) =
(7. 3)
dt/t
I.
To define the Tamagawa measure on G(A) we select in any way an invariant differential form of degree 3 defined over F. In particular, we may take co to be the form whose pull-back through the map
co
(a, x, y) ...... (g ?) (b �) (� ?) (daja) dx dy. f= I I l (g) d = ! g J!(g)l co(g) = I] J f.(g.) co.(g) J ( 7 . 4) = IJ J!.[(ao ?) (b �·)(Y: ?)] �� dx. dy is the form
Then for
TI fv on G(A),
•.
On the other hand, the Haar measure on K is normalized by the condition (7. 5)
J f(g) dg = J 1 [(g ?) (b �)kJ dxa dx dk .
242
STEPHEN GELBART AND HERVE
JACQUET
Similarly the Haar measure on Kv is defined by
(7 .6) Then a simple computation shows that
(7.7)
vol( K") = L( 2 '
Since dk = ;1_ 1 ®v dkv we also have
I
I
) - (lT TV ' ,i. lJ e "f/ V-) '
.
(7.8 ) Now we replace the Mellin transform on Ftc, (introduced in §3) by a Mellin transform on Ax, namely
iex) (g) =
L/[(� ?)g]x- 1 (a) l a l-u z dxa.
Then formula (3. 1 9) should be replaced by .
(7.9)
x
(Fb Fz ) = - 2_! I; s -l- zoo Cf1 (x a '), fz ( x a -s) ) ds X x-ioo 7r: l
Here X runs through the set of characters of Fx(A) trivial on PFt, ; cf. [ Ge, p. 1 67]. B. The "f.p." integra l. In order to compute (6. 34) we have to remove from f � [k- 1 (6 f) k] I a I ' dxa dk the principal part at s = I and then set s = I . So first we write this integral as the product L(s, I F) B (s) where
(7. 1 0)
B(s) =
I L(s, I F )
s �[k-1 (6 nkJ i a l' dxa dk.
At s = I the function B(s) is holomorphic. On the other hand, L(s, I F) = Ld(s - I ) + .:10 + · · · . Therefore the f.p. integral is L 1 8'(1 ) + .:108(1). But dxa = (.:1_ 1 )- 1 ® dxa" and dk = ;1_ 1 ® dkv. So for Re(s) > I ,
and since almost all factors here are equal to one,
I.e., 8(1) is the product of the orbital integral for (6 D with a convergence factor (which is "missing" in the original f. p. integral). Similarly, taking the derivative of B(s) at s = I , we get
243
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
(7.11) u,
Here the sum is extended over all places but only finitely many have a nonzero contribution. C. Recall that vol{P\ F A) ) = = = and H ) = TI .H. .) where
dn ®Comput dn., ation (ogf (6.35). (g Thus
(6. 3 5)
is equal to
(7.12)
where
Since D.
1, dk
,l_ 1
® dk.,
n.k.] dn. dk.
-�- 1 �u � nv=Fu J cp.[k;;In -;;1(() ?) a;t:l
or
O(
--�-1 "�1 !I J cp.[g;1(f g)g.J dg. J CfJu[g;1 (0 ?)guJ Jlu(gu) dgu
/lu Comput ation ofK(6.u,36) and (6.37). -41 � s;� vanishes on
the sum above has only finitely many nonzero terms. can be written as To begin with,
n
x
-•""
(6.36) dy
tr{M(r;)- 1 M'(r;)n�(cp))
where r; = (xa;Y , x- 1 w a-iY), n� is the representation of G(A) on H(r;), and the sum is over all characters X of P\A" whose restriction to F;};, is trivial The derivative is defined by M'(r;)cp = (dfds)M(r;)cp if r; = (xa•, x-1wa-•) and cp /x is independent of s . (To define the derivative we have once again trivialized the fibre-bundle.) Since M(r;) = R.(r;.) {where is the scalar described by we get (by taking the logarithmic derivative) that
m(r;) ®.
M'(r;) M-1 (r;) =
®v
m(r;)
m' (r;)l + �u R;1(r;u)R�(r;u) v®¢u I
(4.15))
•.
But if A = A. where A. is an operator on H(r;.) then tr(A) = TI . tr(A.). Thus is equal to
(6. 36) (7.13)
STEPHEN GELBART AND HERVE JACQUET
244
then = q; for all Thus Recall that if q; = I on sum above once again has only finitely many nonzero terms. As for (6. 37), we write it as
Ku R(r;u)'P 1 - 4
Summing up.
r;u· R'(r;u)ffJ
=
0, and the
X�w tr(M(x . x) 7t'cx. x) (f{J) ) .
E. We have that tr Pcusp( f/J) vol( G (F)\G(A))q; (e),
l:
r elliptic
+
tr Ps p( f{J) is equal to the sum of
JG- (F)\G (A) q;(g- l rg ) dg _
and a complementary term. As in the case of division algebras, ( * ) can be expressed in terms of local elliptic of the form
orbital integrals JGrCF,) \G, .(g g dg., l JZ,N,IG, 'Pv [gl ; (b D g.J dg., ; �)g. dgv, qJ g JA,IG, ,[ (Q J I I 1 is Is=! L(s� J f/Jv[k; (b �·)k.J a. • d"av dk., J .[g (g �) g.J p.(g.) dg. R�(r;.)1t'�.(q;.)) . not special case. q;
-;;I r )
r elliptic.
The complementary term can be expressed explicitly in terms of the following local distributions : (7. 1 4) (7. 1 5) (7. 1 6) (7. 1 7) (7. 1 8) and (7. 1 9)
tr 1t'n.( q;.) ,
I .)
q;
;�
(a #- 1 )
tr(R.(r;.)- 1
The distributions (7. 1 4) and (7. 1 5) are orbital integrals. The distributions (7. 1 6) are invariant and can also be computed in terms of orbital integrals (cf. [DL] for example). The distributions (7. 1 7)-(7. 1 9), however, are invariant. F. A The distributions (7. 14)-(7. 1 6) enjoy the following property. Suppose (7.20)
JA,IG, f/Jv(g- l(g �)g) d"a = 0
for all #- I , i.e., the orbital integrals of vanish for all regular hyperbolic elements . Then each of the distributions (7. 1 4)-(7. 1 6) vanishes. Indeed there is nothing to prove for (7. 1 5), and (7. 1 6) is an integral of orbital integrals of this type. As for (7. 1 4) we note that
a
'Pv
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
245
i.e., the nilpotent orbital integral is a limit of hyperbolic orbital integrals. It follows that if (7.20) is satisfied for a certain place u 1 then the sums in (7. 1 1), (7. 1 2), and (7. 1 3) reduce to the term involving u = u 1 . On the other hand, tr nlrp) vanishes. Thus we have : THEOREM (7.21).
Then
(7. 1 1)-(7. 1 3)
tr P cusp( rp )
+
vaniSuppose sh andcondi we havetion is satisfied for two places vol(G(F)\G(A)) rp(e) S (7.20)
tr p,P( rp )
=
+
I:
r elliptic
G (F) \G (A)
u1
i=
u2 •
rp (g-1rg) dg .
The reader will note that this formula closely resembles formula ( 1 . 1 1). 8 . Applications t o quaternion algebras. As before, G is the group GL(2) regarded as an F-group and Z � GL(l ) is its center. Let D be a quaternion algebra of center and the (finite) set of places of where D does not split. Regard the multi plicative group of D as an algebraic F-group G' and let Z' denote its center. Then for all 1· ¢ the local groups Gv and G� are isomorphic. More precisely, as in § 1 , the isomorphism Dv "" M(2, induces an isomorphism Gv "" G� defined up to inner automorphism. If c,-, 1 � � 4, is an F basis of D then for almost all we can assume that L: Rvci maps to M(2, Rv). Also Kv = GL(2, Rv) maps to the com pact subgroup K� of G� and the isomorphisms Gv � G� give rise to an isomorph ism of the restricted products as = IT v$S G"' G' S = IT v$S G�. A. For the moment, let be a local field and D a division algebra of center so that = Dx ; let )) denote the reduced norm on D. Denote by Iff ( G( ) the set of classes of irreducible admissible representations of G(F) and by 2 ( G( ) the subset of those which are square-integrable (modulo the center). Define G ( ) = 2 ( G ( ) similarly.
F S
F
S
Fvi)
v
Statement ofF results. G'(F) F
e".
e"
7r
e" . (
(8.2)
tr( ' )
=
tr(t),
)
=
<-->
-
= det(t).
This condition implies that the central quasi-characters of and n' are the same. Note that G' is an inner twisting of G and W(G) c W(G') (cf. [Bo]). Thus if = the correspondence n ' .__. is the one specified by Langlands ; in the archimedean case one can at least construct the map using "Weil's represent ation"(as in [JL]). Now let be a number field and D a division algebra with center Let .Cll'( G') be the set of (classes of) automorphic representations of and .Cll' *(G') the subset of those which are not one dimensional. Simi larly let s¥0(G) be the set of cuspidal representations of G( ).
F
G'(A) F. A letrf;n:S,be the represent atiisoomor n ofpG(A) such that, forThen n: S,is n:�in as n:iMoreover, n' is in andtforhe map n:� vi a t h e hism n: i s a bij e ct i o n between and the set of n: in such that for all THEOREM (8 .3) (GLOBAL). If v E
Gv "" G�. .Cll' ,�( G')
<--> n v
st0(G) . .Cll'0( G)
.cY*(G') (8. 1 ), v
n:
nv E Iff2( Gv)
'
nv
>--+
""
vE
S.
STEPHEN GELBART AND HERVE JACQUET
246
We shall give two proofs of this result, both somewhat different from the one in
[JL]. First we shall take Theorem (8. 1) for granted and indicate how it together with
the trace formula implies Theorem (8.3). Then we shall sketch an alternate proof that essentially implies (rather than depends on) Theorem (8. 1). Both proofs use orbital integral techniques (cf. [Sa], [Sh], and [La 2]). B. Let w and w ' be differential forms invariant and of maximal degree on G and G ' respectively. We assume that w and w ' are related to one another as in [JL, pp. 475 and 503]. Then for each rf; S, W v = ± w� and I W v I = I w � 1 . For E S let u s say that I E C�(Gv, w;;- 1) and rp E C:'(Gv, w;;- 1 ) have if
Matching orbital integrals (and consequences).
v v orbital integrals (8.4)
f
Zv\G v
for "any function" Theorem (8. I ), then (8. 5)
det )
g(g)h(tr g, g dg h on F
x
P.
=
J,
, rp( ) ( tr( ), ( )
Zv\G v
matching
g h g v g) dg'
In particular, if nv and
n
� are related as in
On the other hand, tr n.(f) = tr n�(rp)
(8.6)
if nv = x a det and n� = x a ).J. Condition (8.4) can also be expressed in terms of orbital integrals as follows : (i) The hyperbolic (regular) orbital integrals of I vanish, i.e., (8.7) (ii) Let L c M(2, Fv) and L' c
D
be isomorphic quadratic extensions ; then
if there is an isomorphsim Lv -> L� taking in U - px to t ' E L'x - P . (Here we select in any way a Haar measure on F:\L� and transport it to F;\L�x via the isomor phism L" -> L� ; the quotient spaces are then given the quotient measures.) A consequence of the last few identities is
t
e
(8.8)
l(e) = rp( ).
(This follows, for instance, from the Plancherel formula.) Note too that if nv is a representation of G" which is neither square-integrable nor finite dimensional then tr n v( f) = 0.
(8.9)
The following lemma results from the characterization of orbital integrals in
§4 of [La 2] .
matching orbital inGitevgenrals. in LEMMA (8 . 1 0).
rp
C�(G�, w-1)
I
there are (many) in
C�(Gv, w -1 )
with
REMARK (8. 1 1). Fix a representation n� of G� and let nv be the corresponding
FORMS OF
GL(2)
FROM THE ANALYTIC POINT OF VIEW
247
representation of G. given by (8. 1). There is easily seen to be a rp in C�(G �, w;;-1 ) such that tr 7C�(rp) i= but tr a'(rp) = if a' is any irreducible representation of G� (with central quasi-character w.) not equivalent to 1r:�. Thus if f has matching orbital integrals, tr 1r:.(f) i= but tr a ) = whenever a is not equivalent to 7Cv (or finite dimensional). Now suppose f is a function on G(A) such that (zg ) = w-l(z)f(g). Moreover, suppose as before that = n. where f. E C;;"(G., w;;-1) and for almost all is the function defined by
0,
0 {
f f(g) f.(g.) 0,
v
fv(g.)
=
w;;-1(z)
if g.
=
0
f
z.k., k . e K., z. e
0
otherwise.
v¢
f.
f.
z. ,
Suppose rp is a function on G'(A) satisfying similar conditions, rp • ..., via the isomorphism G. :::::: G� for S, and rp. and have matching orbital integrals for V E S.
PROPOSITION (8. 1 2). With rp
andfas above,
tr p�(rp)
(8. 1 3)
=
tr pw, oU)
f.
+
tr Pw, sp(f) .
PROOF. By ( 1 . 1 1) the left-hand side is
vol ( Z(A)G'(F)\G'(A))rp(e) (8 . 1 4) the sum extending over a set X' of representatives for the classes of quadratic extension L' of F embedded in Moreover
J
L' • (A) \G' (A)
D.l rp(g- .;g) dg
=
nJ v
' ' Lv' IG v
t
rp.(g;;- .;g.) dg
•.
Here we have selected for each v a Haar measure on F;\L'; such that (units F�\L�) = 1 for almost all v ; to Ax\£x(A) we have given the product measure. On the GL(2) side, note that S has at least two places vi> v2 and for these places f.,, fv2 have vanishing hyperbolic orbital integrals. Thus by (7.21), the right-hand side of (8 . 1 3) is vol ( Z(A)G(F)\G(A)) f(e) (8. 1 5)
+
1 I; v ol "' Lx\Lx(A)) (A 2 L
I;
<E P \£";< * 1
J
L' (A) \G (A)
f(g-1.;g) d
g
where L runs through a set of representatives for the classes of quadratic extensions of F in M(2, F). The measures are selected as above, and
Note that if v e S and L. is split then the corresponding local integral is by (8. 7) (i). Thus we need only sum over the set X of L which do split at any v e S, that is, which embed in But for every L e X there is an L' in X' and an isomorphism
D.
not
0
STEPHEN GELBART AND HERVE JACQUET
248
L --+ L'. So if we assume that the Haar measures of F�\L� and F;\L�x correspond to one another we have
for all v. Indeed for v E S this is clear, and for v ¢: S it follows from the fact that we may modify the isomorphism G. --+ G� so as to make it compatible with L� --+ L';'. We conclude from (8. 7) and our choice of for v ¢: S that in (8. 1 4) and (8. 1 5) the series on L and L' are equal. Since we have used the Tamagawa measures on G and G' we also know that
q>.,f.
vol(Z(A)G ' (F)\G'(A))
=
vol(Z(A)G(F)\G(A)).
Therefore, taking (8 .8) into account, (8. 1 2) follows. C. Let P�. o be the representation of G'(A) in the orthocomplement of the space spanned by the functions X o with = w. From (8.6) and our choice of for v ¢: S we get
Proofs of the main result. fv, q>. J f(g)x(det g) dg Z (A) \G (A)
=
Thus (8. 1 2) implies that
tr Pw. oU)
(8. 1 6)
J
=
Z (A) \G' (A)
v x2 q>(g)x (v(g)) dg.
tr P�. o(q>) .
FIRST PROOF OF THEOREM (8.2). For each place v E S fix an irreducible repre sentation of G� and let be the corresponding representation of G• . Choose = 1 , and tr 7r�(q>.) = 0 for 77: � inequivalent to E C�(G�, w;;- 1 ) such that tr Thus identity (8 . 1 6) reads
({>o-�.v O"�
O"v o-�(q>.) n(1r') 1r'(q>) n(1r) 1r(f) 1rn(1r) n(1r'))1r' 1T:v O"v 1T:v) 1r 1r vE1r'S 7r�(q>.) vES JT:�(f.) . n(7r') vn$S 1T:�(q>.) n(JT:) vn$S 1rv(fv) O"v 1T:v O"v 1r' 1r, n(1r') n(1r) . [JL] n(1r) 1r'. nv nv 1rv(fv) 1r ®1rv ®1r�) I;
tr
=
I;
tr
where the sum on the right-hand (resp. left-hand) side is over all irreducible repre sentations of G(A) (resp. of G'(A)) such that � (resp. 1r� � for all v in S, and (resp. is the multiplicity of in Pw. o (resp. P�. 0) . Moreover, if and satisfy these conditions, then, because S has even cardinality, we get Thus = TI I = TI
I;
tr
=
I;
(77:�
tr
�
for v E S,
�
for
vE
S),
and Theorem (8 .2) follows from fundamental principles of functional analysis (applied to the isomorphic groups G5 and G' S). Moreover, if corresponds to then = So since we already know from that is at most 1, we have also proved multiplicity one for SECOND PROOF. We rewrite (8. 1 6) as (8 . 1 7)
I;
tr
=
I;
tr 1T: �(q>.),
where = (resp. 1r ' = runs through all irreducible subrepresentations of Pw. o (resp. P�. 0). What we shall do now is manipulate (8 . 1 7) until it equates the
FORMS OF GL(2) FROM THE ANALYTIC POINT OF VIEW
249 one (8. 1 7) an arbiITtr!varyandplace wIToutsareide as and presentatio(8.18). n ofFiGw.x Iff= in (8.12)an irtreduci hen ble unitary """(f) right side to the trace of just representation. (Since we are summing over sub representations rather than classes, two representations on the right side of may, a priori, be equivalent.) LEMMA
S
'rw
1:
tr =®;rv
= 1:
IT tr
n'=@n �
v-:tw
re
IT tr ,.�(
v;;f:;.w
PRooF. Set a,
It suffices to prove a,w
= 1: IT tr ""v(fv) - 1: IT tr ,.�(
w
v-:t:w
v;t:w
0. (8.17) (f ) f 0 But
=
can be rewritten as
1: tr ""w w IT tr ""v(fv) =
1: tr ,.�(
' n'cpw, O
v'l'ow
or, since Gw
�
G� and w
�
1:
<w in of ( Gw)
a,
w
(f .
tr rw w)
Moreover, fw is completely arbitrary in C�(Gw , w;;?). Therefore a,w must be zero by the generalized "linear independence of characters for (cf. [LL]).
GL(2)" and (8.19). have matForchieveryng orbiw ¢tal ifixntegral(whisforchv is unramifiedfor almost all w). Then if LEMMA ,.Jfv) (8.20) ¢ ( IT � in thererff2(Gexiv) forsts alal univ inqS.ue ®""v such that"""(8. 2 1).,.�Given for all v ¢ ®,.Moreover""" GL(2). v ¢ (8.19) (8. 22) (8.22) G� . v0 ¢ 0, (8.9) (8. 22) S
fv
rw
E
1: IT tr
=
vES
(,. c Pw.O> ""w
S,
1: vES IT tr ,.�(
�
'rw , W S) ,. '
C
P�. o , ""w
�
PROOF. Apply the argument above to the restricted direct product cs =
PROPOSITION in P w.o
,. ,
=
S.
�
'rw , W ¢ S). w fi- S Gw .
,.
P� . o
E
=
PROOF. Uniqueness follows from the strong multiplicity one theorem for What remains to be shown is that there is at least one such ,., and that "" " belong to cr2(Gv) for all in S. Suppose no such ,. exists. Then applying with rw = ,. � for all w S we conclude
1: IT tr rJ�(
=
0
(rJ ' c P �. 0, rJ�
�
,.� for w E S).
But is a sum of characters of the group = IT v E S G� Thus by the linear independence of characters for G� we get a contradiction. Now suppose """o rff2(G") for some in S. The fact that ,. is cuspidal excludes the possibility that "" "o is finite dimensional. So by we have ""voUv0) = and again by we get a contradiction.
STEPHEN GELBART AND HERVE JACQUET
250
COROLLARY (8.23). n� ®n� P�. 0 •
and is in The left-hand side of reduces to exactly one term when Given in suchin that such thator all Moreover, for all v for there exi s t s a uni q ue theretioisns of such that ()�.(t') t each time t' and t sateachisfyv the condi v U h h(zl) w.(z)h(l), l h(x) d (x 1: w
(8.20)
=
PROPOSITION (8.24).
n'
e S,
e.
n
=
®n.
Pw.o
P�. o
® n� ±1 (8. 1 ) . =
=
=
n. f e . 8".( )
n�
n.
=
E
tf2(G.) ¢: S. e G.
eS
v
e G.
Later we shall see that e. in fact equals 1 . PROOF. For E S let x. denote a set of representatives for the equivalence classes of quadratic extensions L. of F. contained in M(2, F.) . As in [JL, § 1 5], we can introduce a measure fjv on F�\L� (L. e X.) and we consider the Hilbert space £. Yf'.(m., fj.) of all functions on that union which satisfy = for z e F�, and J 1 2 fj. ) < + oo ; the functions ()J., A. e C2(G., w.), then deter mine an orthonormal basis of £• . Similarly we can introduce a Hilbert space £'� for the group G�. But since any quadratic extension of F. embeds into we may identify £'. and £'�. Recall n = ®vnv c: Pw.o is such that n. E tf2(G.) for all e S. So for each v e S there is O'v e tf(G�, w.) such that (when we regard ()"• and 8u. as elements of £.) (()"•' 8u.) =1- 0. Since both ()"• and 8a. are unit vectors, it follows that I I � 1. Now, fix f!J v to be 8u. when e S. Then for any 7&� E C(G�, w.), =
D.,
=
a.
v
tr n�(qJ.)
= =
v
a.
1 if 7&� � O'v, 0 otherwise.
In particular, if/, e C';'(G., m;1) and f!J v have matching orbital integrals, then (using the notations of [JL, § 1 5]) tr n.(fv)
=
This means (8.20) with 'l:v
J
Zv\Gv
=
(n
fv(g)8".(g) dg
n. for '
c:
v ¢: S reduces to the identity
P�. 0, 7&�
�
O'v for
v
E
S, 7&�
�
'l&v for ¢: S).
v
From this we conclude that the right-hand side has only 1 term (once again proving multiplicity one for G'). We also conclude that I 1 . So since a simple argument involving nv and ir:v implies must be real, we conclude finally that = ± I , i.e., (()"•' 8u.> = ± 1. But this implies that n� = O'v is completely determined by n . and that ()"• = ± ()"� = 8u. (regarded as elements of £). Thus the proposition follows. It remains to complete the local correspondence asserted by Theorem (8 . 1 ) and to show that (J" actually equals - () "· . For this we need to embed an arbitrary local representation ;.� of G� in an auto n'i. orphic representation n ' and exploit the fact that the cardinality of S is even. For details, see the original arguments in [FI].
a.
I a.
=
a.
FORMS OF
GL(2)
FROM THE ANALYTIC POINT OF VIEW
25 1
REFERENCES [Ar 1] J. Arthur, The Selberg trace formula for groups of F-rank one, Ann. of Math . (2) 100 ( 1 974), 326-38 5 . [Ar 2 ] , The truncation operator for reductive groups, B u l l . Amer. Math. Soc. 8 3 ( 1 977), 748-750. [Bo] A. Borel, A utomorphic £-functions, these PROCEEDINGS, part 2, pp. 27-6 1 . [De] J . Delsarte, C.R. Acad . Sci . Paris 214 ( 1 942), 1 47-149. [DL] M . Duflo and J.-P. Labesse, Sur Ia formule des traces de Selberg, Ann. Sci . E cole Norm . Sup. (4) 4 ( 1 97 1 ) , 1 93-284. [FI] D . Flath, A comparison of the automorphic representations of GL(3) and its twisted forms, Thesis, Harvard University, 1 977. [Ge] S. Gelbart, A utomorphic forms on adele groups, Ann . of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J . , 1 975. [GGPS] I . M . Gelfand, M . Graev and I . I. Piatetski-Shapiro, Automorphic functions and represen tation theory, W.B. Saunders Co. , 1 969. [Go 1] R . Godement, Analyse spectrale des fonctions modulaires, Seminaire Bourbaki 278 ( 1 964) ; also The spectral decomposition of cusp forms, Proc. Sympos. Pure Math . , vol . 9, Amer. Math. Soc . , Providence, R. 1., 1 966, 225-234. [Go 2] --, Notes on Jacquet-Langlands theory, mimeographed notes, Institute for Ad vanced Study, Princeton, N.J., 1 970. [Go 3] , Domaines fondamentaux des groupes arithmetiques, Seminaire Bourbaki 257 ( 1 963), W. A. Benj amin, Inc., New York . [JL] H. Jacquet and R. P. Langlands, A utomorphic forms on GL(2), Lecture Notes in Math. , vol. 1 1 4, Springer-Verlag, Berlin and New York, 1 970. [K] T. Kubota, Elementary theory of Eisenstein series, Halsted Press, New York, 1 97 3 . [La] S. Lang, SL(2, R ) , Addison Wesley, Reading, Mass. , 1 975. [La 1] R. P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math. , vol. 544, Springer-Verlag, Berlin and New York, 1 976. [La 2] Base change for GL(2) : the theory of Saito-Shintani with applications, The Institute for Advanced Study, 1 975 (preprint) . [LL] J.P. Labesse a n d R . P. Langlands, £-indistinguishability for S L , , Institute for Advanced Study, 1 977 (preprint) . [Sa] H. Saito, Lecture Notes in Math. , no. 8, Kinokuniya Book Store Co. , Ltd . , Tokyo, Japan, 1 975. [Se] A. Selberg, Harmonic analysis and discontinuous groups, J. Indian Math. Soc. 20 (1 956), 47-87. [Sh] T. Shintani, On liftings ofholomorphic automorphic forms, U.S. -Japan Seminar on Number Theory, Ann Arbor, 1 975 (see also t hese PROCEEDINGS, part 2, pp. 97-1 1 0) . --
--
CoRNELL U NIVERSITY COLUMBIA U NIVERSITY
Proceedings of Symposia in Pure Mathematics Vol . 33 (1 979), part I, pp. 253-274
EISENSTEIN SERIES AND THE TRACE FORMULA
JAMES ARTHUR PART I. EISENSTEIN SERIES
The spectral theory of Eisenstein series was begun by Selberg. It was completed by Langlands in a manuscript which was for a long time unpublished but which recently has appeared [1]. The main references are
On the functional equations satisfied by Eisenstein series, , Eisenstein series, Automorphic forms on semi-simple Lie groups,
1. Langlands, Springer Verlag, Berlin, 1 976. Algebraic Groups and Discontinuous Subgroups, 2. Summer Research Institute ( Univ. Colorado, 1 965), Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R. I. 1 966. 3. Harish-Chandra, Springer Verlag, Berlin, 1 968. ·
·
In the first part of these notes we shall try to describe the main ideas in the theory. Let G be a reductive algebraic matrix group over Q. Then G(A) is the restricted direct product over all valuations v of the groups G(Q. ). If v is finite, define to be G(o.) if this latter group is a special maximal compact subgroup of G(Q.). This takes care of almost all v. For the remaining finite v, we let be fixed special maximal compact subgroup of G(Q.) . We also fix a minimal parabolic sub group of Let be the maximal defined over Q, and a Levi component split torus in the center of M0 • Let be a fixed maximal compact subgroup of G(R) whose Lie algebra is orthogonal to the Lie algebra of (R) under the Killing form. Then TI is a maximal compact subgroup of G(A). For most of these notes we shall deal only with standard parabolic subgroups ; that is, parabolic subgroups defined over Q, which contain Fix such a Let be the unipotent radical of and let be the unique Levi component of which contains Then the split component, Ap, of the center of is con tained in If is the group of characters of defined over Q, define Hom ((XP) , R ) . Then if m n .m. lies in M(A), we define a vector HM(m ) in by
K. any
P0 ,
KR
K = .K.
P, P, Np P A0• X(Mp) M0 • Q Up Up= Q =
exp ( ( HM(m), x > )
=
K.
M0 P0. A0 A0 P0 . P. Mp Mp Mp = Mp
/ x(m) /
IT / x(m.) / t'
•.
Let Mp(A) 1 be the kernel of the homomorphism HM. Then
( A) is the direct
A MS (MOS) subject clas:. ifications ( 1 970) . Primary 22E5 5 ; Second ary 2 2 E 50.
� 1 979, America n Mathematical Society
253
254
JAMES ARTHUR
product of Mp(A) 1 and A(R)O, the connected component of I in A(R). Since G(A) equals Np(A)Mp(A)K, we can write any x e G(A) as nm , n e Np(A), m e Mp(A), e K. We define Hp(x) to be the vector HM (m) in ap. Let .() be the restricted Weyl group of (G, A0) .Q acts on the dual space of a0• We identify a0 with its dual by fixing a positive definite .0-invariant bilinear form ( , ) on a0• This allows us to embed each ap in a0• Let Zp be the set of roots of (P, A) . These are the elements in X(A)g obtained by decomposing the Lie algebra of Np under the adjoint action of Ap. They can be regarded as vectors in ap. Let t:/Jp be the set of simple roots of (P, A). G itself is a parabolic subgroup. We write Z and 0 for A c and a c respectively. Then t:/Jp is a basis of the orthogonal complement of 3 in ap. Suppose that Q is another (standard) parabolic subgroup, with Q c P. Then QP Q n Mp is a parabolic subgroup of Mp. Its unipotent radical is NG No n Mp. We write aS for the orthogonal complement of ap in a0. In general, we shall index the various objects associated with Q P by the subscript Q and the super script P. For example, t:/JS stands for the set of simple roots of (Q P , A0). It is the pro jection onto a S oft:/J�0\t:/J�0• We shall write �G for the basis of the Euclidean space aS which is dual to t:/JG . If P; is a parabolic subgroup, we shall often use instead of P; for a subscript or superscript. If the letter P alone is used, we shall often omit it altogether as a subscript. Finally, we shall always denote the Lie algebras of groups over Q by lower case Gothic letters. If P and P1 are parabolic subgroups, let .O(a, a1) be the set of distinct isomor phisms from a onto a1 obtained by restricting elements in {) to a. P and P1 are said to be if .O(a, a1) is not empty. Suppose that � is an associated class, and that P e f!JJ .
k
k
•
=
=
i
associated chamber
is called the LEMMA I .
a+ = ap = {H e a : (a, H ) > O, a e t:/Jp} of P in a.
Up1ca»UscD(a, a1J s- 1 (a t ) is
a disjoint union which is dense in
a.
D
Before discussing Eisenstein series, we shall define a certain induced representa tion. Fix P. Let .Yf� be the space of functions
t:/J : N(A) · M(Q) · A(R)O\G(A)
�
C
such that (i) for any x e G(A) the function m � t:/J (mx), m e M(A), is �M CRJ -finite, where �MCRJ is the center of the universal enveloping algebra of m(C), (ii) the span of the set of functions t:/Jk : x � t:/J(xk), x e G(A), indexed by k e K, is finite dimensional. (iii)
ll t1J II 2 =
J JA (R)DM(Q) \M(A) lt:/J(mk) l 2 dm dk K
< oo .
Let .YfP be the Hilbert space obtained by completing .Yf�. If A is in ac, the com plexification of a, t:/J e .Yfp , and x, y e G(A), put
(lp(A , y)t:/J )(x) = t:/J (xy) exp ( (A + pp, Hp(xy) ) ) exp( - (A + p p , Hp(x))) . Here pp is the vector in a such that m e M(A). l det(Ad m) CAJ I = e xp( ( 2pp, HM (m)) ) lp(A ) = I¥(A) is a representation of G(A) induced from a representation of u
,
EISENSTEIN SERIES AND TRACE FORMULA
255
which in turn is the pull-back of a certain representation, IM(A) in our nota tion, of We have
P(A), M(A).
/p(A, y) *
=
/p( - ii ,
and
lp(A , f )*
f(y-1) .
=
y-1), y E G(A), fE C�(G(A)), M(Q)\M(A))
lp( - ii, f* ),
wheref * (y) = In particular, lp(A) is unitary if A is purely imaginary. REMARK. It is not difficult to show that IM(O) is the subrepresentation of the regular representation of M(A) on L 2 ( A(R)0 • which decomposes discretely. We can write IM(O) = EB , a', where a' = ®va� is an irreducible repre sentation of If v is any prime and av is an irreducible unitary representation of ( v), define
M Q M(A). G Q , P(Qv)
and then induced to ( v), the result is a representation If av, A is lifted to /p(av, A) of ( v) acting on a Hilbert space J'f'p(av). In this notation,
GQ
lp(A) = EB,®v lp(a�. A) ·
Thus /p(A) can be completely understood in terms of induced representations and the discrete spectrum of are fixed, and Q( a, a1), let w, be a fixed representative of s in the If P and n G(Q) with the normalizer of Ao. For (/) J'f'� , A ac. and intersection of x G( ), consider
S
E A
P1 K M. s E
E
E
f/J( :;- 1 x) exp ( ( A + pp , Hp( w:;-1nx) )) d exp ( - (sA + P � > H1 (x))) .
n wn N0, H(A) N1(A) w,N(A)w:;-1H(Q)\H(A) \N1(A).) to ThenSuppose the abovethatintegral converges >absolforuteeach ly. in such that sa belongs M(s, K 1 ( C� M(s, -sA). Moreover, if f E G(A)) , the K-conjugate invariant functions in C�( G(A)), M(s, E J't'fl, x E G(A) and E with ReA E the series converges absolutely. _1
N 1 (A) nw,N(A) w, \N1 (A)
(We adopt the convention that if H is any closed connected subgroup of dh is the Haar measure on which makes the volume of one. This defines a unique quotient measure dn on n (a, Re A - pp)
LEMMA 2. - l:p1 •
0
a
D
I:p
The integral, for A as in the lemma, defines a linear operator from J'f'� to £'�1, which we denote by A). Intertwining integrals play an important role in the harmonic analysis of groups over local fields, so it is not surprising that M(s , A) arises naturally in the global theory. LEMMA 3. M(s, A) *
=
A)lp(A, f)
=
lp(sA , f) M(s, A).
D
We now define Eisenstein series : LEMMA 4. /f (/J
A
E(x, f/J, A) = D
2:
oEP (Q) \G (Q)
ac,
PP + a+,
f/J(ox) exp (( A + pp, Hp(ox)) )
JAMES ARTHUR
256
The principal results on Eisenstein series are contained in the following :
A)
E .Yt'�.
M( A)
Supposephitchatfuncti !JJ ons toE(x, !JJOn, iaand, E(x, !JJs,, !JJiscanregulbeaanal y ticalMAIs,ly Ncontis iuninuedtary.as Formeromor r, and equatioEns(x,hold : !JJ, f C�( (Ay E(xandy, !JJt, dy, , the following functional E(x, s,!JJ !JJ, s ) E(x, !JJ, !JJ ted class ofparaboliocnssubgroups. Let suchbe thatthe set of collec tions Let be an associaofmeasurablefuncti s, THEOREM. (a)
M(
A)
E
ac. E Q (a� .
) )K
G
(i) lp(A , f) A) = Sc cAJ f( ) (ii) M( A) A = A), (iii) M(ts, A) = M(t, sA)M(s, A ) . (b) flJ
)
or ,
2
i9
Fp : ia -> .Yt'p
Fp1(sA) = M( A) Fp(A) ,
II F II 2 = k /(A ) - I
(ii)
)
A)
F = {Fp : P E flJ}
(i) If s E Q(a,
A)
o
( 2�Jdim
A
J,.o) Fp(A) II 2 dA <
00 ,
F
where is the number of chambers in Then the map which sends to thefunction n(A)-1( )dim S . E(x, declosed finedG(A)-i for ninvaria adense subspaceL�of(G(Q)\G(A)) extends otof Va uni(G(Q)\G(A)). tary map fromMoreover,ontowea nt subspace have an orthogonal decomposition M V(G(Q)\G(A)) E8 L G(Q)\G(A)). G(A) V(G(Q)\G(A)) d s, A)!JJ E(x, !JJ!JJ, !JJ !JJ N(A)M(Q)A(R)O\G(A) = !JJ (mk) l 2 dm dk x G(A), !JJ(nx) dn = Cf(G(A)) is !JJ HiLEMMA lbert-SchmiIffdt operat or onfor some large the map a.
n(A)
I;
PE!!!
F
A
I 2 . n1
Fp(A) , A ) dA,
'"
i9,
=
i9
0
9
The theorem states that the regular representation of on is the direct sum over a set of representatives { P} of associated classes of parabolic subgroups, of the direct integrals S�tlp(A) A . The theorem looks relatively straightforward, but the proof is decidedly round about. The natural inclination might be to start with a general in .Yt'� and try to prove directly the analytic continuation of M( and A). This does not seem possible. One does not get any idea how the proof will go, for general IJJ , until p. 23 1 of [1], the second last page of Langlands' original manuscript. Rather Langlands' strategy was to prove all the relevant statements of the theorem for in a certain subspace �. cu sp of �. He was then able to finesse the theorem from this special case. Let .Yt'P, cu sp be the space of measurable functions on such that (i) II !JJ II 2 < oo , SKSM(Q) A (R)o\M(A) I (ii) for any Q � P, and E SNQ (Q)INQ (A) 0. It is a right G(A)-invariant Hilbert space. a
5.
E
.Yt'G, cusp ·
0
N,
IJJ -> IJJ * J,
E .Yt'G, cusp•
This lemma, combined with the spectral theorem for compact operators, leads
to
EISENSTEIN SERIES AND TRACE FORMULA JfG, cusp
257
of G(A), each occurring widecomposes th finite multintiopliacidityr.ect sum of irreducible representations cusp forms G(A). .9"(G) COROLLARY.
D
JfG cus p is called the spa�e of on It follows from the corollary, , applied to M, that any function in JfP, cusp is a limit of functions in Jf�. Therefore JfP, cus p is a subspace of Jf P· It is closed and invariant under lp(A). Let be the collection of triplets X = (f1JJ , "// , W), where W is an irreducible representation of K, f1JJ is an associated class of parabolic subgroups, and "// is a collection of subspaces
{ Vp c JfM, cusp • the space of cusp forms on M(A)}pEgo, such that (i) if p [1/J, Vp is the eigenspace of JfM. cusp associated to a complex valued homomorphism of fl'M(Rl • and (ii) if P, P' f1JJ , and s E Q(a, a1), Vp1 is the space of functions obtained by con jugating functions in Vp by • . If p [1/J , define JfP. X to be the space of functions (/) in Jf�. cus p such that for each
E
X
E G(A),E
E
w
(i) the function k --+ (/)(xk), k E K, is a matrix coefficient of W, and (ii) the function M(A), belongs to Vp. JfP. z is a finite dimen sional space which is invariant under Ip(A, f) for any C� ( (A)) K JfP, cusp is the orthogonal direct sum over all X = (f1JJ , "// , W), for which P f1JJ, of the spaces JfP .z · Fix X = (f1JJ, "// , W) and fix P f1JJ . Suppose that we have an analytic function
m --+ (/)(mx), m E
A
--+
(J)(A)
=
(J)(A,
/E G E
E x), E E A
.
a0, x N(A)M(Q)A(R)O\G(A),
of Paley-Wiener type, from ac to the finite dimensional space JfP.z · Then rp (x)
=
( )dim S 2 1
. 1C l
A
ReA=Ao
exp{(A + pp, Hp(x)))(J)(A, x) dA
is a function on N(A)M(Q)\G(A) which is independent of the point A0 compactly supported in the A(R)O-component of x.
E a.
It is
For as above, the�function (x) (G(Q)\G(A)). converges ely anded bybelallongssuchto �-L2Then closed subspabsolut ace generat there anLetorthogonal decomposibe titohne E LEMMA 6.
(/)
=
I:
rp(ox)
oEP(Q) \G (A)
LHG(Q)\G(A))
is
L2 (G(Q)\GA))
=
EB Li(G(Q)\G(A)) .
"/.E.9' (G)
D
Suppose that A0 pp + a + . Then �
rp (x)
=
=
( )dim S . (-. )dim S 1
2 1C l 1
21C l
A
A
Re A=Ao
I;
oEP(Q) \G (Q)
exp( ( A + pp, Hp(ox) ) ) (J)(A, ox) dA
E(x, (J)(A), A) dA . Re A=Ao
JAMES ARTHUR
258
(1)1(Ah x)
Suppose that duct formula for
in terms of (/) and
S
is another function, associated to
(1)1 . �
S
G (Q) \G (A)
We want an inner pro
The inner product is
I;
G (Q) \G (A) oEP 1 CQ) \G (Q)
(x)<jJ� I(Ox) dx (X)(h(x) dx
J ( )dim s = =
� � (x) 1(x) dx
P1 .
Pt (Q) IG ( A) 1 At
JJ J E(namk, (f> (A), A) 211:i 2(ph H1(x))) (h(amk) da dm dk. LEMMA then 7. Suppose that PandP1 are ofthe same rank.lf(f> andRe Aepp E(nx, A) dn S ( (s, A)(f>)(x) (sA H1(x)) ). (associ is empty ; that is, if P and P1 are not Of course, ated.) the right-hand side is ReA=Ao
· exp ( -
K A1 (R) 0Mt (Q) \Mt (A)
At(R) O
e Jt'P. x
a+,
+
(/),
Nt (Q) \Nt (A)
I;
sED (o, OJ)
exp(
M
+ Ph
0 if Q(a, n1)
PROOF. Let {Q} be the set of s e Q such that s- 1 a > 0 for every a e (/)b. Then {Q} is a set of representatives in Q of the left cosets of Q modulo the Weyl group of M. By the Bruhat decomposition,
S
NI (Q) \Nt (A)
= I;
tE {Ol
J
· exp (
E(nx, A) dn (/),
I; 1 N1 (Ql 1Nt (A) ("Ew/ P (Q) w, nNo (Q) INo(Q) l
(f>(w1 J..mx)
v nx))) dn, Hp(w A pp, ( 1 w1N1 wl1 s 1 tWt"1Pwt No\No Wt"1Nwt Nl\Nl . (w.N(Q)w 1 N(Q)\w.N(A)w:;-1\N(A)) (f>(w 1 nx) (A pp, Hp(w:;-1nx))) dn. J +
n M is the unipotent radical of a standard parabolic subgroup of M. If the group is not M itself the term corresponding to above is 0, since (/) is cuspi dal. The group is M itself if and only if s = maps a onto n1. ln this case n is isomorphic to n Therefore the above formula equals
I;
vol
sED (o, OJ)
:;- n
· w,N(A) w; 1 nN(A) \N(A)
:;-
exp(
+
The volume is one by our choice of measure. The lemma therefore follows. Combining the lemma with the Fourier inversion formula, one obtains COROLLARY.
ifJ
D
and P1 are associated, and that and (h are as in the discussion precediSuppose ng Lemmathat P Then 7.
EISENSTEIN SERIES AND TRACE FORMULA
S where
G (Q) IG (A)
=
(/J (/J (x) 1(x) dx
1 . )dim (2 71:1
A
s
1:
ilo+ia s r= O (a, a1J
A0 is any point in pp a+. 0 +
259
(M(s, A)W(A), Wl( - sil)) dA,
1-6
The proofs of Lemmas are based on rather routine and familiar estimates. This is the point at which the serious portion of the proof of the Main Theorem should begin. There are two stages. The first stage is to complete the analytic con tinuation and functional equations for a vector in £�. cu sp· This is nicely described in [2], so we shall skip it altogether. The second stage, done in Chapter 7 of [1], is the spectral decomposition of Li(G(Q)\ G(A)) . From this all the remaining asser tions of the Main Theorem follow. We shall try to give an intuitive description of the argument. The argument in Chapter 7 of [1] is based on an intricate Fix X = (BI'x , r X ' 81' x · For J'l'P , , fl' , we are induction on the dimension of for x x assuming that and are meromorphic on In the process of proving this, one shows that the singularities of these two functions are hyperplanes of the form r = (a, = a and that only finitely many + = 3 (a, e ) > 0, a r meet Let L�,.:( G(Q)\G(A » be the closed subspace of Lr( G(Q)\G(A)) generated by functions where comes from a function as above, which vanishes on the finite set of singular hyperplanes which meet + If comes from 81' x• we can apply Cauchy's theorem to the formula for for
W
W). E(x, W, A)W a+ ia {A{AE ac:ac: ¢(x), ¢ W1(A 1), P1 E
M(s,(A/Z),A)W P E WE ac. P E A)R Af1, f1 E C, E EWp}.1,'p}, W(A),a+ ia. ¢1(x) (/J (x)(/J1 (x) dx. S ( )dim S ( M(s, A)W(A), W1(sA)) dA, 2 s.;J, sA ia . ( _1_.)dim n(A)- 1 2nl S.. ( M(s, tA)W(tA), W 1(tA)) dA (-1-.)dim n(A)_1 2nz s. M(rt- 1 , tA)W(tA), W 1(rA)) dA ( 1 )dim n(A)- 1 s ( M(t, A)-lW(tA), M(r, A)- 1W 1 (rA)) dA 'll:l G (Q) IG (A)
The result is
1
A
.
71: 1
since -
=
on
1:
.
• a s E D (a, a1)
Changing variables in the integral and sum, we obtain
A
1:
1:
1:
PzE i!l'x t E Q (az, a)
•
where
1:
1:
1:
Pz rE D (a2, a1)
·
=
az
A
=
sE O (a , a 1)
r
t az
A
t E O (az, a)
(
1: 1: 1: Pz
r
t
.
t az
2 60
JAMES ARTHUR
(I)
Fr, Pi A)
=
I;
r<=O (az, OJ)
M(r, A)- 1 1/> J (rA),
and Fp2 is defined similarly. Define i&x. x to be the subspace of the space i9x (defined in the statement of the Main Theorem) consisting of those collections {Fp2 : P2 E &>x} such that Fp2 takes values in .Yt'h x ·. We have just exhibited an isometric isomorphism from a dense subspace of L9� x to a dense .subspace of L� · x( G(Q)\G(A )) . Suppose that {Fp2} is a collection of functions in L9 . x each of x x which is smooth and compactly supported. Let h(x) be the corresponding function in L� · x( G(Q)\G(A) ) defined by the above isomorphism. We would like to prove x that h(x) equals h'(x)
=
I;
d ( ) im ASlaz. E(x, Fp2(A), A ) dA . 1
Pz E &x
n (A 2 ) - 1 271: 1
.-
If (/J1(x) is as above, the same argument as that of the corollary to Lemma 7 shows that Az I I; n{A z) - 1 _ h ' (x) /jJ J (x) dx 271: 1 G (Q) \G A)
S
=
(
( )dim .
p2
S E
· iaz r<= z, aJ) ( M(r, A)FpiA), 1/> I{YA)) dA.
Since the projection of /jJ1(x) onto L�x· x( G(Q)\G(A) ) corresponds to the collection { Fl, P2} defined by (1 ) this equals Az . ( Fp2 I: n (A z) -1 _ 1 (A), Fl, Pz(A)) dA 271: 1 1az P2 ,
=
J
( .)dim s
G (Q) \G (A)
h(x)(/J1(x) dx .
We have shown that h(x) = h ' (x). This completes the first stage of Langlands' induction. To begin the second stage, Langlands lets Q be the projection of Li( G(Q)\G(A) ) onto the orthogonal complement of L�.X• x( G(Q)\G(A) ) . Then for any (/J(x) and (/J1(x) corresponding to 1/>(A) and I/>1(A1), (Qrp, ¢1) equals
( )dim A(J I; (M(s, A)IP(A), 1/>1( - sA)) dA r - s. I; (M(s, A)IP(A) , 1/>1( - sA)) dA) . A
_
I
Ao+ia sEQ (a, OJ)
71: 1
10
s
Choose a path in a+ from A 0 to 0 which meets any singular hyperplane t of s E Q ( a, a1)} in at most one point Z(t). Any such t is of the form X( t) + t�, where tv is a real vector subspace of a of codimension one, and X(t) is a vector in a orthogonal to tv . The point Z( t ) belongs to X(t) + tv. By the residue theorem (Q(/J, /jJ1) equals
{M(s, A) :
(
I 2 1. _ 71:
) A-l I; J dim
r
Z (r) +irv
I;
s E Q (a, al)
Res r (M(s, A) IP (A), 1/>1( - sA)) dA .
The obvious tactic at this point is to repeat the first stage of the induction with A 0 replaced by Z(t), 0 replaced by X(t), and E(x, 1/>, A) by Res rE(x, 1/>, A).
26 1
EISENSTEIN SERIES AND TRACE FORMULA
Suppose that rv {H E a : (a, H) 0} for a simple root a E f/)p. Then rv aR , for R a parabolic subgroup of G containing P. If (/) E .Yt'P.x• define ER(x, (/), A) P (Q)I;\R (Q)rf)(ox) exp( ( A + pp, H(ox) ) ) . =
=
=
=
This is essentially a cuspidal Eisenstein series on the group MR( A) . It converges for suitable and can be meromorphically continued . It is clear that
A E ac
I; R (Q) \G (Q)
ER(ox, (/), A) whenever the right-hand side converges. Suppose that AEt, and A AV E r�, Re AV E PR + a]i. Then for any small positive c, RestE(x, (/), A) 271:1 1. J2"0 E (x, (/), A + te2"iOX(r)) d() 21 . J2"0 ER (ox, rf), A + ce2"iOX(r)) do) I; (R (Q) \G (Q) I; rpv(o x) exp( ( Av + PR • HR(ox) ) ) , R(Q) \G (Q) E(x, (/), A)
=
=
X(r) +
AV,
=
=
7r:l
where
rpv(y) 2�i J:"ER (y, (/), (1 =
+ t)X _(r)e2"iO) dO,
the residue at X( r) of an Eisenstein series in one variable. One shows that the func tion MR(Q) \ MR(A), is in the discrete spectrum. Thus
m --> rpv(my), m E
RestE(x, (/), A) E(x, rpv, AV), =
rpv t , A),
the Eisenstein series over R(Q)\G(Q) associated to a vector in .Ye}.,.Yt'R, cusp · Its analytic continuation is immediate. Let be the finite dimensional subspace of By examining Res M(s one obtains the £' consisting of all those vectors operators
rpv,
R
.Yt'R,x
Their analytic continuation then comes without much difficulty. for s Let f?/J ' be the class of parabolic subgroups associated to R. In carrying out the second stage of the induction one defines subspaces L'fp.' x(G(Q)\G(A)) c L i(G(Q)\G(A) ) and i'fp., x c i'fp., and as above, obtains an isom orphism between them. In the process, one proves the functional equations in (a) of the Main The orem for vectors The pattern is clear. For R now standard parabolic subgroup and flJ irP, x and associated class, one eventually obtains a definition of spaces L'fp, x(G(Q)\G(A)) . By definition, is unless R contains an element of and the other two spaces are unless an element of flJ contains an element of flJ If flJ is the associated class of R , there corresponds a stage of the induction in which and part (b) for one proves p art (a) of the Main Theorem for vectors in the spaces LrP, x and L'fp, x(G(Q)\G(A)) . Finally, one shows that
E Q(aR , aQ).
rpv E .Yt'R. x·
{0}
any .Yt'R.x {0}
Li (G(Q)\G(A))
.Yt'R .x•
rpv .Yt'R,x
=
E8 tP
L'fp x(G(Q)\G(A)) . •
any
f/Jx,
x·
262
JAMES ARTHUR
Notice that if R and
f!lJ
are fixed, i9 = E9 i9' X•
J't'R = ffiJ't'R, ...Y • )!
and
)!
L�(G( Q)\G(A) ) = ffiL�' x(G( Q\G(A) ) . )!
The last decomposition together with the Main Theorem yields V(G ( Q ) \G( A) ) = ffiL�' x(G( Q ) \G(A )) .
(2)
9, )!
This completes our description of the proof of the Main Theorem. It is perhaps a little too glib. For one thing, we have not explained why it suffices to consider only those t above such that tv = {A e a : (a, A) = 0} . Moreover, we neglected to mention a number of serious complications that arise in higher stages of the induction. Some of them are described in Appendix III of [l]. We shall only remark that most of the complications exist because eventually one has to study points X(t) and Z(t) which lie the chamber a + , where the behavior of the functions A) is a total mystery.
outside
M(s,
PART II. THE TRACE FORMULA
In this section we shall describe a trace formula for G. We have not yet been able to prove as explicit a formula as we would like for general G. We shall give a more explicit formula for GL3 in the next section. In the past most results have been for groups of rank one. The main references are
Automorphicforms on Sur Ia formule des traces de Selberg, The Selberg trace formula for groups of F-rank one,
4. H. Jacquet and R. Langlands, Berlin, 1 970. 5. M. Duflo and J. Labesse, Ecole Norm. Sup. (4) 4 (197 1 ), 1 93-284. 6. J. Arthur, (2) 100 (1 974), 326-385.
GL(2), Springer-Verlag, Ann. Sci.
Ann. of Math.
We shall also quote from 7. J. Arthur, 32
The characters of discrete series as orbital integrals,
Invent. Math.
(1 976), 205-26 1 .
Let R be the regular representation of G(A) on V(Z(R)O · G(Q)\G(A)) . If � e i0, recall that Re is the twisted representation on V(Z(R)O · G(Q)\G(A) ) given by R e(x) = R(x) exp( (�, Hc(x)) ) . We are really interested in the regular representation of G(A) on £2( G(Q)\G(A)) ; but this representation is a direct integral over � e i0 of the representations Re , so it is good enough to study these latter ones. The decomposition (2), quoted in Part I, is equivalent to V(Z(R )O · G(Q)\G( A)) = ffi L�. x(Z(R)O · G(Q)\G(A )) . 9, )!
Suppose that f e C�( G(A) ) K. Then this last decomposition is invariant under the operator Re (f). Re (/) is an integral operator with kernel K(x, y) = I; fe(x- 1 ry), r oc G (Q)
EISENSTEIN SERIES AND TRACE FORMULA
where
JZ (R)O.f(zu) exp((�, Hc(zu)))dz,
263
E
u Z(R)O\G(A). For every we can expressfas a finiKte sum offunctions of theform f fiE C!"(G(A)) . "1"", W) !/(G)(Z(R)0G(Q)\G(A)).{G}. EG fe(u) =
The following result is essentially due to Duflo and Labesse. N�0
LEMMA 1 .
D
l * J2,
Let !/ ( ) be the set of X = (flJJ , in be the restriction of Re(f) to ffigoffize.9'E (G) L�. x
such that flJJ -:1-
Let Re. e(f)
Then
Rcusp, e (f) = Re(f) - RE. e(f)
is the restriction of Re(f) to the space of cusp forms. It is a finite sum of composi tions Rcusp, e(fl)Rcusp, e(J2) of Hilbert-Schmidt operators and so is of trace class. For each P and X let fJBp, x be a fixed orthonormal basis of the finite dimensional space Jft'P , x · Finally, recall that a c = a� is the orthogonal complement of 3 = ac in a . If A is in iaG, we shall write Ae for the vector A + � in ia.
2. is an integral operator with kernel KE(x, n(A)-I(-
LEMMA
RE(f)
y)
=
.E .E
XE.9'£ (G)
2
p
1
.
1C l
)
dim (A/Z)
·faG te�p, ;(x, lp(Ae , j)f/J, A) E(y, f/J, A)} dA.
The lemma would follow from the spectral decomposition described in the last section if we could show that the integral over A and sum over X converged and was locally bounded . We can assume that/ = fl * f 2 . If Kp, if, x, y) = a
I[Je_Eglp,
X
E(x, lp(A, f) f/J , A)E(y,
finite sum, then one easily verifies that J Kp,z(f, x, y)
J :5: Kp,z(f l * (fl) *, x, x) I I2 Kp, z((J2) *
f/J, *
A),
J2,
y,
y )1 12 .
By applying Schwartz' inequality to the sum over X • P and the integral over A, we reduce to the case that .f = l * (fl ) * and x = y. But then Re. e
f Ke(x, x)
x,
Kcusp(x, y) = K ( y)
One proves without difficulty LEMMA 3.
The trace of
Rcusp, e(f)
equals
- KE x, y) .
J Z (R) OG (Q) \G (A) Kcusp(X, X) dx.
(
D
E
264
JAMES ARTHUR
Of course neither K nor KE is integrable over the diagonal. It turns out, however, that there is a natural way to modify the kernels so that they are integrable. Given let 'rp (resp. fp) be the characteristic function of { H E a 0 : a (H) > 0, a E C/Jp ( resp. p(H) > 0, p E �P )} . (Recall that �P is the basis of ac which is dual to C/Jp.) Then 'rp ;:;:; fp. Suppose that T E a 0 .
P,
any
rp( H(ox) - T) finite. D
4. partForicularP, sum is P P J
LEMMA x E G(A). In
L: oEP (Q) \G (Q)
the
n e fun t n of
is a locally bou d d c io
We will now take T E a t. We shall assume that the distance from T to each of the walls of at is arbitrarily large. We shall modify K(x, x) by regarding it as the term corresponding to = G in a sum of functions indexed by If x remains within a large compact subset of Z(R) 0 G(Q)\G(A), depending on the functions corre sponding to # G will vanish. They are defined in terms of KP(x, y)
=
T,P.
1: h(x- l npy) dn,
N (A) p.cM (Q)
the kernel of R[(f), where RP is the regular representation of G(A) on L2 (Z(R) 0N(A)M(Q)\G(A)). Define the modified function to be kT (x)
=
1: ( p
-
J )dim (A/Z)
1:
o c P (Q) \G (Q)
KP (ox, ox)rp(H(ox) - T) .
For each x this is a finite sum. The function obtained turns out to be integrable over Z(R) 0 G(Q)\G(A). We shall give a fairly detailed sketch of the proof of this fact because it is typical of the proofs of later results, which we shall only state. We begin by partitioning Z(R) 0 · G(Q)\G(A) into disjoint sets, indexed by which depend on T. Fix a Siegel set r in Z(R) 0\G(A) such that Z(R) 0 \G(A) = G(Q)r. Consider the set of x E r such that p(H0 (x) - T) < 0 for every p E �0 . It is a compact subset of r. The projection, G( T), of this set onto Z(R)OG(Q)\G(A) remains compact. For any we can repeat this process on M, to obtain a compact subset of A(R)OM(Q)\M(A), which of course depends on T. Let T) be its characteristic function. Extend it to a function on G(A) by
P,
P
FP(nmk, T)
=
FP(m,
FP(m,
T),
n E N(A), m E M(A), k E
This gives a function on N(A)M(Q)A(R)O\G(A). If Q be the characteristic function of
c
P,
K.
define -r� (resp. r�) to
{H E a 0 : a (H) > O , a E C/J� ( resp. p(H) > O, p E ��) } .
The following lemma gives our partition of N(A) M(Q)A(R)O\G(A). It is essentially a restatement of standard results from reduction theory. LEMMA 5. Given
P, equals !for almost all 1:
1:
{Q: PocQcP) oEQ (Q) \P (Q)
x E G(A).
FQ(ox, T)-r� (H(ox)
D
We can now study the function kT(x). It equals
-
T)
EISENSTEIN SERIES AND TRACE FORMULA
�
(Q, P:PocQcP)
�
( - l ) di m (AIZ)
oEQ (Q) \G (Q) ·
265
FQ(ox, T)1:�(H(ox) - T)
fp (H(ox) - T)KP(ox, ox).
Now for any H E a0 we can surely write =
7:�(H) rp(H)
�
{P�o Pz:PcP tcPz)
( - l ) d im CA tiAz) 7:�(H)f2 (H)
where (J MH)
=
(J� t(H)
=
�
{Pz: Pz=>Ptl
( - l )dim CAtiAz ) 7:� (H)f 2 (H).
To study (Jh, consider the case that G = GL3 . Then a0/3 is two dimensional, spanned by simple roots a1 and a 2 • Let Q = P0, and let P 1 be the maximal para bolic subgroup such that a1 = {H E a0 : a 1 (H) = 0} . Then (Jb is the difference of the characteristic functions of the following sets :
The next lemma generalizes what is clear from the diagrams. LEMMA 6. Suppose that H is a vector in the orthogonal complement of 3 in a Q If (Jb(H) # 0, and H = H* + H * , H* E a� , H * E ab then a(H* ) > 0 for each a E f/J¢ and II H * II < c ii Hd for a constant c depending only on G. In other words H * belongs to a compact set, while H* belongs to the positive chamber in a¢. D .
We write kY(x) as
�
QcP1
�
oE Q (Q) \G (Q)
FQ (ox, T) (J�(H(ox) - T)
�
{P:QcPcPtl
( - I )di m (A IZ) KP(ox, ox).
The integral over Z(R)0 G(Q)\G(A) of the absolute value of kT(x) is bounded by the sum over Q c P 1 of the integral over Q(Q)Z(R)0\G(A) of the product of FQ(x, T)(Jb (H(x) - T) and
(1)
I � ( - l )d i m (A/Z) I {P:Qcpcpl)
f
N (A)
�
f1EM (Q)
j
fe(x- 1 n px) dn .
266
JAMES ARTHUR
We can assume that for a given x the first function does not vanish. Then by the last lemma, the projection of HQ(x) onto a � is large. Conjugation by x- 1 tends to stretch any element M(Q) which does not lie in Q(Q). Since f is compactly supported, we can choose T so large that the only which contribute nonzero summands in ( I ) belong to QP(Q ) = Q(Q) n M(Q) = MQ (Q) NS(Q). Thus, ( 1 ) equals
f1 E
I
I;
f1
I;
11EMQ (Q) IP:QcPcP,l
which is bounded by
I;
I
I;
11EMQ (Q) IP:QcPcP,l
( - l )d i m CAIZJ
J
( - I )dim (A/ZJ
J
I;
N (A) v e N
� (Q)
I; N (A)
vE 't?o (Q)
I,
fe(x- lvJ)nx) dn
l
fe(x-1.uvn x) dn .
Recall that n0 is the Lie algebra of N0• Let < , ) denote the canonical bilinear form on n0. If X n0, let e(X) be the matrix in N0 obtained by adding X to the identity. Then e is an isomorphism (of varieties over Q) from n0 onto N0• Let ¢ be a nontrivial character on A /Q. Applying the Poisson summation formula to nS, we see that ( I ) is bounded by
E
I
j
J
I; I; ( - l )di m CA/ZJ I; fe(x-l.u e(X)x)cj;((X, 0) dX . P 11 CEn � (Q) "Q (A)
If nb(Q ) ' is the set of elements in nb(Q ) , which do not belong to any n{;(Q ) with
Q
c
P�
P 1 , this expression equals
I
1:
f:
11 CEnQ \Q) '
J
nQ (A)
fe(x- 1,u e(X) x)¢ ((X, 0) dX I .
Therefore the integral of lkT (x)l is bounded by the sum over Q c 1 and MQ (Q) of the integral over (n * , n * , a , m , k) in Ni (Q)\N1 (A) X N�(Q) \ Nb( A) X Z(R) O\AQ ( R)O X AQ (R) O MQ (Q )\ MQ( A) X K of
FQ(m , T) ab(H (a)
fc I J
• CEn Ql '
- T) exp ( -
"Q CAl
P f1
2(pQ,
H(a) ) )
I
fe (k- 1 m-la-1n ;,/ n* - ' · .ue(x) · n*n * amk)cj;( (X, O ) dX .
E
The integral over n * goes out. The integrals over k and m can be taken over com pact sets. It follows from the last lemma that the set of points { a-1 n * a }, indexed by those n* and a for which the integrand is not zero, is relatively compact. Therefore there is a compact set C in Z(R) 0\ G(A) such that the integral of iF(x) l is bounded by the sum over Q c MQ (Q) and the integral over x C of
S
PI > f1 E
Z (R) 0\AQ (R) 0
· =
I;
CEnb (Q) '
J
exp( - 2(pQ , H(a))) a�( H(a) - T)
jJ
I
fe(x-1.ua- 1 e(X)ax)cj;( <X, �)) dX da
"Q (A)
Z (R) 01AQ (R) 0
E
a� (H(a)
- T ) L;
[ J"Q (A) .fe (x- 1 .ue(X)x)
C I
I
· cp((X, Ad(a)O ) dX da .
2 67
EISENSTEIN SERIES AND TRACE FORMULA
Since f is compactly supported, the sum over f.1- is finite. If Q unless of course Q = = G, when it equals 1 . If Q � Pb
Y -->
J
P1
I1Q (A)
=
Ph (J� equals 0,
Y ))
fe(x-1w(X)x) ¢(< X, dX,
is the Fourier transform of a Schwartz-Bruhat function on n¢{A), and is con tinuous in x . If H(a) = H* + H * , H* E a¢, H * E a d a , H * must remain in a com pact set. Since H* lies in the positive chamber of ab, far from the walls, Ad(a) stretches any element � in n�(Q)'. In fact as H* goes to infinity in any direction, Ad(a)� goes to infinity. Here it is crucial that � not belong to any nG (Q), Q c P � It follows that if Q � Ph the corresponding term is finite and goes to 0 exponen tially in Thus the dominant term is the only one left, that corresponding to Q = P 1 = G. It is an integral over the co mpact set We have sketched the proof of
P1 . T.
THEOREM 1 . We can choose c
\\'al/S,
S
Z (R) DG (Q) \G (A)
>
G(T).
s ch thatj01· any
0u
x dx
kT ( )
=
J
G
kT(x)
a(j,
T E sufficiently K(x, x) dx + O(e -•I ITII ) .
far from the
0
We would expect the integral of to break up into a sum of terms corre sponding to conjugacy classes in G(Q). It seems, however, that a certain equivalence relation in G(Q), weaker than conjugacy, is more appropriate. If f.1- E G(Q), let f.l-s be its semisimple component relative to the Jordan decomposition. Call two ele ments f-1. and in G(Q) equivalent if f.l- s and f.l-; are G(Q)-conjugate. Let !fl be the set of equivalence classes in G(Q). If o E !fi', define
f.J- '
Kf(x, y)
and
Then
k;(x)
=
I;
I'EM (Q) n o
I; ( - 1 ) dim (AIZ) P
J
fe(x-1f.J-ny) dn, Kf(ox, ox)ltp (H(ox) - T). kT(x) k�(x).
=
N (A)
I;
/i E P (Q) \G (Q)
=
I;
or=W
If f.1- E G and H is a closed subgroup of G, let H" denote the centralizer of f.1- in H. LEMMA 7.
Fix
P. Then for f.1- E M(Q) and rp E C :,"'(N(A)) ,
I;
C E Np, (Q) \N (Q)
I;
" "' N1,, (Q)
rp(f.J-- 1 � - I f.l-))�)
=
I; rp(v).
vc N (Q)
0
This lemma is easily proved . It implies that for o E !fi', P(Q) n o = (M(Q) n o)N(Q). If this fact is combined with the proof of Theorem 1 we obtain a stronger version. THEOREM I * . There are positive constants C
I; oE 'l!'
s
Z (R) DG (Q) \G (A)
such that
and \kr(x) - FG(x, T)K�(x, x) dx c
I
� ce-eiiTI .
0
268
JAMES ARTHUR
The integral of kf(x) cannot be computed yet. What we must do is replace kHx) by a different function. Define Jr(x, y)
=
J
f�(x- 1 (-Ip.n(y) dn. I: I: p E M (QJ n o CENp, (Q) \N(Q) Np, (A)
It is obtained from K{(x. y) by replacing a part of the integral over N(A) by the corresponding sum over Q-rational points. Define j f(x)
=
I: J : (ox, ox) ?p(H(ox) - T) . I: ( - l )di ( A/Z ) p IJEP(Q) \G (Q)
m
The proof of the following is similar to that of Theorem
1 *.
THEOREM 2 . There are positive constants C and c such that
I: DE-l
sz (R) OG (Q) \G (A) l jr(x) - FG(x, T)K�(x, x) I dx � ce-•IITII.
D
Suppose that T1 is a point in T + at . By integrating the difference of k;[(x) and kr1(x) one proves inductively that the integral of kHx) is a polynomial in T. The same goes for the integral ofj;[(x). Since the integrals of kf(x) andj;[(x) differ by an expression which approaches as T approaches oo , they must be equal. Summariz ing what we have said so far, we have
0
THEOREM 3.
S (R) OG (Q) \G (A) kT(x) dx Z
whe re J f(f�)
=
Jz cRJ OG (Q) \G C AJ H(x) dx .
=
I: Jr(J�),
D
Suppose that the class o consists entirely of semisimple elements, so that o is an actual conjugacy class. The centralizer of any element in o is anisotropic modulo its center. The split component of the center is G(Q)-conjugate to the split com ponent of a standard parabolic subgroup. Thus, we can find fl.o e o and a standard parabolic P0 such that the identity component of Gl'o is contained in Mo and is anisotropic modulo A0• We shall say that the class o is unramified if Gl'o itself is contained in M0• Assume that this is the case. We shall show how to express J;[(f�) as a weighted orbital integral off�. Given any P, suppose that fl. E 0 n M(Q). Then by the same argument, we can choose an element s in U p1 Q(a0, a1) and an element '1} e M(Q) such that sa" contains a, and p. = '1}Wsf1.ow;1'1)- 1 . Let Q(a0 ; P) be the set of elements s in U p p(a0, a1) such that if a 1 = sa0, a1 contains a, and r 1 a is positive for every root a in (1)�1 . Then if we demand that the element s above lie in Q(a0 ; P), it is uniquely determined. Thus J!'(ox, ox) equals
Therefore H(x) equals
EISENSTEIN SERIES AND TRACE FORMULA
Since the centralizer of W5,U0w,;-1 in G is contained in
I:
oEGp0 (Q) IG (Q)
fe(x- 1o- 1,uox) I: ( - J ) dim (A /Z) P
Then I[(fe) equals
(2)
vol(A o(R) O GI'0(Q)\GI'0 (A))
where
v(x, T)
=
M,
269
this equals
I:
sEQ(a0;P)
rp (H(wsox) - T) .
JGp0 (A) IG (A) fe(x-l,ux) v(x, T) dx,
JZ (R) DIA0(R) 0 { I; ( - J )dim CA/ZJ sEQI;(a0 ;P) rp(H(w5ax) - T) da } . p
The expression in the brackets is compactly supported in a . In fact it follows from the results of that v(x, T) equals the volume in a0/3 of the convex hull of the projection onto a0/3 of { s - 1 T - r1H(w5x) ; U p p(a 0 , a 1 ) } . It was Lang lands who surmised that the volume of a convex hull would play a role in the trace formula. By studying how far J[(fe) differs from an invariant distribution, I hope to express J[(fe), for general o , as a limit of the distributions for which o is as above, at least modulo an invariant distribution that lives on the unipotent set of G(A). However, this has not yet been done. The study of KE(x , x) parallels what we have just done. The place of { o E
[7, §§2, 3]
sE
E
E
where nP(AQ) is the number of chambers in aQ/a. Then if P
KP(x, y)
=
f=
G,
I; Kf(x, y) .
XE.5PE (G)
The convergence of the sum over X and the above integral over the argument of Lemma Define
ki(x)
=
2.
A
is established by
I; ( - l )dim (A!Zl OEP(Q)I;\G (Q) Kf{ox, ox)fp(H(ox) - T) . p
Then
Kcusp(x, x)
=
I; k'[(x) - XE.5PE(G) I; k{(x).
o E�
We would like to be able to integrate the function ki(x). But as before, we will have to replace it with a new function JI(x) before this can be done. To define the new function, we need to introduce a truncation operator. In form
270
JAMES ARTHUR
•
it resembles the way we modified the kernel K(x, x), but it applies to any continuous function ¢ on Z(R)O · G(Q)\G(A). Define a new function on Z(R) 0 G(Q)\G(A) by
d I: ( - I ) i m A /Z
=
(AT rp) (x)
J
(
P=>Po ·
N (Q) IN (A)
)
I:
ocP
(Q) \G (Q)
rp( H( o x)
- T)
rp(nox) dn.
AT has some agreeable properties. It leaves any cusp form invariant. It is a self adjoint operator. These facts are clear. It is less clear, but true, that AT oAT = AT. If X .'/'E( G), define
E
( )dim (A/Z)
�
J{(x, y) =
] n(A )- l -2 i n
S tc�p, 1E(x, Jp(A, ,f )C/J, A) AT iaG
·
E( y,
}
(/), A) dA .
If we apply AT to the second variable in KE(x, y) we obtain I: xcc9'E (c ) Jf(x,y). Define j {(x) J'[(x, x) . THEOREM 4. The funct ion I: xc9'E (G) lji(x) l is integrable over Z(R) 0 G(Q)\G(A) . For any X, the integral ofji(x) equals =
I; n(A) - l p
·
J
iaG
(-1 )dim (A/Z)
I:
2
.
Jr l
'i! E fHp, X
J
Z (R) OG (Q) IG (A)
ATE(x, lp(Ar;, f)C/J , A) ATE(x, C/J, A) dx dA.
The first statement of the theorem comes from a property of AT. Namely, if ¢ is a smooth function on Z(R)0 • G(Q)\ G(A) any of whose derivatives (with respect to the universal enveloping algebra of g(C)) are slowly increasing in a certain sense, then AT¢ is rapidly decreasing. The proof of this property is similar to the proof of Theorem The Poisson formula can no longer be used, but one uses [3, Lemma 1 0] instead. Given the fact that AT oAT AT, the other half of the theorem is a statement about the interchange of the integrals over x and A . By the proof of Lemma we can essentially assume the integrand is nonnegative. The result fol lows. 0
2.
=
2
THEOREM 5. There are positive constants C and c such that 9'E (G) XE I:
s Z (R) OG (Q) IG (A) l k{(x) - j{(x) l dx ;;;:;
ce-tiiTI I .
It turns out that this theorem can be proved by studying the function F(x) AIK(x , y) , at x = y. This is essentially the sum over all X of the above integrand (without the absolute value bars) . The point is that the new function is easier to study because it has a manageable expression in terms off 0 Combining Theorems 4 and 5, we see that Jz (R) OG (Q) IG ( A) I: x E9'E (Gl lk{(x) l dx is finite. In particular, each k{(x) is integrable. With a little more effort it can be shown that the integrals of k�(x) and .i{(x) are actually equal. We shall denote the common value by J[(f,). It is a polynomial in T. We have shown that
EISENSTEIN SERIES AND TRACE FORMULA
27 1
for any suitably large T E at. The right-hand side is a polynomial in T while the left-hand side is independent of T. Letting l.(f,) and life) be the constant terms of the polynomials JT;, (fe) and li(/e) we have THEOREM 6. For any fe C�(G(A))K,
tr Rcusp, e(/) = 'L J.(fe) oE\6'
-
1:
XES"E(G)
life ) .
D
Suppose that x = 1', and that P e PJ>. Then if (/J e .Yl'p, x , ATE(x, (/), A) equals the function denoted E"(x, (/), A) by Langlands in [2, §9] . This is, in fact, what led us to the definition of AT in the first place. If (/J' is another vector in .Yl'P,x , Langlands has proved the elegant formula
(PJ>, W)
S
Z (R) O·G (Q) \G (A)
= 1:
E"(x, (JJ ' , A')E"(x, (/), A) dx
I;
I;
e
p2 tED (a, 02) SED(a, 02) n aE(/)/ tA' + sA, a)
(M(t, A')(JJ ' , M(s, A)(JJ)
(see [2, §9]. Actually the formula quoted by Langlands is slightly more complicated, but it can be reduced to what we have stated.) In this formula, we can set A' = A and (/J' = Ip(A, f)(/J. We can then sum over all (/J in &ip,x and integrate over A E iaG. The result is not a polynomial in T. To obtain l i(fe ) we would have to consider all P, not just those in the associated class PJ>. The best hope seems to be to calculate residues in A and A' separately in the above formula. For GL3 the result turns out to be relatively simple. PART III. THE CASE OF GL3
In this last section we shall give the results of further calculations. They can be stated for general G but at this point they can be proved only for G GL3 • Our aim is to express the trace of Rcusp, e(f) in terms of the invariant distributions defined =
m
8. J. Arthur, On the invariant distributions associated to weighted orbital integrals, preprint.
A trace formula for K-bi-invariant functions has also been proved in 9. A. B. Venkov, On Selberg 's trace formula for SL 3 (Z), Soviet Math. Dokl . 17 ( 1 976), 683-687.
First we remark that the distributions J'[(f,) and JI(fe ) are independent of our minimal parabolic subgroup so there is no further need to fix P0 . If is any Q split torus in G, let denote the set of parabolic subgroups with split component They are in bijective correspondence with the chambers in In fact, if P0 is a minimal parabolic subgroup contained in an element P of then
&i'(A)
A.
If P'
E &i'(A),
PJ>(A)
= U A1
U
Si00 (a, OJ)
w:;-1P 1 w,.
and P' = w:;-1 P 1 w., define
(Mp• 1p(A)(JJ)(x) = (M(s, A)(JJ ) (w,x),
PJ>(A),A.
(/J E .Yl'(P).
A
2 72
JAMES ARTHUR
Then Mr i P (A) is a map from .Yt(P) to .Yt(P'), which is independent of P0. In fact, if Re A E PP + at, (Mp , 1 p( A) q)) (x) =
(1 )
J
N (A) n N ' CA) \N' (A)
q)(nx) exp(
· exp( -
We have changed our notation to agree with that of [8]. A is said to be a special subgroup of G if .?JJ ( A) is not empty. Suppose that A and A1 are special subgroups, with A :::> A 1 . We write QM1(a, a)reg for the set of elements s E Q(a, a) whose space of fixed vectors in a is a1 . Suppose that P1 E .?JJ (A 1) and Q E &MI(A), the set of parabolic subgroups of M1 with split component A. Then there is a unique group in .?JJ(A), which we denote by P1(Q), such that P1(Q) c P1 and P1(Q) n M1 = Q. LEMMA 1 . Suppose that A and A1 are as above and that P = P1(Q) for some P1 E .?JJ (A 1) and Q E &MI(A). Then if A E iab the limit as ). approaches 0 of
I;
P 1 E&' (AJ)
Mp ! (Q) I P(A) - I MP! (Q) I P(A + ).)
(aE) -!
exists as an operator on .Yf'p. We denote it by M(P, A I > A).
This lemma follows from [8] . D Fix a maximal special subgroup A0 of G. From Langlands' inner product formula, quoted at the end of Part II, one can prove LEMMA 2 . For any X E .9"E( G), JifrJ equals the sum over all special subgroups A 1 and A of G, with A 1 c A c A 0, and over s E QM1(a, a)reg of
cs
J...�
A e ) M(s
, 0) lp(A J)q) , q)) dA . e
Here Cs is the product of
( )dim J
(AJIZ)
211:i
n M(A o) · n(A o) - l . i det( l - Ad(s)) a;a1 l - 1
with the volume of afo modulo the lattice generated by q)],, and P is any element in
&(A 1 ) which contains some group in &(A).
D
Recall the decomposition /p(A) = EB ,@vfp(a �. 11) . If q)v is a smooth vector in and Re A E pp + at, define
.Ytp(a�)
( Mp , 1 p(a�' A)q)v)(x) =
J
N (Q,) n N ' (Q,) \N' (Q,)
W.(nx) exp( (A + pp, Hp(nx)) )
· exp ( -
for x E G(Qv). This is the usual unnormalized intertwining operator for a group over a local field. Then MP' I P(A) = EB ®Mr f p(a�, A) . '
v
LEMMA If O" v is an irreducible unitary representation of M(Qv), we can define meromorphic functions rp , 1p(O" v, A ) P, P'E .?JJ (A), A E ac, so that the operators
3.
,
EISENSTEIN SERIES AND TRACE FORMULA
273
Rr 1 p ((J v, A ) = Mr i PCav, A ) · rp • 1 p ((J v, A ) - l can be analytically continued in A, with the following functional equations holding :
P, P', P" E &' (A), and Rp, j p((J v, A ) * = RP I P ' ((J v, _ jj).
Moreover if (J v is of class and 1/J v is the Kv-invariant function,
1
Rr i P((J v , AY/J v = 1/J v .
0
Suppose that (J = ® v(J v is an irreducible unitary representation of M( A). If 1/J = @ vi/J v is a smooth vector in .Yf'p((J) = @ v.Yf'p((J v) , define RP' I P((J)I/J = @ Rr i P((J v)I/Jv·
v
For almost all v, the right-hand vector is the characteristic function of Kv . Define
Ml to be the kernel of the set of rational characters of M defined over Q . Then M l is defined over Q, and M(A) = M1(A)A(A). Let f be a function in C�(G( A))K, as in Part II. LEMMA
4.
There is a function rp A(f) in Coo(M(A) )KnMCA) such that m E Ml(A), a E A(A),
is compactly supported in m and a Schwartz function in a, so that the following pro perty holds. If (J = ® v(J v is any irreducible unitary representation of M(A),
· tr(RP' I P((J)-l Rp, 1 p(a;.)lp((J, j)) .
(The limit o n the right exists and is independent of the fixed group P E &I' (A).) 0 Suppose that o is an equivalence class in G(Q). Then M(Q) n o is a finite union (possibly empty), or u . . . u oi1, of equivalence classes relative to the group M(Q). If F0M is a function defined on the equivalence classes relative to the group M( Q), let us write
Fo
=
n
nM(A o)n(A o) -l I; F0M . t"=l
t
Now we shall define an invariant distribution /0 for each o E �- The definition is inductive ; we assume that the invariant distributions /� have been defined for each special subgroup A, with � A c A0. We then define
Z
LEMMA 5. 10 is invariant.
This is essentially Theorem 5.3 of [8]. Note that this lemma is necessary for our inductive definition, since ¢A(f) is only defined up to conjugation by an element in
M(A).
0
Suppose that (J = ® v(J v is a unitary automorphic representation of M(A) . Then
JAMES ARTHUR
274
P, P' E
�'P' 1 P(a) = IT �'r i P(
.9'(A),
is essentially a quotient of two Euler products and is defined by analytic continua tion. If A ia , let rr iP(A) be the operator on £'p which acts on the subspace deter mined by Ip(O'') by the scalar rr iP(a�). If A ::J A 1 define the operator A I > A) on .Yt'p by the limit in Lemma 1 , with Mp1 cQJ IP(A)-1Mp1 (Q) IP(A + A) replaced by rp 1 (QJ IP(A)-1rp1 cQJ IP(A + A). It commutes with the action of G(A). Finally, for X E Y'E(G), define iift) by the formula for Jx (f�) in Lemma with M(P, Al> A�) replaced by A I > A�) . Then ix is an invariant distribution.
E
r(P,
2,
r(P,
THEOREM 1 . For any jE C� ( G(A) ) K,
tr Rcusp, � (f) = L Jo(f� ) oE'if
I;
XE.'/'£ (G)
i/Je ).
D
REMARK. It follows from formula of Part II that if o consists entirely of semi simple elements, I. is one of the invariant distributions studied in [8]. Moreover since G = GL3, it is possible to show that for any o, is a sum of limits of the invariant distributions in [8]. Suppose that/ = TI v!v and that for two places v ,
(2)
J
T (Qv)
\G (Ovl
f.(x- 1 tx) dx =
I.
t
0, E
T{Q v) reg•
for all maximal tori T in G such that Z(Q.)\ T(Q.) is not compact. It follows from the last theorem of [8] that if there exists an element r in a given o which is Q elliptic mod Z, I.(f�) = vol (Z(R) O · Gr (Q)\Gr (A) )
J
Gr (A) \G (A)
f�(x-Irx) dx,
and that I.(f�) = if no such r exists. Moreover, it is easy to see that each ix (f�) = 0. From this it follows that if {r} is a set of representatives of G(Q)-conjugacy classes of elements in G(Q) which are elliptic mod Z,
0
tr R cus p, �(f) = .E vol ( Z(R) 0 · Gr (Q)\Gr (A)) for f a s above. DUKE UNIVERSITY
�
J
s�w�
f�(x- 1 rx) dx,
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 275-285
0-SERIES AND INVARIANT THEORY
R. HOWE 0. Introduction. Historically 0-series have been one of the major methods of con structing automorphic forms. In [Wi 1] Weil reported the discovery that a certain unitary representation, also important in quantum mechanics, underlay the theory of 8-series, especially as developed by Siegel [Si 1], [Si 2]. This note is a sum mary of work which extends Weil's representation theoretic formulation of the theory of 8-series. So far, this formulation provides a systematic framework in which to place recent work by several authors as well as several important older computations (see [G] for examples). A key feature of this approach is its strong ties with Classical Invariant Theory, as exposed in [Wy]. Indeed, it would seem that the theory of 8-series is in a sense a transcendental version of Classical Invariant Theory. Convention. Throughout F will denote a field, never of characteristic More precise specifications on F will vary from paragraph to paragraph, and will be stated at the beginning of each.
2.
1 . The Heisenberg group and the Stone-von Neumann Theorem ; local case. Let F be a local field. Let W be a symplectic vector space over F, with form ( , ) . Define H( W), the Heisenberg group attached t o F by : H( W) = W $ F as set, and has group law (w, s)(w', s') = (w + w', s + s' + t (w, w' ) ). Let x_ be a nontrivial character of F.
STONE-VON NEUMANN THEOREM. There is only one equivalence class of irreducible unitary representations of H with central character X . We denote this representation by Px: The corresponding representation on the smooth vectors [C 2], [Wa] will be written p';. Although this representation is unique up to equivalence, it has, as already noted by Cartier [C 1], many different concrete realizations which can be organized into smooth families. This circumstance seems to be at the heart of the theory of the oscillator representation. 2. The oscillator representation, local case. F continues to be a local field. Let Sp be the isometry group of ( , ) . The action g ((w, s)) = (g(w), s) for g e Sp, and (w, s) e H( W) embeds Sp into the automorphism group of H( V). According to [M] there is a unique nontrivial two-fold cover S p of Sp (except A MS (MOS) subject classifications (1 970) . Primary 22E5 5 . © 1 9 79, American Mathematical Society
275
276 if F = map.
R. HOWE
C,
when we will understand Sp = Sp). Let g -->
g
denote the covering
THEOREM (SHALE-WEIL). There is a representation wx (unique up to unitary equi valence) of S p on the space of Px such that (2. 1 ) for g E S p and h E H. To (l)X' there is a corresponding smooth representation wr. The smooth vectors of Px and of wx are equal (in any common realization) so Px and wX' act on the same space. We call wx , or the corresponding multiplier representation of Sp, the oscillator representation. 3. Adelization. Let F now denote a global or A-field. Let v be a typical place of F and F" the completion of F at v. Let A be the adeles of F. If W = WF is a sym plectic vector space over F then we can form H( Wp) and also its adelization H( WA). As usual H( Wp) is embedded as a discrete subgroup of H( WA), with com pact quotient. We will suppress W, and just write H( Wp) = Hp, and H( Wp) = Hv , etc. Let { e,-, be a symplectic basis for W. Let Fv be the closure in H" of the sub group generated by the elements { (e,-, of Hp. Then HA is the restricted direct product of the H" with respect to the r". Let X be a nontrivial character of A with F � ker X · Let { x v} be the local factors of X · At almost all places, there will be a unique vector
/;}
0), (/;, 0)}
[F].
(2.1)
(3. 1 ) (3.1)
The products i n are restricted products with respect to the J". (For almost all v (in fact for v not dividing there is a unique lifting of J" to S p". We use this lifting to consider J" a subgroup of S p" when convenient.) For almost all v the vec tor SpA such that the triangle
2),
0-SERIES AND INVARIANT THEORY
277
commutes, where the horizontal and vertical arrows are the canonical injection and projection respectively.
Thus we may speak unambiguously of w ig) , for g belonging to SpF. 4. The 8-distribution. F continues as in
§3.
THEOREM 4. 1 . There is a linear functional 6) on (the space Y of) Px which is invariant by HF. That is, B(p';(h)(y)) = B(y) for h E HF and y E Y. The functional e is unique up to multiples.
This result is essentially a strong form of the Poisson Summation Formula. Results of the same type are in [C 1]. Existence of e is implicit in Weil [Wi 1], and exploited classically. Existence of 6) is more or less responsible for the lifting (3 of
§3.
Since SpF normalizes HF, we see by formula (2. 1 ) and the uniqueness of e that w x <SpF) preserves e up to multiples. Since SpF is its own commutator subgroup, 0 will in fact be invariant by SpF. Therefore, if u is a vector in the space of p';, the function on S pA defined by O(u)(g) = O (w; lg)u) , g e SpA, actually factors to the coset _ space X = SpF\SPA· It may also be computed that O(u)(g) is of "moderate growth" at oo on X, so that actually O(u) is an automorphic form on S pA- We call O(u) the theta-series attached to u. If dim W = 2 and u is chosen appropriately, then O(u) will be the classical 0-series of Jacobi. With somewhat more general u, but still a restricted class, one will obtain "0-constants with characteristic" [I]. Thus the 0-distribution gives rise to automor phic forms which generalize well-known 0-series. However, the functions O(u) are still very special automorphic forms. But if G £ SpF is a reductive algebraic sub group, then we may restrict O(u) to GA (where this is the inverse image in S pA of GA £ SpA) and thereby obtain automorphic forms on GA · If G is a relatively small subgroup, this will produce fairly general automorphic forms on GA- For general G it will be difficult to be very precise about the nature of O(u) I GA• but if G belongs to a reductive dual pair as defined below then the structure of the O(u) I GA should be specifiable in considerable detail. It is in this circumstance that the relevance of 0-series to the theory of automorphic forms lies. 5. Reductive dual pairs. Here F is arbitrary (except still not of characteristic Let (G, G') be a pair of subgroups of Sp. We say (G, G') form a reductive dual pair if (i) G and G' act absolutely reductively on W; and (ii) G is the centralizer of G' in Sp and vice versa. Reductive dual pairs may be classified as follows. If (G, G') is a reductive dual pair in Sp, and if W = W1 Ee W2 is an orthogonal direct sum decomposition where W1 and W2 are invariant by G G', then we say (G, G') is reducible. The restrictions (G;, G;) of (G, G') to the W; define reductive dual pairs in the Sp( W,·) . We say that
2).
·
278
R. HOWE
(G, G') is the of the (G;, G;) . If (G, G') is not reducible, it is Any pair is a direct sum in an essentially unique way of irreducible pairs. The irreducible pairs may be described as follows. Typ II. There are maximal isotropic subspaces X, Y £ W with X EEl Y = W. and X and Y invariant by G G' . In this case there exist : (a) a division algebra D over F (not necessarily central) ; (b) a right D-module X1 and a left D-module X2 ; such that (c) X � X1 ®v X2 in such fashion that (G, G') are identified to (GLv(X1 ), GLv(Xz)) . I. The j oint action o f G · G' i s irreducible o n V. Then there exist (a) a division algebra D, (b) with involution q, and (c) D-modules V1 and V2 (d) with forms ( , ) 1 and ( , )z, one 1:1 -Hermitian and the other 1:1 -skew-Her mitian ; such that (e) W � V1 ® V2 in such a way that (G, G') are identified to the isometry groups of ( , )1 and ( , )z, and (f) < , ) = trv1F(( , ) 1 ® ( , )�) . Here trv1 F is reduced trace. The algebra D is not necessarily central over F. Although the ten sor product of forms over D does not make sense, when you take traces you get a good £-bilinear form. REMARKS. (a) Essentially the same classification can be found in Weil [Wi 2] and elsewhere. It essentially goes back to Albert. It may also be deduced from the results in [Sa]. (b) Over a local field F a division algebra with involution is either a quadratic extension or a quaternion algebra over some extension field of F. (c) Consider a pair (G, G') over a number field F. Suppose (G, G') is irreducible of type I, that q is a positive involution on D, that V2 � D with q -Hermitian form (x, y)2 = xyq . Then V1 � W has a q-skew-Hermitian form ( , ) 1 . The data D, q , V1 and ( , ) 2 are o f the things Shimura [Sh] uses to specify his PEL-types. The rest of Shimura's data arises further on in the development of the theory of reduc tive dual pairs. Thus Shimura's PEL-types arise naturally in the context of reduc tive dual pairs. It would be an interesting study to see how much of the work of Shimura and others on these objects can be understood in terms of the oscillator representation.
irreducible.
direct sum
e
·
Type
4
7
6. Local duality. F is now again a local field. Let £ S p be a closed subgroup. Define
H
fJ£(H) = {0' : o- is an irreducible smooth representation of and there exists a nontrivial H-intertwining map
H,
a:
(J)x
--+ 0'
}.
As before, if G £ Sp, then G is the inverse image of G in S p. If ( G, G') £ Sp is a reductive dual pair, then G and G' commute in S p, and so an irreducible admissible representation of G G' will pull back to an irreducible admissible representation of G x G ' . Such a representation has the form 0' o- ' where 0' and o- ' are irreducible admissible representations of G and G' respectively. Thus fJ£(G G ') defines (is the graph of) a correspondence between representations of G and representations of G'. ·
®
·
279
0-SERIES AND INVARIANT THEORY
Conjecture (local duality). If (G, G') is a reductive dual pair in Sp, then f!ll(G · G') is the graph of a bijection between f!ll(G) and f!ll( G'). REMARKS. (a) If duality holds for irreducible pairs, it holds for all pairs. (b) By invariant theoretic methods, this conjecture can be shown to "almost" be true. See § 1 1 . (c) For various pairs (G, G') such that G or G' is anisotropic, the duality con jecture has been considered by many authors (see [G) for some examples). It can be established for such pairs by the method of (b). For F = R , the i n finitesimal methods of Classical Invariant Theory apply directly to prove duality when G or G' is anisotropic. See [H 2]. (d) Duality holds also if G or G' is G L 1 (D) or 02 , or if G = G' = T is a torus. The theory in these cases amounts more or less to the local theory for L-functions at tached to grossencharaktere. (e) For the pair (GLn(F), GLiF)) the duality conjecture amounts essentially to the conjecture made by Weil [Wi 3] about the left and right actions of GLn on Mn. Weil formulated his conjecture in order to provide a distribution-theoretic rationale for the theory of zeta-functions of local algebras.
7. The spherical case ; correspondences between Heeke algebras. F will be a non Archimedean local field of odd residual characteristic. In general the local duality conjecture may be very difficult ; however for appli cations to the theory of automorphic forms weaker results may suffice. We will discuss one such partial result that is particularly relevant for the global theory. Let (!} be the integers of F. Let U be a vector space over F, with a form ( ), either skew, or symmetric, or Hermitian (with respect to some automorphism of F of order 2) . If S £ U is a set, then we put ,
{
S* = u e U : (u, s) e (!} for all s in S } . A lattice in U is a compact open {!}-module. If L is a lattice, so is L* . We say L is self-dual if L = L* . We say ( , ) is unramified if self-dual lattices for ( , ) exist. If G is the isometry group of an unramified form, then a standard maximal compact subgroup is the isotropy group of a self-dual lattice for ( , ) In GLn{D), with D a division algebra not necessarily central over F, a standard maximal compact is the isotropy group of any lattice which is invariant under the integers of D. By an unramified classical group over F we mean (i) GLn(D), with D unramified over F, or (ii) the isometry group of an unramified form over F' where F' i s an unramified extension of F. Returning to our symplectic vector space W, we say an irreducible pair (G, G') £ Sp is unramified if both G and G' are unramified. A general pair is unramified if each of its irreducible summands is. The group Sp is always unramified, and the group J defined in § 3 (there sub scripted J.) is a standard maximal compact subgroup of Sp. As in §3, we can write j J x { 1 , - 1 } unambiguously. If (G, G') is an unramified reductive dual pair in Sp, and (K, K') are standard maximal compacts in (G, G'), then up to conjugacy we can assume K and K' are contained in J. Thus we may write K = .
=
280
R. HOWE
K
K x { 1 , - 1 } unambiguously, and likewise for K'. So we may consider and K' to be well-defined subgroups of S p. For an unramified pair (G, G') with standard maximal compacts K, K', put
8£( G, K)
=
O"
{E
8£( G) : O" admits a K-fixed vector}
and similarly for G', and for G · G' . We assume the character X defining w x is unramified, in the sense that 0 � ker X 12 1C- 1 o, where 1C is a prime of F. Let C'('(G//K) be the (Heeke) algebra of K-bi-invariant functions on G. Define C'('(G' //K') similarly. Let l(K, K') be the vectors fixed by w';(K) and by w';(K'). Then obviously w';( C';'(G//K)) leaves l(K, K') i nvariant so that we may consider the restrictions w';( C'('(G//K)) [ l(K, K'). THEOREM 7. 1 . (a) :Jf(G · G', K') is the graph of a bijection betll'een :Jf(G, K) and fJ?(G', K'). (b) If O" ® O" ' :Jf(G · G') and O" is in fJ?(G, K), then O" ' is in :Jf(G', K'). (c) The restrictions w';(C'('(G//K)) [ l(K, K') and w';(C'('(G'//K')) [ I(K, K') are the same algebra of operators.
E
K
·
8. Global duality. F is now again a global field. Associated paraphernalia are as defined in §3. If G � Sp is an algebraic group over F then we can consider GA � S pA- In an alogy with the local case, we can define
:Jf(GA)
=
O" is an irreducible admissible representation of GA and there exists a nontrivial G A intertwining operator a : w�
{CT:
-->
O" } .
Each O" i n :Jf(GA) will b e a tensor product o f local representations i n the same sense that w'; is, and it is easy to see that 8£(GA) itself will be the restricted product of the :Jf(Gv) with respect to the compact subsets :Jf(G", Kv) where Kv = G n Jv. Conjecture (global duality, part I) . If (G, G') � Sp is a reductive dual pair, then :Jf(GA G�) is the graph of a bijection between !J?(GA) and :Jf(G�). We note that this conj ecture would follow directly from the local duality conjec ture together with the theorem of §7. Let G � Sp as above. By the theorem of §3, we can consider G to be a subgroup of GA - Let d(GA) denote the slowly growing smooth functions on G\ GA- Let O" be an irreducible admissible representation of GA- We will call O" automorphic if there exists a GA embedding O" d(GA . (This notion of automorphic representa tion is slightly more restrictive than the usual one [P].) A natural complement to the conjecture just above is Conjecture (global duality, part II). In the above bijection between :Jf(GA) and fJ?(G�), an automorphic representation of GA is matched with an automorphic re presentation of G� and vice versa. This conjecture is in fact more than j ust a convenient addition to the first con jecture. Like the local duality conjecture, the theorem of §7, and indeed, the whole of the theory, it has a basis in invariant theory. ·
{3: --> )
9. Transcendental classical invariant theory (non-Archimedean case). Here F will be a non-Archimedean local field of odd residual characteristic and integers 0.
0-SERIES AND INVARIANT THEORY
28 1
We will sketch here how the conjectures and results of §§6, 7, 8 are related to in variant-theoretic considerations. For econ omy of space, we will only consider a non-Archimedean field . Unfortunately, this case is more than one short step re moved from the algebraic classical invariant theory of [Wy], and the strong rela tions may not be immediately clear. For a recasting of classical invariant theory in terms of reductive dual pairs, including a proof of the local duality for pairs ( G, G') over R where G or G' is compact, see [H 2]. The first necessity for invariant theory is a supply of "natural" invariants. To provide these we return to the symplectic vector space W and the Heisenberg group H H( W). Let X be a nontrivial character of the center of H. For any sub group A c;;; H, we define A* to be the centralizer of A in H modulo the kernel of X· If A A* we say A polarizes X · If A polarizes X• then F (here considered as the center of H) is contained in A, and A (A n W) E9 F. The set A n W will be a subgroup of W, hence a -module. If A n W is an @-module, (I) being the inte gers in F, we will say A is an £-polarization or a polarization over F. Let Q denote the set of all polarizations and QF the subset of £-polarizations. The action of Sp on H permutes the polarizations. If dim W n , then QF consists of n + 1 orbits for Sp. If A c;;; H polarizes X· then there is a unique extension XA of X to A such that A n w c;;; ker XA · =
=
=
Zp
=
THEOREM 9. 1 . Let the representation p'{' be realized on a vector space polarization A of X there is a linear form AA on Y such that
Y.
For each
(9. 1 ) The functional A A is specified u p t o multiples b y equation (9. 1 ) . The resulting map fl : Q --> embeds Q as a compact subset in The map f2 is equivariant for the natural actions of Sp on Q and on Since has a natural topology, the map f2 implicitly topologizes Q. In this topology QF is a closed subset of Q and each Sp orbit in QF has its natural topology. We can label the Sp-orbits in QF by {Q�}7=o in such a way that Q� is open and dense in U'l=; Q�. Thus Q'j;- is the unique closed orbit. It consists of polarizing A such that A n W i s a maximal isotropic subspace of W. Also Qfj;. is the unique open orbit. It consists of A such that A n W is a lattice (an open compact 0-module) in
P(Y*) P(Y*)
* P(Y * P(Y ). ).
w.
The way QF is used to produce invariants is as follows. If G c;;; Sp, and A E QF, and A is invariant by G, then AA must be invariant up to multiples by w';:.(G) . Thus there is a quasi-character ¢A on G such that
(9.2) Thus the existence of G-fixed points in QF leads to the existence of at least quasi invariant linear forms on and these forms will often be invariant. The analogy with construction of automorphic forms (0-series) by means of the B-distribution as formulated in §4 is clear. This analogy will be developed further in § 1 2. This is also essentially the basis of the First Fundamental Theorems of classical invariant theory, though the parallel is perhaps not so immediately seen. Let ( G, G') c;; Sp be a reductive dual pair. If A D'F is G-invariant, let ¢A be the
Y,
E
R. HOWE
282
quasi-character of G attached to A as in (9.2). For any g' in G', we see that g'(A) will again be G-invariant, and that ¢A and ¢g' CAl will be equal. Thus if we let Q'F(G, ¢) be the set of points A in Q'j;. which are G-invariant and such that ¢ = ¢A, then Q'}(G, ¢) is a union of G' orbits. Since !!2(Q'}(G, ¢)) is a compact "subvariety" of P( Y*), the restriction of the hy perplane section bundle defines a line bundle L(G, ¢) = L
1
f2(fJ'f(G, ¢)) Let Secoo(L) denote the space of smooth sections of L. The evaluation of points of the inverse image in Y* of !!2(Q'}(G, ¢)) defines a linear map e : Y -+ Secoo(L(G, ¢)) . The evaluation map e is clearly G' equivariant. We have the dual to e, e * : Secoo•(L) -+ Y*. Clearly the image of e* will consist of eigenfunctionals for w'X'(G), with eigencharacter ¢. THEOREM 9.2. JfQ'fr(G, ¢) is nonempty, then it consists of a single G' orbit. Further L(G, ¢) is then a G' homogeneous line bundle over Q0(G, ¢). Finally, iff-l is any linear functional on Y such that w'X·(g)f-l = cp(g)fJ. for g G, then f-l = e * ( v) for some dis tribution v in Secoo•(L).
E
10. The spherical case. F and other objects are as in §9. The subgroups of H comprising Q� are compact modulo the center ; because of this, Theorems 9 . 1 and 9.2 have analogues for Q� which are somewhat more direct. We formulate them.
THEOREM 0. 1 . For every A E Q�, there is a vector y A in Y, unique up to multiples, such that p'f'(a)yA = XA(a)yA for a E A . The map A -+ YA is equivariant for the relevant actions of Sp on Q� and P( Y).
I
Let (G, G') be an unramified reductive dual pair in Sp and let (K, K') be a pair of standard maximal compacts of K and K', both contained in the standard maximal compact J of Sp. As explained in §7, we can consider J, hence K and K', to be em bedded in Sp. Let L £ W be the self-dual lattice fixed by J. Assume that X is unramified in the sense of §7. Then A L EEl F £ H polarizes X · Since J normal izes A , the vector YA will be w'X(J)-invariant up to multiples. Since (except when the residue class field of F is 3 and dim W = 2, a case which can be dealt with sepa rately, but which we will exclude) J is perfect, we see YA will actually be invariant under w'X by J, and a fortiori by K and K' . =
THEOREM I 0.2. For f C;:"( G' IK'), define r(f) = w';(f)(yA)· Then r is a surjection from C;:'(G' /K') to I(K), the space ofK-fixed vectors in Y.
E
Since every irreducible admissible representation of G' with a K' fixed vector is a quotient of C;:'(G'/K') in a unique way, and since C;:'(G'/K')K' = C;:'(G'f/K') is a commutative algebra with 1 (hence a cyclic module for itself) Theorem 7. 1 follows from Theorem 9.4.
0-SERIES AND INVARIANT THEORY
283
11. Doubling variables and local duality. F and other objects continue as in §§9 and 1 0. We will now describe how Theorem 9.2 can be used to attack the local duality conjecture. For an irreducible admissible representation (J of G, let Y11 be the quo tient of Y by the kernel of all G-intertwining maps from Y to (the space of) (J. Then wX'(G) evidently factors to a multiple of (J on Y11, and it is not hard to see that the kernel of the quotient map from Y to Y11 is G'-invariant, so Y11 is actually a G x G' module. As such, it must have the form (J -. where -. is some smooth module for G'. The condition that Blt(G G') is an injection from Blt(G) to Blt(G') (terminology as in §6) amounts to saying that whenever Y11 =1- {0} , the representation -. should have a unique nontrivial irreducible admissible G'-module (J 1 as quotient. We will consider the question of uniqueness of (J , which is more crucial than existence. We will suppose that (J * , the (smooth) contragredient of (J is a quotient of w'f.', where X = x- t is the character of F inverse to X · This will be automatically so if (J is unitary or even quasi-unitary. Let yv be the space of w'f.' and let Yd' be the maximal quotient of yv on which G acts by a multiple of (J * . Then Yd' is a G x G' module of the form (J * -.v where -.v is another G'-module. Therefore Y11 Yd' is a (G G) x (G' G')-module of the form ((J (J * ) (-. -.v) . Let G11 be the * diagonal subgroup in G x G. Then the restriction of (J (J to G11 contains the trivial representation uniquely as a quotient. Therefore if ( Y yv) t is the maxi mal quotient of Y yv on which G11 acts trivially, we see that -. -.v will be a G' x G' quotient of ( Y yv)t. Therefore if we can analyze ( Y yv)t as G' x G' module, and in particular if we can show that the only irreducible quotients it ad ' '* mits are of the form (J 1 (J , where (J is an irreducible admissible representation of G', then we shall have established a reasonable approximation to the local duality conjecture. Indeed, passing back through the above analysis, one can easily extract 3 or so technical conjectures which together would be equivalent to the local duality conjecture. The point of the above reduction is that ( Y YV)t can be described by means of Theorem 9 .2. Specifically, let w- be the symplectic vector space whose underlying space is W, but whose symplectic . form is - ( , ), the negative of the form on W. By way of contrast, we will then also write w+ for W. Put 2W = w+ Ei3 w as symplectic vector space. We call 2 W the double of W. We can form H(2 W) according to the recipe of § 1 . We may define injections it and i2 of H( into H(2 by the formulas ·
®
®® ®® ®
®
® ® ®® ® ® ® ®
®
it(W, t ) = ((w, O)t ) ,
(1 1.1)
iz(w, t) = ((0, - w), - t )
W)
W)
for w E W, t E F.
Thus i t x i2 : H( W) x H( W) -+ H(2 W) i s a surjective homomorphism with the diagonal of the centers as kernel. Let X be a character of F and let 2px be the cor responding representation of H(2 W). Then it is easy to see that (1 1 . 2)
(outer tensor product). Let Sp(2 be the symplectic group of 2 W, and Sp( the group of W. We have two embeddings of Sp( W) into Sp(2 W), compatible with the maps of ( 1 1 . 1 ) :
W)
W)
284
R. HOWE
= g( ) , =
h wz) ( wt wz) (I 1. 3) iizt((g)(w g)(wh wz) (wh g(wz)) ! (2.1). 1.2),
for g E Sp( W), and
(wh w2)
E 2 W.
hese maps lift uniquely to maps between S p's. Let cvx be the representation of Sp(2 W) defined by Just as for ( I it is easy to see that
2
( 1 1 .4)
(outer tensor product). Let be a reductive dual pair in Sp( W). Then it can be shown there is a group form a dual reductive pair in Sp(2 W). The such that (i1 x group 2G' is to G' as Sp(2 W) is to Sp( W). In particular 1 ( ) x f; From this doubling construction, we see that the space (Y yvh defined above is described as a 2 G '-module by Theorem By restriction we can investigate its structure as a G ' x G ' -module. Doing so we find that the duality conjecture is cer tainly true if or is compact, and in general is "almost" true. It remains in the general case to remove the "almost". For more details in the case of finite fields. see [H 1].
(G,2G' G')
iz(G4), 2G')
9.2.
G G'
i G'® i2(G') 2G'.
12. Global invariants. In this section, F is a global field, and has associated struc
§3.
tures as in Given an F-module
M f;
WA, put
M * = {x E WA : (x, m) E F, all m E M } .
I f M = M*, w e will call M a global polarization o f W. T o each global polarization M we can associate the closed subgroup M Et> F f; HA( W). The following analogue of Theorems 9. 1 and extends Theorem 4. 1 . Let the space of global polariza tions be denoted {h
10.1
THEOREM 1 2. 1 . Let the representation Px of HA be realized on a space Y. For every M E OA, there is a linear functional on Y, unique up to multiples, such that
()M p';'(x) ()M ()M =
The resulting mapping f2 : QA compact image.
--+
for x E M Et> F
f;
HA -
PY * is an SpA equivariant embedding of QA
ll'ith
Evidently WF f; WA is a global polarization and fl( WF) = e is the {}-distribution of §4. The isotropy group of wF in Sp A is SpF• so the SpA orbit of e is - SpA/SpF• a global version of the Siegel upper half-plane. Also the SpA orbit of e is the unique open orbit in fl(QA) and is dense, so that QA is a compactification of SpA/SPF · The embedding f2 is a global analogue of the projective embeddings of the Siegel modu lar varieties studied by lgusa [1], and they can be derived from it. F. Herman is studying this problem. It seems it may also be fruitful to study the canonical models of Shimura [Sh] from this viewpoint. As in the hyperplane section bundle over PY* induces a line bundle L over fl(Q A), and there is an evaluation map from Y to sections of L. These sections are just the 8-series of §4 and antiquity. (Since () is actually invariant by SpF, the bundle L is S pA-equivariant1y trivial over the orbit of e, and an S pA-invariant cross-section of L results by taking the S pA orbit of e. The restriction of sections of L to this cross-section is the {}-series.) The fact that the {}-series extend to sections over all of fl(QA) guarantees their good behavior "at oo " on SpA/SPF·
§9,
0-SERIES AND INVARIANT THEORY
285
Let (G, G') be a reductive dual pair in S pF. The 0-distribution is Sprinvariant, hence G-invariant. Thus the whole G� orbit of e in !2(QA) consists of G-invariant functionals. In analogy with Theorem 9. 1 , which itself is in analogy with classical invariant theory, we might expect that we would get all G-invariant functionals this way. We formulate a weak version of this precisely. Conjecture (weak global invariants) . Let Y1 c Y be the kernel of all Grinvariant functionals on Y1 and let Y2 be the intersection of the kernels of w'X*(g')8 for all g' in G�. Then Y2 = Y1 . If G�\ G� is compact, then this conjecture implies that every automorphic repre sentation in �(GA) is obtainable by taking 0-series on GA · G� and integrating against some function on G�\ G�. In particular, the weak global invariants conjec ture plus the first global duality conjecture would imply that if q fJ£(GA) is au tomorphic then so is q ' �( G�), proving one direction of the second global dual ity conjecture. Still when G� is anisotropic, another point of view towards the global invariants conjecture is that it provides a local-global principle : if q is an au tomorphic representation whose local components are everywhere quotients of the local oscillator representation, then q is obtained by 0-series. In conclusion, it seems fair to say that the major facts one would like to know about correspondences be tween 0-series are intimately related to invariant-theoretic considerations.
E
E
REFERENCES [CI] P. Cartier, Quantum mechanical commutation relations and theta functions, Proc. Sympos. Pure Math., vol . 9, Amer. Math . Soc., Providence, R . I . , 1 966, pp. 3 6 1 -3 8 3 . [C2] , Representations of p-adic groups, these PROCEEDINGS, part 1 , pp. 1 1 1- 1 5 5 . [F) D . Flath, Decomposition of representations into tensor products, these PROCEEDINGS, part 1 , pp. 1 7 9- 1 8 3 . [G] S. Gelbart, Examples of dual reductive pairs, these PROCEEDINGS, part 1 , p p . 287-296. [HI] R . Howe, Invariant theory and duality for classical groups over finite fields, preprint. [H2] , Remarks on classical invariant theory, preprint. [I] J. lgusa, Theta functions, Die Grundlehren der Math. Wissenschaften , Band 1 94, Springer Verlag, Berlin, 1 972. [M] C. Moore, Group extensions of p-adic linear groups, Inst. Hautes Etudes Sci . Publ. Math. 35 0 968). [P] I . I . Piatetski-Shapiro, Classical and adelic automorphic forms. An introduction, these PRo CEEDINGS, part 1 , pp. 1 8 5 - 1 8 8 . [Sa] I . Satake, Symplectic representations of algebraic groups satisfying a certain analyticity condi tion, Acta Math. 1 17 ( 1 961), 2 1 5-279. [Sh] G. Shimura, Moduli of abelian varieties and number theory, Proc. Sympos. Pure Math. , vol . 9, Amer. Math. Soc., Providence, R . I . , 1 966, p p . 3 1 2-332. [Si I] C. L. Siegel, Indefinite quadratische Formen und Funktionen Theorie. I, Math. Ann. 124 ( 1 9 5 1 ) , 1 7-24 ; II, I24 ( 1 952), 3 64-3 87. Uber die Zeta functionen indefiniter quadratischen Formen, Math. Z. 43 ( 1 938), [Si 2] 682-708 . [Wa] N. Wallach, Representations of reductive Lie groups, these PROCEEDINGS, part 1 , pp. 7 1 -86. [Wi I] A . Weil, Sur certains groupes d'operateurs unitaires, Acta Math. 1 1 1 ( 1 964), 1 43-21 1 . [Wi 2] , Sur Ia formule de Siegel dans Ia theorie des groupes classiques, Acta Math. 113 ( 1 965), 1 -87. [Wi 3] A. Weil, Fonction zeta et distributions, Sem . Bourbaki, expose 3 1 2, June 1 966. [Wy] H. Weyl, The classical groups, Princeton Univ. Press, Princeton, N.J. , 1 946. --
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YALE UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 1, pp. 287-296
EXAMPLES OF DUAL REDUCTIVE PAIRS
STEPHEN GELBART* This note is meant to complement Howe's paper [Ho 1] by giving concrete down to-earth examples of dual reductive pairs and the automorphic forms and group representations which arise from them ; I am grateful to Howe for explaining his theory of the oscillator representation to me. CONTENTS
. . . .. . . . . ..
1 . Historical Remarks 2. SchrOdinger Models 3 . The Pair (Sp( V0), O( U0)) 4. Adelization 5. Local Examples 6. Global Examples .
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289 290 291 293
1. Historical remarks. In 1964 Weil singled out a natural representation of the symplectic group (the Segal-Shale-Weil-metaplectic oscillator representation) in order to give a group theoretical foundation for the classical theta function
{)(z)
00
nnz
= I; exp(2 i 2 ) . n::::: - oo
For SL(2, F) this representation acts in L2 (F) according to the formulas
and
rx(6 f) ([J(x)
=
x
( 01 O1 )W(x)
Tx
_
Here X is a fixed character of F and
r
�
= r W (x ) .
is an eighth root of unity which depends on
X· For Sp(2m, R) the oscillator representation acts in L2 (Rm) through similar
operators. In 1966 Shalika and Tanaka tensored Tx with Tx, to produce a representation of SL(2, F) in L2(£ 2) which commutes with the natural action of 0(2) and decomposes AMS (MOS) subject classifications ( 1 970) . Pri mary I OC30, 1 0D l 5, 22E50, 22E55 . *Alfred P. Sloan Fellow. © 1 979, A merican Mathematical Society
287
STEPHEN GELBART
288
according to the latter group's representation theory. In this way Shalika and Tanaka constructed discrete series representations of SL(2, F) and reformulated earlier results of Heeke and Maass on the construction of theta series with gros sencharacter. For Sp(2m, R) an analogous 2m-fold tensor product of oscillator re presentations acts in V(RZmxm) through the operators (1 . 1 )
( t) W(X) = exp(�i tr XNX1)W(X),
illx �m
illx
(� (.-i\- 1) IP(X) = / det A j m IP(XA).
and (1 .2) This representation decomposes into discrete series representations of Sp(2m, R) indexed by certain irreducible representations of 0(2m, R) (namely those which occur in the natural representation of 0(2m, R ) in L2(R2mxm) given by left matrix multiplication) ; cf. [Ge 1] and [Sa] . Howe's recently developed theory of the oscillator representation reformulates and refines Weil' s theory so as to "explain" the above results and point the way towards new ones. A key notion of the theory-the concept of a dual reductive pair -formalizes that natural duality (already experimentally observed) between the symplectic and orthogonal groups. To describe this phenomenon more concretely the notion of a Schrodigner model is crucial. 2. SchrOdinger models. We adopt the notation and definitions of [Ho 1]. Thus F is a local field not of characteristic 2, V is a symplectic space defined over F,
H(V) is the Heisenberg group attached to V, X is a fixed nontrivial character of F, Px is the irreducible unitary representation of H attached to X• and Sp( V) is the isometry group of the symplectic form ( , ) on F. Recall that H( V) is a central extension of V by F. Let Sp denote the nontrivial 2-fold cover of Sp( V) (whose existence and unique ness is assured by [Mo]). The oscillator representation illx is the unitary representa tion of Sp in the space of Px such that (2. 1 ) for all g E S p and h E H. This representation i s unique up t o unitary equivalence, and it is "genuine" in the sense that it does not factor through Sp( V). The cor responding multiplier representation of Sp( V) is again denoted by illx . To describe a Schrodinger model for ill x (and Px) we need the notion of a com plete polarization of V. This is a pair of subspaces ( X, Y) of V satisfying the follow ing properties : (i) X and Y are isotropic for ( , ), i.e., ( , ) is trivial on X and Y; and (ii) X EB Y = V. With respect to this polarization, the space of Px is V(X) and the action of H( V) is given by
289
DUAL REDUCTIVE PAIRS
p r_((x, 0))/(x ' ) = f(x ' - x), Px (( y, O)) f(x ' ) = x( (y, x ' ) )f(x'), Px ((O , t )) f(x ' ) = x ( t ) f(x ' ),
(2.2)
x, x' E X, Y E V, x' E X, t E F, x' E X.
The corresponding Schrodinger model for wx is more difficult to describe. Thus we first restrict our attention to some distinguished subgroups of Sp( V). Let P( Y) (resp. P( X)) denote the subgroup of Sp( V) which preserves Y (resp. X), and let N( Y) denote the subgroup which acts trivially on Y. If M = M(X, Y) = P(X) n P( Y), then P( Y) is a semidirect product of M and N( Y). Using (2. 1 ) and (2.2) we find that
n
wxC )f(x) = x( - t (x, nx)) f(x),
(2.3) and
x E X,
n E N( Y),
x E X, m E M. wxCm)f(x) = Jl.(m)f(m- 1 x,) In the Schrodinger model for wx then, N acts by multiplication by a quadratic
(2.4)
character, and M simply acts linearly (the factor Jl.(m) being roughly the square root of the modulus of m acting on X; cf. ( 1 . 1 )). Rather than attempt a general description of wx outside P( Y), we refer the reader to the many examples of this paper ; recall that matrices of the form [b n and [-� bl generate SL(2, F). 3. The pair (Sp( V0), O(U0)). The general notion of a dual reductive pair is de scribed in [Ho 1]. The basic example we treat in this paper simply formalizes the special features of the Sp(2m, R) example just discussed. Let ( V0, < , )0) denote any symplectic vector space, and ( U0, ( , ) ) any orthogonal one. Put V = V0 ® U0, and define a form < , ) on V by '
(v
® u, v' ®
u' ) = ( v,
v ')0 (u , u') o
for v, v E V0, and u, u' E U0• Then ( V, < , ) ) is a symplectic space, and (Sp( V0) . O( U0)) is a natural dual reductive pair in Sp( V). In particular, the groups Sp( V0) and O( U0) are each other's centralizers in Sp( V). ((Sp(2m, R), 0(2m, R)) is such a pair in Sp( V), with V = R2m ® R2m = V0 ® U0. If (X0, Y0) is a complete polarization of V0, let X = X0 ® U0 and Y = Y0 ® U0. Then (X, Y) is a complete polarization of V. Moreover-according to §2-the Schrodinger model for w" in £2 (X) is such that O(U0) acts linearly. On the other hand, if (X0• Y0) is a complete polarization of O( U0)-which is possible only when U0 is split-then the spaces X = V0 ® X0 and Y = V0 ® Y0 provide a polariza tion of V with respect to which wxC Vo) (not O( U0)) acts linearly. In either case, the problem is to describe the decomposition of wx restricted to Sp( V0) • O(U0) . Over R one has the following generalization of the results alluded to in § 1 ; cf. [Ho 2] . Let Sp0 and 00 denote the inverse images of Sp( V0) and O(U0) in Sp( V). Then wxC Sp0) and wx
�
� 'r; ® a; '
290
STEPHEN GELBART
where '&'; is an irreducible representation of Sp0 of holomorphic type, and (J; is an irreducible finite-dimensional representation of 00 ; the representations (J; and '&'; determine each other, and the resulting bijection (3 . 1) may be computed using classical invariant theory. These results, already valid for the pairs (i) (Sp(2m, R), O(n, R)) in Sp(2nm, R), with n, m positive, also extend to cover the pairs (ii) ( U, Up, q) in Sp(2n(p + q), R) , and the pairs (iii) ( Sp( p , 0 ), 0 * (2n) ) in Sp(4np , R) . Thus-in addition to generalizing [Ge 1]-these results also contain earlier works of [GK] (which treats (i) with n � 2m, (ii) with n � 2p and p = q , and (iii) with p � 2n) and [KV] (which treats (i) and (ii) with no restrictions). We shall call the correspondence which results from the "duality" of the pair (Sp, 0) the duality correspondence. Now consider an arbitrary local field and an arbitrary subgroup H of Sp. Let R(H) denote the set of irreducible smooth representations of H for which there is a nontrivial H-intertwining map from the smooth vectors of wx . Then if (G, G') is a dual reductive pair in Sp, R(G · G)' should be the graph of a bijection (or duality correspondence) between R(G) and R(G') ; cf. (3. 1 ) and [Ho 1]. Though unproved in general, this conjecture has been established when G or G' is com pact. The examples later on give more motivtiaon for the theory.
F
4. Adelization. Now let F denote an A-field not of characteristic 2, v an arbitrary place of F, Ov the ring of integers of F0, A the adele ring of F, and X = I1 x. a non trivial character of F\A. If V is a symplectic space defined over F, then for each v the groups and representations Hv = H( V/F0), Spv = Sp( V/F0), Spv, P x•• and Wx,. are defined as in §2. As explained in [Ho 1], one can make sense out of the restricted direct product @wx. and use it to define a multiplier representation w of SpA = Sp( VA). If SpA x denotes the 2-fold cover of SpA determined by the product of the cocycles defining Spv then wx also defines an Ndinary representation of SpA. A fundamental property of the global representation wx is that it splits over the rational points Sp(F) = Sp( V/F) ; cf. [We]. If (X, Y) is a complete polarization of V and (} is the distribution on the Schwartz-Bruhat space 9' ( X(A)) defined by (} (cf>) = L: ,::x (FJ C!> (�) , then (} is Sp(F)-invariant. In particular, the functions (4. 1 )
(} "' ( g) = O(w x ) = I: ( w ig) C!>) (�)
t;cX ( F)
are automorphic functions on SpA (slowly increasing continuous functions which are left-Sp(F)-invariant) . The basic goal of the global theory of the oscillator representation is to describe the automorphic forms and representations which arise from the {}-distribution through functions of the form (4. 1 ) . As in the local theory, a great deal of structure is introduced by considering dual reductive pairs in SpA. If (G, G') is such a pair, it follows from the local theory that there should be a duality correspondence
29 1
DUAL REDUCTIVE PAIRS
between representations of G and G' which intertwine with wx . Moreover, this bijection should pair representations of G with repre sentations of G'. For a precise description of what to expect in general, see [Ho 1]. Rather than pursue the general theory, we refer the reader to the examples of §6.
automorphic
automorphic
5. Local examples. Throughout this section, F is a local field not of character istic 2, X is a fixed character of F, V0 is the space equipped with the skew form U0 is the orthogonal space Fn with quadratic form q, and V = V0 U0• Then Sp( V0) = SL(2, F), O( U0) = O(q), and the problem is to describe wx restricted to SL(2, F) · O( U0). Recall that we may choose a polarization in V so that wx acts in according to the formulas
F2
x1 y2 - x2yt. (5 . 1 )
L2(Fn)
wx
and (5.2)
®
(6 f)J(X) = x (bq(X)) f(X),
wx
( - 0I o1 )f(X) = r(q, x)f- (X) .
L2(£n)
The orthogonal group acts linearly in through its natural left action in P . A. Assume that F is nonarchimedean and q is anisotropic. Then � 4, O(q) is compact, and
nThe anisotropic case. 11 Case n
) (8) 17. .E Here the sum is over in R(O(q) ) and the multiplicity of each ) is one (cf. [RS 1] ) . It remains to describe R(O(q)) , R(SL(2, F)), and the resulting duality correspondence. with O(q) = { ± 1 } , and the (i) : = 1 . In this case, = corresponding representations of SL(2, F) act on the space of even or odd func tions in L2(F). More precisely, if is the trivial representation of O(q), then ) (defined by formulas (5. 1), (5.2) restricted to the space of even functions) is the unique subrepresentation of an appropriate nonunity principal series representa tion of SL(2, F) at = - 1/2; if 11 is the nontrivial representation of O(q), then ( ) (defined by (5. 1), (5.2) acting in the space of odd functions) is a supercuspidal representation of SL(2, F) which is "exceptional" in the sense explained later in § 6 ; for more details, see [Ge 2], [G-PS], and [Ho 3]. Note that and lead to equivalent oscillator representations if and only if ab -1 In case F = R , the pieces of wx are square-integrable with extreme vectors of weight 1 /2 and 3/2. (ii) : = 2. Let U0 denote a quadratic extension K of F equipped with the inner product derived from its norm form q. Then O(q) is the semidirect product of the norm 1 group Kl in K with the Galois group of K over F. In this case each in R( O(q) ) can be described in terms of a character of K I and most of the repre sentations of R(SL(2, F)) are supercuspidal. For further details, see [ST], [Cas], [G] , or [RS 1] ; in [ST] the correspondence is 2-to- 1 because K I (the special or thogonal group) is used in place of O(U0). This example is historically important because Shalika and Tanaka were the first to use the osciiiator representation to construct an interesting class of irreducible representations. (iii) : = 3 [RS 1]. Let H denote the unique division quaternion algebra over F, n o the subspace of pure quaternions, and q3 the restriction of the reduced (5.3)
n- 11
Case n
Case n
W X l sL(2. F) ·O(q) =
1r(11
n-(11 ® 11
q(x) ax2 a E P, 11
n-(11
s
2 2 ax bx 2 E (P) •
11
STEPHEN GELBART
292
norm form on H to H 0 . Then every anisotropic ternary form q over F is equivalent to one of the forms aq3 with a in P/(P) 2 , and every irreducible representation 0' of O(q) belongs to R(O(q)) (SO(q) is isomorphic to (H 0 )x) . In [RS 1] it is also shown that the corresponding representations x (O') of SL(2, F) are square-integrable, and in fact exhaust the class of all such "genuine" representations (at least when the residual characteristic of F is odd). Case (iv) : n = 4. Let U0 denote the division quaternion algebra H defined over F and equipped with the inner product derived from the reduced norm form q. Then though O(q) is not the same as Hx, each 0' in R(O(q)) corresponds naturally to an irreducible representation of Hx, all such representations of Hx thus arise, and each x (O' v) in R(SL(2, F)) is square-integrable. Actually, x (O"v) extends in a unique way to a representation x ' (O' v) ofGL(2, F") such that
and the resulting duality correspondence 0' v +---> x ' (O' ) is onto the set of classes of square-integrable irreducible representations of GL(2, Fv) ; for further details, see [JL]. Note that Case (iv) is analogous to Case (iii) in that the x(O')'s of the local duality correspondence are characterized by their square-integrability. Only in Case (ii) is a neat characterization of R( 0(q)) lacking. B. The noncompact (real) case. Assume that F = R and x (u) = e2triu. Let O(q) = O(k, denote the isometry group of the standard quadratic form of signature (k, on Rn , with n = k + /. In [ST] Strichartz has essentially given a decomposi tion of the natural action of O(q) in V(Rn), i.e., of wx restricted to O(q). Using results of [Re] on the tensor product of representations of SL(2, R), Howe has proved the following version of his local duality conjecture : "
I) I)
SL (2, R) · O (q)
where ds is a Borel measure on the unitary dual of SL(2 , R), O"s and 0'� are irre ducible unitary representations of O(k, I) and SL(2, R) respectively, and O's and 0'� determine each other almost everywhere with respect to s. The resulting corres pondence O's --> x (O's) = 0'� is particularly interesting for the discrete series repre sentations in R(SL(2, R) ) and R( O( q)) . For complete details, see [Ho 4] and [RS 2] ; the case (k, I) = (2. 1) is described in [Ge 2]. C. Noncompact p-adic case. An interesting example arises when we replace the quadratic extension of example 5.A(ii) by the hyperbolic plane £2 equipped with inner product ((x, y), (x ' , y')) = xx ' yy' . In this case O(U0) splits. Thus we can find a polarization of V left fixed by Sp( V0) = SL(2, F) (take X0 = {(x, x) :x F} and Y0 = {(x, - x) : x F}). The corresponding Schrodinger model for wx then makes SL(2, F) act linearly in V(£2). More precisely, w/g)f(�) = f(rl(�)) . So since functions on £2 {0} can be identified with functions on G invariant by N - W m. the oscillator representation in this case leads to a direct integral of principal series representations of SL(2, F). The orthogonal group, however, no longer acts linearly. Indeed the operator corresponding to reflection (the element -
E
-
E
DUAL REDUCTIVE PAIRS
293
taking (x, y) to (y, x) ) is the Fourier transform (in £2(£2)) taken with respect to the skew form in £2 . The significance of these facts for the representation theory of SL(2, F) has already been discussed in Cartier's lectures (see also [G]). 6. Global examples. A. The anisotropic case. Case (i) : 0(1 ) . By tensoring the pieces of the oscillator
representation in V(F v) we obtain some very interesting automorphic representa tions of SL(2) and (by inducing) GL(2). The idea that nontrivial pieces of the oscillator representation could define interesting cusp forms was first communi cated to me by Howe (cf. [Ho 3]) . We shall be content to merely sketch the sub sequent development and refer the reader to [G-PS] for details. If a = ®a" is an automorphic representation of 0(1) then a" is the trivial repre sentation for all v except in a finite set S whose cardinality is even. If X = IT Xv is a character of A/F, let w xv denote the oscillator representation of SL(2, Fv) in V(Fv) corresponding to Xv · Fix a character ?. = IT Av of Ax/Fx such that ?.vC - 1) = 1 (resp. - 1) if v ¢ S (resp. S) and let w �" denote the even (resp. odd) piece of W xv · Then extend w �v to G: = {g GL(2, Fv) : det(g) (F;) Z }
vE
E
E
by defining w�" (b gz) to be the operator 1J(x) >--> J. (a) J a J - 1 12 ifJ(a- 1 x), and induce w �v up to GL(2, Fv) to obtain the representation w� of the double cover Gv of GL(2, Fv). This representation w� is an irreducible unitary "genuine" repre sentation of Gv which is independent of Xv fo r all v and class 1 for almost all v. The representation n(J.) = ® w� is an automorphic genuine representation of GL(2, A) which is cuspidal precisely when S #- 0 . In general, when S is empty, these n(A) generate the discrete noncuspidal spectrum of GL(2, A) (cf. [GS]) ; in particular, when F = Q, n(A) generalizes the classical theta-function 8(z) = I; �= -oo exp(2nin 2z) . On the other hand, when S #- 0 , n(A) generalizes such clas sical cusp forms as Dedekind's 17-function. In this case, n(A) is exceptional in the sense that the Fourier expansion of any function in its representation space con tains "only one orbit of characters", generalizing the fact that the expansion of the 17-function has "only square terms" : 1J(24z) = 1: ;;"=1 (3/n) exp(2nin 2z). Since the Fourier expansion of a cusp form 1t on GL(2, A) is carried out with respect to a group which is isomorphic to the product of F with the double cover f'x of P , we can say 1t is exceptional if-given X on F-the Whittaker model for 7t with respect to ( x , p.) exists for only one character p. of f'x. Thus the notion of exceptionality makes sense locally too ; see [G-PS] for details. Our conjecture is that the oscillator representation produces all possible exceptional representations, both locally and globally. In other words, for 0( 1) the image of the duality corre spondence a >--> n(a) should be all the exceptional genuine representations of G. The classical analogue of the global part of the conjecture just described is that every cusp form of weight k/2 whose Fourier expansion involves only square terms must be of weight 1 /2 or 3/2 and (a linear combination of translates of functions) of the form
STEPHEN GELBART
294 00
I; ¢(n) n" exp(2n-in 2z),
n =l
).1
= 0 or 1 .
At this conference we learned that M.-F. Vigneras has proved this assertion in [V].
Case (ii) : 0(2). Let K denote a quadratic extension of the field F, X a character of F\A, and O" = @O" v a character of Kl(F)\Kl(A). Piecing together the resulting local representations n-(O".) of SL(2, F.) (described in §5.A(ii)) produces automor phic cuspidal representations of SL(2, A) (provided O" is nontrivial). These automor phic forms are related to the theta series with grossencharacter constructed earlier by E. Heeke [H] and H. Maass [M] ; for more details, see [ST] and [Ge 3, Chapter 7] .
Case (iii) : 0( 4). Let D denote a division quaternion algebra over F and SD the set of places where D ramifies. Suppose (J = ®O"v belongs to R(D"(A)) . If v E SD then D� is isomorphic to the unique division quaternion algebra defined over F. and 11:'((1.) is defined as in §5. On the other hand, when v Ft SD, D� is isomorphic to GL(2, F.), and the natural duality correspondence is the identity map. Now suppose (J is actually automorphic, i.e., there is an embedding of (J into the space of automorphic forms on G'(A) = D" (A) . Then according to [JL] the repre sentation n-'((J) = ®.n-'((J.) of GL(2, A) will also be automorphic. Conversely, if n- = ®n-. belongs to R ( GL(2, A)) , i.e., n-. = n-'((J.) for some (Jv in R(D�) (equi valently n-. is square-integrable for each v E SD), then n- will be automorphic only if (J = ®(J. is automorphic ; cf. Theorems I4.4 and I 6 . I of [JL] . These results con firm the main global conjecture of Howe's theory in the special context of division quaternion algebras. Whereas the proof in [JL] uses the trace formula, Howe ap parently has an independent proof by different, more elementary, methods. B. 0(2, I ). Let U0 denote the three-dimensional space of trace zero 2 x 2 ma trices over F equipped with the inner product derived from the determinant func tion. Set V0 = (F2, ( , ) ) and V = V0 U0 • Then for each place v of F, the pair ( SL(2, F.), O( U0)) is a dual reductive pair in Sp( V). For F. = R, the duality correspondence (Jv .... n-((J.) pairs discrete series repre sentations of PGL(2, R) = S0(2, R) of weight k I with discrete series repre sentations of SL(2, R) of weight k/2. For F = Q it is conjectured in [Ge 2] that the resulting global correspondence should generalize Shimura's correspondence [Shm] between classical forms of weight k/2 and k I . Evidence for this is pro vided by Howe's recent description of the local duality correspondence for class I representations (in [Ho 1]), and by Shintani and Niwa's work on the global cor respondence using integrals of theta-functions ([Sht] and [N]). For further work on this problem, see also [G-PS]. If F. is nonarchimedean, and (Jv is a discrete series representation of PGL(2, F.) of the form n-'((J�), with (J� an appropriate finite-dimensional representation of the orthogonal group of an anisotropic ternary form, then n-((J.) should coincide with the representation n-((J�) of SL(2, F.) described in § 5A, Case (iii). C. 0(2, 2) (F = Q). Let K = Q( ,.,! > 0 the discriminant, and let r: denote the Galois automorphism of K/Q. Let
®
-
Ll), Ll
U0 =
{x = (x3x1 - xxl�)
:
x1 E
K, x3, x4 e Q
}
�
Q4
DUAL REDUCTIVE PAIRS
295
and define q : U0 --> Q by q(X) = 2 det(X). Then q has signature (2, 2), and the corresponding global duality correspondence should pair together modular forms of integral weight with Hilbert modular forms (since SL(2, R) x SL(2, R) has a natural representation in S0(2, 2)) . Work on this correspondence (in the classical language of forms in the upper half-plane) has been carried out by Oda [0], Asai [A], Kudla [K], Rallis and Schiffmann [RS 3], Zagier [Z], and, in a somewhat dif ferent light (and earlier) by Doi-Naganuma [DN]. Note added proof We have now proved the global "exceptionality conjecture" described at the bottom of page 293. The local conjecture has been proved by James Meister and will appear in his forthcoming Cornell Ph.D. thesis. -
in
REFERENCES [A] T. Asai, On the Doi-Naganuma lifting associated to imaginary quadratic fields (to appear). [Car] P. Cartier, Representations of p-adic groups : A survey, these PROCEEDINGs, part 1 , pp. 1 1 1 1 55. [Cas] W. Casselman , On the representations of SL,(k) related to binary quadratic forms, Amer. J . Math. 9 4 ( 1 972), 8 1 0-834. [DN] K . Doi and H . Naganuma, On the functional equation of certain Dirichlet series, Invent. Math. 9 ( 1 969), 1 - 1 4 ; also H . Naganuma, On the coincidence of two Dirichlet series associated with cusp forms ofHeeke's "Neben"-type and Hilbert modular forms o ver a real quadratic field, J. Math. Soc. Japan 25 ( 1 973), 547-554. [Ge 1] S. Gelbart, Holomorphic discrete series for the real symplectic group, Invent . Math. 19 ( 1 973), 49-5 8 . [Ge 2 ] Wei/'s representation and the spectrum of the metaplectic group, Lecture Notes in Math . , vol. 530, Springer-Verlag, Berlin, 1 976. [Ge 3] --, Automorphic forms on adele groups, Ann . of Math. Studies, no. 83, Princeton Univ. Press, Princeton, N. J . , 1 975. [G-PS] S. Gelbart and I . I. Piatetski-Shapiro, Automorphic L-functions of half-integral weight, Proc. Nat. Acad. Sci . U . S . A. 75 ( 1 978), 1 620- 1 623 . [G-S] S. Gelbart and P. J. Sally, Intertwining operators and automorphic forms for the metaplectic group, Proc. Nat. Acad. Sci . U.S.A. 72 ( 1 975), 1 406-1 4 1 0 . [G] P . Gerardin, Representations des groupe S L , d'un corps local (d'apres Gelfand, Graev, et Tanaka), Seminaire Bourbaki, 20e annee, 1 967/ 1 968, n o 322, Benjamin, New York and Amsterdam, 1 969 . [GK] K. Gross and R. Kunze, Bessel functions and representation theory. II, J. Functional Analysis 25 (1 977), 1 -49. [H] E . Heeke, Mathematische Werke, Vandenhoeck, Gottingen, 1 959. [Ho 1] R . Howe, &-series and invariant theory, these PROCEEDINGS, part 1, pp. 275-28 5 . [Ho 2 ] ----, Appendix to remarks o n classical invariant theory, preprint, 1 976. [Ho 3] - --, Review of [GS], Zentralblatt fur Math . , Band 303, Review 220 1 0, January 1 976, p . 98. [Ho 4] , On some results of Strichartz and of Rallis and Schiffmann, preprint, 1 976. [Ho 5] , Invariant theory and duality for classical groups o ver finite fields, with applications to their singular representation theory, preprint. [JL] Herve Jacquet and Robert Langlands, Automorphic forms on GL(2), Lecture Notes in Math . , vol. 1 1 4, Springer-Verlag, Berlin, 1 970. [KV] M. Kashiwara and M. Vergne, On the Segai-Shale- Weil representation and harmonic polynomials, preprint, 1 976. [K] S. Kudla, Relations between automorphic forms produced by theta-functions, Modular Functions of One Variable. VI, Lecture Notes in Math. , vol. 627, Springer-Verlag, Berlin, 1 977. [Kz] D . Kazhdan , Applications of Wei/'s representation, preprint, 1 976. [Ma] H . Maass , Uber eine neue Art von nicht analytischen automorphen Funktionen und die �-,
--
---
STEPHEN GELBART
296
Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung, Math. Ann . 121 (1 949), 1 4 1 - 1 8 3 . [Mo] Calvin Moore, Group extensions of p-adic linear groups, lnst. Hautes Etudes Sci. Pub! . Math. 35 (1 968), 1 5 7-222. [N] S. Niwa, Modular forms of half-integral weight and the integral of certain theta-functions, Nagoya Math. J. 56 (1 975), 1 47-1 61 . [OJ T. Oda, On modular forms associated with indefinite quadratic forms of signature (2, n-2), preprint. [RS 1] S . Rallis and G . Schiffmann, Representations supercuspidales du groupe metaplectique, p reprint, 1 977. [RS 2] , Discrete spectrum of the Wei/ representation, preprint, 1 975. [RS 3] , Automorphic forms constructed from the Wei/ representation, holomorphic case, preprint, 1 976. [Re] J. Repka, Tensor products of unitary representations of SL,(R), prepri nt. [Sa] M . Saito, Representations unitaires des groups symplectiques, J. Math. Soc. Japan 24 ( 1 972), 232-25 1 . [ST] Joseph Shalika and S . Tanaka, On an explicit construction of a certain class of automorphic forms, Amer. J. Math. 91 (1 969), 1 049-1 076. [Shz] H. Shimizu, Theta-series and automorphic forms on GL(2), J . Math. Soc. Japan 24 (1 972), 638-68 3 . [Shm] G . Shimura, On modular forms of half-integral weight, A n n . o f Math. (2) 97 (1 973), 440-481 . [Sht] T. Shintani, On construction ofholomorphic cusp forms ofhalf-integral weight, Nagoya Math. J. 58 (1 975), 83-1 26. [Sie] C. L. Siegel, Indefinite quadratische Formen und Funktionen Theorie. l, II, Math. Ann. 124 (1951 /52), 1 7-54, 364-387. [St] R . Strichartz, Harmonic analysis on hyperboloids, J . Functional Analysis 12 (1 973), 341-383. [V] M .-F. Vigneras, Facteurs gamma et equations fonctionnelles, Modular Functions of One Variable. VI, Lecture Notes in Math . , vol. 627, Springer-Verlag, Berlin, 1 977. [We] Andre Wei!, Sur certaines groupes d'operateurs unitaires, Acta Math. lll (1 964), 1 43-21 1 . [Z] D . Zagier, Modular forms associated to real quadratic fields, Invent. Math. 30 ( 1 975), 1 -46. --
--
CORNELL UNIVERSITY
Proceedings of Symposia in Pure Mathematics Vol. 33 ( 1 979), part 1 , pp. 297-3 1 4
ON A RELATION BETWEEN
St2
CUSP
FORMS AND AUTOMORPHIC FORMS ON ORTHO GONAL GROUPS
S. RALLIS This lecture is devoted to showing how, starting from the ideas of A. Weil in [22], one can construct a correspondence or lifting between SL 2 cusp forms and auto morphic forms on orthogonal groups. 1 . Siegel formula and the lifting of modular forms.
Let Rk be k dimensional be the bilinear form on = Let Q be the
x,.y,. el > .Ef=1 x,.e,. ] .Ef=1 y,.e,.. x,-y,. x,.y,. a .EI.r=1x7 (a, a .E = x�.integral . exponent n L nL nL discriminant g x y y g
Euclidean space with standard basis , ek . Let [ , Rk given by [X, Yj = .E 7=1 where X = Y quadratic form on Rk given by . . .
Q(X, Y)
=
a
I;
i=l
k
- I;
i=a+l
with
<
k.
Then for every X E Rk we have Q(X, X) = I X+ l l 2 - I X- 11 2 , with I I X+ 11 2 = and II X- 11 2 = f a+I Thus Q has signature b) with + b = k and b � A lattice L £:; Rk i s Q (Q-even, resp.) if Q(L, L) £:; Z, the integers ( Q(L, L) £:; 2 · Z, resp.). The Q dual lattice to L is L * (Q) = {.; E Rk I Q(.;, L) £:; Z } Then be the of this group, i.e. L * (Q)/L is a finite Abelian group. We let the smallest positive integer .; E L for all .; E L * (Q). If .; 1 > . . , .;k is so that any Z basis of L, let DQ 1 2 , -
.
with z = + .J - 1 E H the upper half-plane (i.e. > 0) and orthogonal group of Q. Let r; E L * (Q) and consider the () series (1-2)
E
O(Q), the
Then let r; run over a set of representatives of L * (Q)/L. We define the column matrix ( 1-3) Then Siegel proves the following functional equation :
{L,
Q-even)
AMS (MOS) subject classifications (1 970) . Primary 1 0D20, 22E45 ; Secondary 43A80. © 1 9 79, American Mathematical Society
297
S. RALLIS
298
1/f� (F · z, gr) = c(F, r )(cr z + dr) Ca-b> 1 2 1f!�(z, g)
(1 -4)
{for F E SL2(Z), 7 E O(Q)L = {g E O(Q) I g(L) = L } ), where c(F, r) is a [L*(Q)/L] 2 unitary matrix depending only on F and T · In general we know that the map (F, r) - c(F, r ) defines a of the group SL2(Z) X O(Q)L. The main difficulty in studying 1/f�(z, g) is that this function is not automorphic as it is not an eigenfuncti on for the center of the enveloping algebra of SL2 x O(Q). However this can be remedied by integrating 8�. �(z, g) against the constant function on O(Q)/O(Q)L. More precisely, the main point of the analytic part of Siegel's f rm ul is to introduce a suitably normalized O(Q) invariant measure dfJ.� on the homogeneous space
projective unitary representation
o a
O(Q)/{ r E O (Q)L I rrJ
=
7J
mod L } � O(Q)/O(Qk �
so that we have the identity (valid for k � 5 and L, Q-even [20])
S
(1 -5) where (1-6)
O (Q) IO (Q) L, �
B�. �(z, g) d"� (g) = E71(z),
E71(z) =
(lm z) b / 2
I; ME SL2 (Z) �\SL2 (Z)
c71, o-((M, l )) (cMz + dM) Ch-a> 1 2 1 CMZ + dM �-b
with SL2(Z)� = {F E SL2(Z) I cr = and c71, 0- {(M, 1)) represents an element of the first column of the matrix c{(M, 1 )) { 1 , identity element of O(Q)) . Then we know that E71(z) is an eigenfunction of the center of the enveloping algebra of SL2 ; in concrete terms this means that £71 satisfies the equation :
0},
(1-7) with L1
(
02 02 k / 2 - y2 ox2 + oy 2
)
+
I 2
k
0 + I 1 2I (b y oi v -
- a)y
0 ax·
In terms of representation theory, Siegel's formula expresses the fact that the identity representation of O(Q) corresponds (in a sense to be made precise below) to the representation of SL 2 determined by the nonanalytic Eisenstein series (£71)71• We then note that it is easy to extend this correspondence to other automorphic representations of O(Q). Namely we can do this in two ways. First we can vary the kernel function. We let P be a homogeneous polynomial on Ra ( = R span of , ea � ) of degree satisfying o(Q)(P) = where o(Q) is the differential operator Q(o/ox l > . . . , o/oxn). Then we form (Im z) t1 2 P (g · (lm z) l 1 2X) cf; (X, z, g) cj; ( X, z, g)
{eb . ..
t
*
0,
=
and as above define 1/f�.(z, g). Thus 1/f�. satisfies a functional equation similar _ to ( 1 -4). The second step is then to jntegrate 8� 71 against a cusp on O(Q)/O(Qk 71• This is possible because 8� 71 is a "slowly increasing" function on •.
•.
form {3
299
CUSP FORMS AND AUTOMORPHIC FORMS
denote the Hilbert space inner H x O(Q)/C!(Q)L,�. Let
. � {3 � (z, g)dp,{3�().g))o(zQ)!O (QJ L. � � (z,g) Wo({3(g))Q) {3 }.{3, z ( {3, g) {3(g)) z ( , g) {3(g)) {3 unitary z ( , g) {3(g)) •.
=
•.
•.
SL2
h(z)
h( �( )
(1-8)
i
•.
)
'
=
••
(with v, the of Then it is simple to deduce that g ...,.... I ) is an eigenfunction for the center of the enveloping algebra of O(Q). Again the cuspidal behavior of this function is another question. Niwa in [10] shows (for the case 0(2, 1 )) how such a correspondence explains the construction of Shimura in [18] of cusp forms of integral weight from cusp forms of semi-integral weight. In any case the main object of study is the kernel function B� We �dopt the following point of view. _9 ne can analyze the "cuspidal components" of e�.. 7J (that is, the "projections" of e�.. � on the various spaces of cusp forms discussed above) by studying the spectral decomposition of the Weil representation. Then knowing precisely the "cuspidal components" gives a correspondence or lifting between modular forms on the two groups in question. •.
•.
·
�·
in detail
fE
2. Discrete spectrum of the Weil representation. V n L2 , then
defined as follows : if
'lCQ
J(r) =
J
Rk
Also if
A,
Fourier transform, be
f(X)e2rr.J-HX, Yl dX.
Then let be the Weil representation of the two-fold cover of SL2(R). In particular, if
then
Let
(([g �_1J. l ), g) E SL2
x
SL2
O(Q)
x
O(Q) in L2(Rk), where SL 2 is
with a > O,
300
S. RALLIS
then I!Q(w0) q;(Z) = OQI{!( - MQ(Z)) with MQ E Aut( Rk) , so that Q(X, Y) = MQ( Y)] for all Y E Rk and OQ a certain eighth root of unity. Then n"Q is determined by the above formulae. We let k be the maximal compact subgroup of SL2 given by
[X,
X,
Also let
{k(O, c) = ( [ _ �f� � ��s �J c)[ - n- � 0 < n-, c = ± I }. {
A = a(r ) =
and
( [� �-IJ 1 ). r > 0}
n, and t are the infinitesimal generators of A , N, and k, then WsL = 2 a 2 + (n + Ad(w0) n)2. We let K ( � O(a) x O(b)) be the maximal compact subgroup of O(Q). If n- is a
Then if a, - f2
+
differentiable representation of SL2 x O(Q) on a Banach space )8, then let )BJ?x K be the space of k x K finite vectors in )8. Let FQ be the space of coo vectors of n"Q in L2(Rk). An easy use of the Sobolev regularity theorem shows that if q; E FQ, then for any y belonging to u (SL 2 x O(Q)), the enveloping algebra of SL 2 x O(Q), n"Q(y)q; is a coo function on Q+ and Q_, where Q+ = {X E Rk I Q(X) > 0} and Q_ = {X E Rk I Q(X) < 0} . Then by a simple exercise in invariant theory, one can verify that the centers of the enveloping algebras of SL 2 and O(Q) collapse on each other in the representa tion n"Q · That is, n-Q(3( SL 2) ) = n"Q(0(0(Q))), where 3( ) = the center of the envel oping algebra of ( ). In particular we have that n"Q(WsL,) + n"Q (Wo
(!-k2 - k).
THEOREM 2. 1 ([12], [14]). (a) Spec(n-Q) = { s 2 - 2s I s = mod 1 and s > 1 } . (b) Let FJ"(.A.) =_ { q; E FQ(A) I q; 1rL = 0} and FQ"(A) = {q; E FQ(A) I q; 1fh = 0} . Then FJ"(A), FQ"(.A.) are SL 2 x O(Q) topologically irreducible modules ; there is a direct sum
!k
FQ(A) = FJ"(.A.) Ef> FQ"(A). Moreover FJ"(.A.) and F0(A) are inequivalent SL2 x O(Q) irreducible representations. (c) FJ"(s 2 - 2s) is SL2 x O(Q) equivalent to the tensor product L2(Whit) ,q, ® .s;�;, where L2(Whit),z_2, is the eigenspace of WsLz (with eigenvalue s 2 - 2s) in the space of coo vectors of the unitarily induced representation of SL2 (from the unitary character
([6 �]. 1 )
�
)
e� �-lx
CUSP FORMS AND AUTOMORPHIC FORMS
301 z and i s the ei g ens p ace (wi t h ei g enval u e s of attheionLaplace Belt r ami operat o r on the space o f C"' vectors o f the st a ndard represent imeasure n L2(F+Ion• dp Here where is determined by the separationandof varidt-ta+blanes of inofvariant a(Q) - atazz -tk -- Tta with t s z s C"' ( dp-I) · pf(t)pf(t;) p[, p"t C"' t·l;, t di
- 2s - tkZ
+) , F+ I ·
+
k)
F+l = {X e Rk l Q(X, X) = 1 } Wl _
+
1
Wl,
O(Q)
O(Q)
1 W+, e
7I
X = · l; , I; E F+ l · REMARK 2. 1 . F0 ( 2 - 2s) is SL 2 x O(Q) equivalent to the tensor product 2 L (Whit);1_2, ® d-;. where L2(Whit);1_2, is the representation of SL 2 on L2 (Whit),z_Zs twisted by the unique outer automorphism A of SL 2 , and d-; is the - 2s tkZ + k) of We-, the Laplace-Beltrami eigenspace (with eigenvalue operator on the space of vectors on the standard representation of O(Q) in L2 F- l > The first point in analyzing the discrete spectrum of 71:Q is simply to observe that a k eigenfunction rp (also K finite) in FQ(A) has a separation of variables property on D±, i.e. rp(X) = with functions on D+ (Q_ respectively) satisfying certain differential equations (with X = t; E r+ l , E R+). The second . point is to realize that the map rp '""-+ [G -+ n:Q(G-l )(rp)(t;) with G E SL 2] defines an SL 2 infinitesimal intertwining map of the k finite functions in FQ(A) to the "square-integrable" functions (L2(Whit)± space, the positive and negative Whit taker models) of the unitarily induced representation of SL 2 from the unitary character n (x) '""-+ e" v'-l x sgn QC<. fk, then, in fact, such functions are continuous at 6. These results are deduced by studying the explicit form of the function rp. In a similar fashion we deduce the following growth property of rp, a k x K finite func tion in Ft(sZ - 2s) :
0.
f
one
s {0}.
p{-
p"t
S. RALLIS
302
(with Z E th), where Q(Z) = II Z+ II 2 - II Z- 11 2 , II I l k the Frobenius norm of the linear operator g E O(Q), and rc the A-part of the lwasawa decomposition of G, i.e. G = K(G)A(rc)N(xc). 3. SL 2 cusp forms. The key property of the Wei! representation ""Q as set forth in [22] is the following one : given a lattice L s;:; Rk so that Q(L, L) s;:; Z, there exists an "arithmetical discrete" subgroup F(Q, L) of SL 2 x O(Q) such that for any Schwartz function rp on Rk (3-1)
is a coo function on SL 2 x O(Q), transforming on the right according to a finite dimensional unitary representation of F(Q, L). (Note ( 1-4) is a special case of this. ) The main analytic content of this statement is that the family of Poisson distribu tions G1J = L: eE L oH1J defines a continuous intertwining map ( relative to SL2 x O(Q)) of S(Rk), the Schwartz space, to the coo sections of a certain vector bundle over SL 2 x O(Q)/F(Q, L). However the above methods do not directly apply to the K x K finite functions on F"/j(s 2 2s ) since such functions are not Schwartz functions. But the growth and continuity properties discussed above for such rp coupled with the classical Poisson summation formula give the following theorem. -
,
THEOREM 3 . 1 [13] . Let s Fc[(s2 - 2s)
>
-!k. Then for any rp , a K
K finite function in
x
(3-2)
is an absolutely convergent series. Then for any (F, r) E FL(Q) FL(Q) =
O(Q) L with
x
{( [� �J c)/ a, b, c, d E Z, ad - be = 1 , c
=
0 mod 2NL, b
=
}
O mod 2 .
we have the functional equation :
8�(GF, gr) = c(F, r) fJ�(G, g),
(3-3)
with c(F, r). a unitary character on FL(Q) x O(Q) L , taking values in = {z E C l = 1 } . Moreover e� is a coo function on SLz X O(Q) satisfying D * !}�(G, g) = e�Q (Dlrp (G, g) for any D in the universal enveloping algebra of SL2 x O(Q) (* denotes differentiation on the left).
S4
z4
In particular it follows from Theorem 3 . 1 that 8� is an eigenfunction of the center 0(SL 2 x O(Q)) of the universal enveloping algebra of SL2 x O(Q). That is, 2 Wsf4 * e�(G, g) = (s - 2s ) e�(G, g). On the other hand, from the growth estimates (2-1 ) we have if r > 1 rcs-1!2 + . G (3-4) $ M ll g-l ll k• k/2-2 l fJL'P (G g) r(; e -•
j
{
-
=
'
'
(with M and c positive constants independent of (G, g)). This allows us to show
COROLLARY TO THEOREM 3 . 1 (WITH THE SAME HYPOTHESES AS IN THEOREM 3 . 1 ) .
e�(G, g) is a cusp form in the SLz variable.
CUSP FORMS AND AUTOMORPHIC FORMS EXAMPLE 3 1 . An example of a K -
rp(X ) = 0 on
303
K finite function in FJ"{s 2 - 2s) :
x
{}_,
= Q ( X ) • -le->rQ CXJ II X+ Jl - <s+st +s2 +kl2-2) Q(X, .;+ )' '
(3-5)
with X E {)+ ,
Q(X, ,;_) •2
where .;+ <e- resp.) is a complex isotropic vector in Ca (Cb resp.). Here isotropic is relative to I + 11 2 on Ca ( I - 11 2 on C6 resp.), and and s2 are positive integers satisfying S ] - s2 s - t (a b). Then we put into classical coordi nates relative to the upper half-plane H by setting
nonzero =
e�(z, g) =
(3-6)
-
e�(([ � �-�J ) )
e� s1
t , g (Im z)'12 ,
with z = v' - l jx2 - yjx E H. Then e�(z, g) =
(3-7) where rp�'(g) =
�
1:
CM<= L I Q (M, M) =n!
holomorphic cuspformI ca ( f: : � )
e� -
where
· rp�'(g),
+ Q(M, g-l e+ )'t Q( M, g-1 ,;_)'2 11 (gM) + JI - <•+•t +•2 k12-2J .
Then e (z, g) is a arithmetic group .dN L = { G E SL2(Z) (3-8)
e" ../-lzn
n•-1
1:
ne.Z, n�l
.g =
in z of weight s with multiplier V Q for the = 0 mod 2NL , b e = 0 mod 2}, that is
vQ ( G)
( z + d) • e� (z, g),
c
and
with
ca
¥-
0 (where if =
e�
v' - 1
if
m m
=
1 mod 4,
=
3 mod 4),
and where ( - ) is the Legendre symbol. The expression of above is precisely the Fourier expans ion of e� i n the z variable at { oo } ; thus n• '-1rp�t (g) , the nth Fourier coefficient, is an automorphic form on 0 (Q) relative to the group 0 (Q)L.
e�
4. Constant term of in O(Q) variable. The main problem is to determine the Fourier coefficients of B�(G, g) in the O(Q) variable. In general a formal answer can be given, which resembles the Fourier coefficients of Poincare series, i.e. an infinite sum involving Bessel functions and certain trigonometric sums (similar to Kloosterman sums). Such an expression is unsatisfactory from the arithmetic point of view.
304
S. RALLIS
However, the zeroth Fourier coefficients of 8� ( , ), the constant term along the different unipotent radicals of rational O(Q) parabolics, can be determined in a rather elementary fashion. If Q is the set of rational numbers, then let O(Q)Q-L = {r O(Q) I r(Q L) = Q, L} ; Q · L is a Q vector space Q · L}, where Q · L = the Q span of of Q-dimension k. We know that every parabolic subgroup of O(Q) is 0 (Q) conjugate to a group of the form where PF = O(Q) g(F) = F} with F a nonzero Q iso tropic subspace of Rk, i. e. Q(X, X') = 0 for all X, X' F. Then there exists (by Witt's Theorem) another Q isotropic subspace F' so that dim F = dim F', F n F' = (0), and Q is nondegenerate on the direct sum F + F'. The Levi decomposition of PF is given as follows : PF = GI(F) x O( Q, (F + F')J.) · NF • where Gl(F) is the group of all R-linear transformations of F, O( Q, (F + F')J.) is the orthogonal �roup of Q restricted to (F + F')J., and NF, the unipotent radical of PF· Here GI(F) embeds + with into O(Q) via the map + + = F, + (g1 )- 1 F', and (F + F') l- . with the lattice L if there exist F ' and Z lattices LF, We say that PF is LF', and L LF' ® L cF+F ' ) j_ in Rk. In particular this means that PF n O(Q)L is an arithmetic subgroup of PF (relative to the Q structure Q · L on Rk). In order to study the cuspidal behavior of we must examine integrals of the following form :
{r· � IrE � E pP, maximal {g E I E
(4- 1 )
with
e� (G, g), B�(G, da(v) J E compat e�(G,ibg)le with cusp form s> E F(f(s2 -compat 2s)RxibKle s U) dU, dU gvr)
NFINFn rO (Q) a- 1
g
E
O(Q), r
·
(u2) u3 u1 E u2 E
g(u1 u2 u3) g(u1) compatible
u3 E
E
O(Q)Q·L and da, some NF invariant measure on
NFfNF n rO(Q)a-1. Here we assume that pF is NFfNF n rO(Q)a-1 is compact. We know that
L. In such a case is a on O(Q) relative to the group O(Q)L if the family of above integrals vanish for all g O(Q), all r E O(Q)Q·L• and all PF with L (see [5]). and define (for On the other hand let rp tk)
(4-2)
(/)�( W) =
E
E
F rp( W +
with W (F + F') J. , some Euclidean measure on F. We know that the affine plane W + F (with W E (F + F')l- ) is the NF orbit of the nonzero vector W in Rk. So, in particular, (/)� can be interpreted as the integral of rp over the NF orbit of (the orbit here carries an NF invariant measure which can be identified to above). Then we have SL 2 x LEMMA 4. 1 [13]. > tk. rp ..._. (/)�
dUW
Let sning mapThen the 2map- 2s)KxK ontoisF(fanF(sin2fini- 2s)KxK tesimalF' where intertwi of FJ'(s anded represent QF is atrestionsriocftedtheto universal envel(Inofipiningtesialmgale means relat i v e to t h e associ a t bra.) 2 Ft (s - 2s) F F(f/-
O(Q, (F + F')J. ) KF = K n O(Q, (F + F')j_ )
Q
REMARK 4. 1 . If QF is positive definite, then ) (when Q is indefinite).
way as
. .
(F + F')J. .
is defined in a similar
CUSP FORMS AND AUTOMORPHIC FORMS
305
FQ"is2 - F!F(s2 Let g) 2 O(Q). Then and bb, and and and bb (wi- th(wib th ab - a orbor a a), rp e
The most important case of Lemma 4. 1 occurs when is the case when QF has signature 1). Then 2s)
(m,
E
SL
=
2s) = {0} . Such {0} . Thus
rtJ:Q CCG, g) - l ) (tp)
in thefolrplowiFQ"(s ng cases: FQ"(s222 --rprp Ft(s rp Ft(s2 gO(Q,O(Q), rpeFt(s2 polarpdecomposition g k · k e p (p malr paraboli cQ L· ofLetO(Q)Sr. icompat iblelatwititche L.in Let O(Q) · L) be the Let rp Ft(sobt2 a-ined byLetprowijtecth sibengartk.maxi S (relative L onto to the Q split ing of Then for any g) e L2 O(Q) CoROLLARY TO LEMMA 4. 1 . E
(a) (b) (c) (d)
E
E
E
2s)KxK 2s)KxK 2s)KxK 2s)KxK
(G,
dim F = dim F = dim F = dim F =
x
=
0
1,
=
1
=
= = 2).
1
We emphasize here that in the statement of the Corollary to Lemma 4. 1 , although i n Lemma 4. 1 the map -+ r[J: i s intertwining only for (F + F')l- ) . The reason is that f/J:( W) = 0 for all W (F + F')j_ and all 2s)ilxK implies that
E
rtJfQ Cg) - ltp ( W )
=
rtJfQ Ck) - l (tp) (( p W) cF+F' ) l_ )
=
0.
p with (We use the of = K, E PF• and the projection of W on (F + F')l_.) Thus we can determine the cuspidal behavior of <9� completely.
THEOREM 4.2 [13]. E
(F + F')l_
2s) K xK Rk
=
PF
>
E
( ) n (F + (F + F')l_)
F + F' + (F + F')l_ ) .
W) cF+F' ) \
(F + F')l_
(G,
x
(4-3)
where c1 is a positive constant depending only on p
F·
A schematic way to convey the inductive nature of (4-3) is the following com mutative diagram :
Q( L 'IC l�:;g)------.... ;:• ..., rO( ) Lr-l k 1 · g1 · n k 1 Q g 1 O ( (
-0:!;;L::
(4-4)
e�(G,
) (�)
constant term along NFINF n
S7. F
......
(
"" �
)
e�vc�� (G, •Q Ck1 ) 'P
( g,)-'
n0,( G,
) ( !!>f,oc•>,)
gt)
rO(Q)LT-l
Here g = with E K, E Q , F + F')l_), and NF. From (4-4) we see that the constant term of <9�(G, along NF/NF n is a 8 series of a function on a space of smaller dimension. The main idea in the proof of Theorem 4.2 is to substitute (3-2) for <9� in (4-1 ) and reorder the summation and integration t o obtain a sum o f "Nrhorocyclic" projection maps of over equivalence classes of NF n in L. Here a horocyclic projection map is the integral of over an NF orbit (carrying an NF invariant measure) in Rk. An example of a horocyclic projection map is given by
rp
g) n e
rO(Q)LT-l
rp
S. RALLIS
306
(4-2) . Essentially there are two types of NF orbits in Rk . The first type of NF orbit corresponds to a "curved conic section", called a regular horocycle. The projection map of cp corresponding to a regular horocycle is zero. (This statement is called the First Cusp Vanishing Theorem in [13] .) This follows essentially from the fact that rp is continuous on Rk and that rp vanishes on the light cone. The second type of NF orbit is the affine type discussed above. The contribution of the horocycl ic projec tion maps of cp corresponding to the affine type NF orbit is then determined by Lemma 4. 1 . Then using Corollary to Lemma 4. 1 and Theorem 4.2, we have CoROLLARY TO THEOREM 4.2. Let cp E F(J(s2 - 2s)KxK with s > !k . Then if PF is compatible with L and dim F = b - 1 or b, the constant term of B�(G, g) in the direction of NFINF n rO(Q)Lr-1 (r E O(Q)g.L) vanishes. In particular if b = 2 and cp E FQ"(s2 - 2skxK• then B�(G, g) is a cusp form on O(Q) relative to O(Q)L.
Let PF be compatible with REMARK 4.2. Let cp E Ft(s2 - 2s)KxK with s > L. If dim F = b - I (when either a = 2 or a = b) or dim F = b (when either b = a - I or b = a), then the constant term of B�(G, g) in the direction of NFINF n rO(Q)r-1 (r E O(Q) Q - L) vanishes. In particular if a = 2, b = I or 2, and cp E Ft(s2 - 2s)KxK • then B�(G, g) is a cusp form on O(Q) relative to O(Q)L. The case b = 2 is thus the main case of interest. In particular we know that O(Q)IKh where K1 = O(a) x S0(2), is then a Hermitian symmetric space. Let ff' = f + p be the Cartan decomposition of the Lie algebra of O(Q) . Then com plexifying ff to ff'c = ff' ® R C, we have the direct sum ffc = fc EEl p + EEl p-, where p+ and p- span the holomorphic and antiholomorphic tangent vectors at the
-!k.
"origin" in O(Q)IK1 • We recall the construction of a family of holomorphic dis crete series representations of O(Q) . We let Xn : K1 --+ S' be the unitary character on K1 which is trivial on O(a) and maps S0(2) =
to
r ..J-tn
{[ - csmc;>s ()() cos sin ()]\ ()
7r:
<
()
:::;; }
-
7r:
o . Then we form the holomorphically induced representation space
.Yt'( O(Q)IKh Xn ) =
{ cp :
O(Q )
--+
cI
cp E coo ( O(Q)) , cp(gk) = cp(g) xn (k)
for all g E O(Q), k E K1 o cp * W = 0 for all W E p+ and
with
*
J O(Q) !K1 I cp(g) I Z dr(g) < oo }
W convolution on the right and d-. some O(Q) invariant measure on
O(Q) IK1 • Then we have
THEOREM 4.3 ([13], [15]). The representation of O(Q) in d-; (see Remark 2. 1) is equivalent to the "holomorphic " induced representation of O(Q) in .Yt' ( O(Q)IKh Xs2)oo" where s 2 = lsi + tk - 2 (( ) oo denotes coo vectors) . 5. The Shimura-Niwa correspondence. Having determined the cuspidal proper ties or e� in both variables (cp E F!-(s2 - 2s).Kx K with s > tk) , we now go back to
CUSP FORMS AND AUTOMORPHIC FORMS
307
the question of using e� as a kernel function to set up a correspondence between modular forms on the groups involved. In particular we let be as in Example 3-1 , and consider S� , (see (3-7) ) Then w e let b e a of weight tk) on satisfying = with r e LJNL · + As in § I we consider the Petersson inner product of/with 8� :
q> phic cusp form (z g) . f holomor s (s > H f(r·z) vQ(r)(c7z d7)• f(z) (5-1) (S�( ' g) if( ) s e�(z, g)f(z) 1 ·-2 dx dy. (5, -g)1)I f( ) ), e �. ( " X X , P(X)e [ J. 1, e �. 'l . ) L, , 2 ) ) =
)
I Im z
IJNL\H
The definition given in does not, at first sight, coincide with ( 1 -8), the Peter sson inner product < where e�. is the o series constructed from the Schwartz function (In keeping with the notation of § we note that In particular we must determine if TJ E then we dr�p the subsc�ipt TJ from the relationship of e�.( , to 8�( , defined above. But we know that the map cf; -+ e�( ) is an SL X O(Q) infinitesimal intertwining map from ce�tain SL2 X O(Q) stable subspaces of V(Rk) to the space of C"" functions on SL2 x O(Q) . Thus perhaps the appropriate problem to analyze is the following : for a given K x finite function cf; and the projection ¢D of cf; onto the discrete spectrum of SL2 X O(Q) in V(Rk), what is the relation between the () series er/> and er/>D " This we now do partially in
K
LEMMA . 1
5 [16]. Let be a forKfiniallte Schwartzfunctiandon in andLet suppose that s' > Let P"j; be the projection ofFQ onto the subspace Fct(s'2 -i 2s'). Then (5-2) ( Sf( g) I f ( )) ( S�,t cFl g) f ( )) . 5P-;;, 2 FQ F(i(s' (5- 1) 2s'). f-'"+ (S�(f(r·z) , g) ))vQ(r) (c7z d7)• f(z) r s, vQ]z o H -+ vQ. f(z) Gn(z) ( 1 dr )s vQ(r) g) Gn( )) Cl ·q>�'(g), c1 g n. s> G s, vQ]o. S�( , g) i f ( n s, vQ]o} q>�' n (n q>�' q>�' s, vQ]o. g) i f( )
71: m(k((), c))(F)
=
F e ..!-l s'O ckF
- 71:
=
•
V(Rk)
< () �
11:
(
c
=
± 1.
fk + l .
•
REMARK . 1 . If s ' < - Hk + 1 ), then a similar statement is valid where Pt is
replaced by the projection of onto + Thus the two correspondences ( l -8) and are essentially the same. The next main problem is to characterize the image of the map .I as f varies in the space [LJNL • = {! : C l fholomorphic, f( = + for all e LJNL ' e H, and f vanishes at the cusp points of LJNL on Q U { oo} }, the holomorphic cusp forms of weight s and multiplier The first trivial observation is that if we let =
=
I; rE (.dNL) ""\.tJNL Cr Z +
e" .J-l nr C•l ,
the Eisenstein-Poincare series, then
= (�( , I with a nonzero constant independent of and However since tk , we know that the functions span [LJNL ' )) I fe and hence the space { ( [LJNL ' is exactly the complex linear span of the as ;:;;; 1 . From Corollary to Theorem 4.2 we know the cases when all rp�' ;:;;; I ) are cusp forms on O(Q) relative to O(Q)L. And the most general Fourier coefficient of is difficult to describe arithmetically, since is essentially very much like a Poin care series. However using the results of §4, it is possible to determine for which fe [LJNL • <8;( , ) will be a cusp form on O(Q) relative to O(Q)L.
S. RALLIS
3 08
Fi maximal
L.
We let be a Q isotropic subspace so that PFi is compatible with Then we consider a flag of Q isotropic subspaces so that each � � ••• � has dimension 1 . Then we know that O(Q) o·L • F is compatible with L and ,. P the set of rational points in O(Q), has the following decomposition : O(Q)0.L O(Q)L · BF . · (pF .)o· L• where BF · is some subset of O(Q)0. L (see [2, p. 104]). ) to be a cusp form. Then we have the following c ;iterion for (B�( ,
F1 F2
F.-+1/F;
Fi
=
finite g) If( ) fk. Let(/) be given by LetfE s, (sVQ]20. Let givbeen theby thunie compl que ex liLets ( F ; FY) i n vari a nt su p ace o f h_! relative ntoearthespangroupof {!P:� Ckll" I andk E onlK}.y Then , g) I f( )) is a cusp form for all E all E and all g1 E (F; F;)l -) as i · · · , g (F,.s,-f.vQ]0.F;)l.) F; CLl (;, g) gh (F F')l.) (F F' ) l. ) [1] g) I f ( finite f ...... (�A;j. F;CLl ( , g,.) If( ) b Rk-2 e2, e3,signatureek_2, ek Rk-2. restricted to{WE Rk-2 Rk-21 e{k) } future Eforward Rk-2 E Rk-2 E R}, phimsms1 E (F1 FFi1 )l.), A(r) E {t·(e1 -me1k-A(r) 1)/analyt· n1iIcEtautomor 1 E E Rk-2 (F1 F;)J.r{m. 1(Z !Zl{m1 A(r)nh r, unique mhol1Aomor (r)nE d;p1 hiE c automorb phy factors tk) THEOREM 5.2. KF,.
on
=
>
(3-5).
K n O(Q,
+
YF,. - 2s)
F(JFi
(8�(
if
O(Q)L if
O(Q)
[LINL •
(5-3)
Ajj
Bii,.\
cp
YF.-•
O(Q,
+
=
j.
I,
We note that YF · is a finite dimensional space, BF · is a finite set, and that as varies in O(Q, , the functions B�•ii· span a finite dimensional In particular, if Q has signature (k - 1 , 1), then we need subspace of [LINL • the identity element of O(Q, (in this only the validity of (5-3) for + case O(Q, + � KF). This fact probably explains the orthogonality condi tion of required to get a lifting from st2 cusp forms to cusp forms on 0(3, 1). We note that the condition for (B�( , ) ) to be a cusp form on O(Q) is de termined essentially by the fact thatf belongs to the kernel of a family of cor ) ) coming from lower dimensional respondences ( i.e . quadratic forms and lattices.
6. "Zagier polarization identity". We recall that if 2, then O(Q)/K1 is a Hermitian symmetric space. In fact we can realize O(Q)/K1 as a tube domain. Let be the span of (see § I ). Thus Q is of · · ·, Lorentz type, i.e. Q has (k - 3, I ) on Then let C+ Q( W, W) < 0 and Q( W, < 0 ( the or light cone). The tube domain of c+ is T(C+) X + .v - 1 y I X, y with y c+} + .y' - 1 C+ · Then let O(Q)# be the subgroup of O(Q) which is the connected component of the full group of of T(C+). If denotes the -/2) then Ph PFl n O(Q)# acts on Q isotropic line T(C+) as follows : if q · p� 1 with O(Q, + SF1 , and n NF, then q (Z) + Y)} where n 1 is determined uniquely by an element Y + We recall that there exists a � on T(C+) so that Z) 1 / where Ph as given above. Then, in light of Theorem 4.3, we let 'P (with 2 and > be the unique (up to scalar multiple) "lowest weight" vector of O(Q) relative to K1 • Then 'P can be given explicitly as follows : =
=
=
=
=
=
=
=
=
=
fP(X)
(6- 1 )
=
=
on {)+ ,
0
e"Q <X> ek S2 S tk
Q(X, e+)-•2 on {)_ ,
Q(X, X)s-1
with e+ and + V- 1 + - 2. We are going to consider a kernel function (constructed from f{)) which is slightly =
ek-1
=
CUSP FORMS AND AUTOMORPHIC FORMS
309
twisted from (5-1 ). We assume that the Q integral lattice L has a Q orthogonal direct sum decomposition in the form .P ED (Ztv ® Zii) with .P s;;; Rk - 2 ( the R span of {e2, e3, - · · ,ek _2, ek } ) , satisfying the condition that Q(.P) £ 2 · Z and n!i' · Q(p., p.) E 2 · Z for all p. E .P *(Q), the Q dual to .P. Moreover v and span a 1 . Also assume t is an hyperbolic plane with Q(v, v) Q(ii, ii) 0 and Q(v, ii) integer so that n!i' I t and 4 1 t. Thus the exponent nL of L equals t. Then we let X be a Dirichlet character mod t and consider the formal sum =
=
Bi, x (G, g)
=
�
r mad t
ii
=
=
x(r) B&, ,.(G, g).
Then following Example (3- 1 ) we define B�. x(z, g)
(6-2)
B&, x
=
=
(( [� �-1], 1 ) , g) (Im z)- l sl t2
� ,B,,x(g) l r l s- 1 e" v-1rz,
r�-1
with z = - yfx - ,.,j - I x 2 e B, the lower half-plane, and ,8 , , x (g) =
�
�
u mod I {MELIQ (M+uv, M+uv) =r)
x(u) Q(M + u v, g-l . '+ ) -52. ·
Then using the holomorphic automorphy factor � . we let
[�(g, -./ - 1 ek)]'2 ,B, ,x ( g-1 ) with Z Then we immediately have �, .xCZ)
(6-3)
=
�, .i Z)
=
�
{
�
u mod l {MELIQ (M+uv, M+uv) =r) ·
=
g -./ - 1 ( ,.,j2ek) E T(C+).
x(u)
{ - t Q(M, ; ) Q(Z, Z) + Q(M, Z) 2
+ Q (M,
-./ 2 ii) +
,.,j2ur'l
Moreover �'· X is a holomorphic function on T(C+) satisfying �r. ir · Z) [�(r, Z)]•z · �'· X (Z) for all Z e T(C+ ) and all r_ e O(Q)l . {r e O(Q)! I r(v) = v mod L} . On the other hand we note that e;. X is a holomorphic modular cusp form in the z variable (z e H), i.e. =
=
B&, x< r · z, g)
=
1 j(r . z)2 1' A6, :. 1 (d7 ) x (d7) B&, x (z, g)
for all r e F0(t) where
and _
L AQ, I sl (m) -
=
{ [� �}
SL2( Z) I
c
·=
0 mod
t},
_ 1 s - k/ 2 ( (2m )k (�) m) ' m
a Dirichlet character mod t. Then we define (as in (5-1 ) ) the Petersson inner product ( B&, x ( , g), h1( ) ) , (i) ¢ holomorphic on where ( f(z)) with / e S21 ,1( F0( t ), ,8) {¢ : (ii) !fo (r · z) = j(r . z)21•1 ,8 (d7)¢(z) for all r e F0(t) and z e (iii) ¢ vanishes at
h1(z)
=
=
H- Cl H;
H;
S. RALLIS
3 10
cusps of To(t) on Q U { CXJ } } and {3 (m) = 0.�. l s i ® ,X) (m) ( - l /m)2 1 s 1 • We then define (6-4)
Fi Z i qJ, L, v, X· Fo(t) ) =
(with g( .J
1 .J2 ek) = Z E T(6'+) and s2 = s + tk - 2 > 2). Then s all for E T(tfi'+) Fo(t)) • F0(t L, qJ, i { r, = ) Z)} ) v, FtCZ Z z L, qJ, v, i Z · X .0J( F1(r X· and all r E 0( Q)i_ v ·
-
k-
The main problem is to give an adequate characterization of the image of this map. The difficult problem is to determine the Fourier coefficients of �r.x · As shown in [23], [11], and [13] such Fourier coefficients are highly transcendental, involving an infinite sum of Bessel functions and certain trigonometric sums, which behave like Kloosterman sums. We know that NF , ( � Rk - 2) acts by translation on T(6"+). Thus there exists a lattice !l'F , in Rk- 2 = (F1 + F�)j_ so that if � E !l'F , , then F1(Z +
� I qJ, L, v, X · Fo(t))
= F1 (Z I qJ, L, v , X · F0(t))
for all Z E T(tfi'+).
A simple computation shows that !l' F t = .J 2 · { � E !l' I Q (�, �) = 0 mod 2t, Q (!l', �) = 0 mod t } . Hence we have an expansion of F1(Z qJ , L, v, X • To(t)) of the form : I; a ( fJ. , f) e-z" ..;-IQ C". zJ , (6-5) F1 (Z i qJ, L, v, x, F0(t)) = 11"' (5!'FI) , (Q) n6'+ where (!l'F ) * ( Q) = {fl. E (F1 + F� ) j_ I Q(fJ., !l'F ) � Z}, the Q dual to !l' F, and
I
(6-6) a( fJ. , f ) = S
F1(X + .J - 1 Y l qJ, L, v, x, Fo(t)) ez" ..;- I Q CXU-IY, "l dX.
Rk - Zf5fF1
The problem is then to determine a( fJ., f) in terms of the Fourier coefficients off at { w }. We note immediately from the Corollary to Theorem 4.2 that a(fl. , f) = 0 if fl. rt C!l'F )i Q) n tfi'+ · We begin with a heuristic discussion. First we know that the space S2s (F0(t), {3) admits a reproducing kernel function K : H x H --> C, a holomorphic (antiholo morphic) function in the first (second) variable which satisfies K( n (z J ), r2 (z2 )) = J(n , ZJ ) 2 s J (r 2, Zz) 2 s {3 (dT! ) {3 (dT2 ) K(ZJ, Zz)
for all Zj , Zz E H and
n,
rz E Fo(t) .
Moreover K satisfies the formula (f(z), K(z, w)) = f(w) ( ( , ) the Petersson inner product) for all j E S2 s{F0(t), /3). Then we know for s sufficiently large, there is a convergent expansion of K(z, w) = .E n;,; I ns-1 e-Zrr .J- Inw Gn(z) where Gn is an Eisenstein-Poincare series belonging to S2s (F0(t), {3) . On the other hand, we know that K(z, w) = K(w, z) . Thus we have another expansion of K(z, w) = I; ns- 1 Gn(w) ez" ..; -Inz. n �l
The second expansion can be interpreted as the Fourier expansion of K(z, w) in z at { w } (i.e. z ..-.. K(z, w) is an element of S2s (F0(t), {3), where the nth Fourier coefficient ns-I Gn(w) is a modular form of antiholomorphic type. We, however, note the analogy of this second expansion with the Fourier ex pansion of B�(z, g) in z given in (3-6), where each Fourier coefficient is a modular
311
CUSP FORMS AND AUTOMORPHIC FORMS
Thus it is reasonable to ask if there is a "polarization identity" of form on C where $� of the form : �,.;;:; 1 for all is a function which satisfies by an averaging is related to n PF 1 • Also we should require that n P l ' i.e. over .E rEO(Q) L!O (Q) LnpF (qJ�1 ) * (gr) . Zagier in proved such a "polarization identity" i J the case when However such a formula can be proved generally (by Oda in for the cases k � 6, and in a very general version of the formula is shown to be valid without restriction to the case 2).
O(Q). e�(z, g) = ns-1(qJ�l)* (g)G,.(z), (qJ�1)* : O(Q) --> *(gr) = qJ(qJ�1 �1)(g) (qJg�e1)*O(Q) and all r e ) t � (qJ O(Q)O(Q) L dO(Q)L F ql�1(g) = [23] a = b = 2. [11] [16] b = x(z, Z)Then ewe�. x(z,haveg-th1)e formul [.@(g, a : - l ek)]•z (see witLeth z =sg 2k.- I( vLet2ek) T(C+). I x(z, Z) = n 1 ·-1,8!, x(Z)G,.(z, where G,. is the Eisenstein-Poincare series on the lower half-plane, given by THEOREM
6. 1
.v
(ZAGIER
IDENTITY) .
s;.
(6-7)
Ei_
>
v'
(6 - 2)) �
E
=
x).
{nEZ in:;;; - 1 )
fi,
(6-8)
with (note here that e�. x(r·z, g) = s6, x(r)(c7z d7)5 e�. x(z, g) for Fo(t)) and {Q(�, Z) .V2Ut - ]} •z} x(Z) = { ,8,., x {� e 2 x = n,L e (singular horocyclic Q(�, �)O(Q) a(p,f) rE
+
(6-9)
�
� x(v)
,13:,
v - .
+
{o E SI'I Q (o, o) =n) ;jEZ
v mod t
REMARK 6. 1 . The function ,13:, is obtained from by restriction of the sum mation in (6-3) to precisely those lattice points of the form + tjv I � with and j Z} . This latter set lattice points) is stable under n P Fl' Then using Theorem 6. 1 , it is possible to determine (see (6-6)). The main idea is to apply the Lipschitz identity to the inner sum of the right-hand side of (6-9), i.e.
(
)sz
l z r•z-1 w m Lets,.;;:;>1 a1(n)Letfe be the Fouriwitehr expansi a Diriocnhletof charact e r t. Then letf( z ) = fatAssume (2Fl) (Q) and fl.¢ 2/(.V2·t). Then a(p,f) = * let (2F)iQ) with fl. = (m/(-v'2·t))·� with �. a t h e ot h er hand, On primitive element of the lat ice 2 (m, a positive integer). Then � /....1
m EZ
1
+
(2 7C vI - ) S - (s 2 - I) ! _
r =+oo
�
e2" ,J-1 wr
r=1
With W E H.
Then we deduce
COROLLARY TO THEOREM 6. 1 .
{ oo} .
mod fl.
E
2k.
.E
n c+ fl.
E
S2.(F0(t), u)
0',
e 2" ,;-1 "'
n c+
0.
S. RALLIS
3 12
a(p., f) c ( 11 t)t-•z iGx ,
=
a1 ( m: I with X11 the Dirichlet character given by X11(x) a(x)( -1 /x)2• ,(x) and G(x11, t), the Gauss sum given by e t) (here ci is a nonzero constant depending only on s). 1( L, v, X11, 5. (6-10)
. :E x"
=
1).
A�.
=
G(x" '
Q<e . e)
])
:E X"(v) ztr ..;-� vtt
• mod i
Then following the well-known methods in automorphic form theory, it is possible to associate a Dirichlet series to the automorphic function 1�. F0(t)). The cases k = 3, 4 have been studied extensively in [10] F and [6], so in the ensuing discussion we assume that k � In particular we let
(6- 1 1)
where F,h(Q n c+ } denotes the of O(Q, F! + Fi)# n O(Q)L in (.PF)*(Q) n C+ and C(O) = the order of the finite subgroup of O(Q, F + Fi)# n O(Q) L which fixes 0. Then from (6-10) we deduce immediately
{(Sf ) set ofequivalence classes 1 5). Lets C so that Re(s) is sufficiently large. Then we have the identity : f) d x t)t2ii-sz 11 s2) aj(-n) n)j n j -•, Sf, where i s the Si e gel mass number o f t h e form Q rest r i c t e d to 1 relat i v e to t h e lat t i c e on the quadri c o f level n (i. e . n) 1 1 where run through a setn})oandf represent ativesconst of ant dependent orbi t s i n Sf d a nonzero 1 s2 s er Also L(o,s) is the classical Lfunction only onatse2d(recall associ to the Dihererichletthat charact f) a•), D f)n) In 1 -•). a1(n)n-• f) f) f)}, PROPOSITION 6.2 [16] (WITH THE SAME HYPOTHESES AS IN THEOREM 6. 1 AND k � e
R(s,
(6-1 2)
F
+
=
1 G(
2"L(X, , 2s + 1 M(QI> Sf,
11 ,
:E
(neZin&-ll
M(QI>
n)
Fi)
:E7l'f e(e;) - , n O(Q) L
-
(= Q M(Ql> Sf, = O(Q, F1 + Fi)#
Sf
el > . . . , eh n {X e R'.-2 1 Q(X, X) = = + tk - 1). o.
Thus we have expressed R(s, as the product of elementary functions (i.e. an L function, and the Rankin convolution of 2 Dirichlet series (i.e. the Dirichlet series (s, = :E,.-., 1 and Siegel's zeta function C -(Ql> Sf, s) = .E {neZi n&-ll M(Ql> Sf, The analytic nature of the function R(s, can then be determined easily from = {1r- 2 1 F(s - tk + 2) F(s) R(s, then Proposition 6. 1 . If we let R * (s, R*(s, f) can be analytically continued to the whole s plane (F, the gamma function). REMARK 6-2. Using (6- 1 2), it is possible to deduce a type of Euler product expan sion of R(s, f). It suffices to study the possible Euler product properties of the Ran kin convolution of and C- (QI> Sf, s). However if k is even and both and C -{Ql> Sf, s) admit the usual Euler product of the theory, then the Rankin
D(s,f)
GL2
D(s, f)
CUSP FORMS AND AUTOMORPHIC FORMS
313
convolution of these series can be expressed as an Euler product with numerator of degree 2 and denominator of degree 4 for almost all primes p. (For suitable is a finite sum of Euler products of the G� choice of Q1 and 2, ,_(Qh 2, theory, see [7].) On the other hand, if is odd then and ,_(Qb 2, do not have the usual GL2 type Euler product. However in [18] a modified theory of Euler products is set forth for St2 automorphic forms of semi-integral weight. In partic ular if/E S2 s (F0(t), m is a Heeke eigenfunction in the sense of [18], then the partial the discriminant of an imaginary quadratic Dirichlet series .E n ;,;1 extension of Q) can be expressed as an Euler product with numerator of degree 1 and denominator of degree 2 for almost all primes p. And it is possible for suitable Then by purely 2 to find a similar Euler product for .E n ;,;1 M(Qb 2, algebraic methods, one can show that the Rankin convolution .E n;,;I is an Euler product with numerator of degree 3 and denominator M(Qb 2, of degree 4 for almost all primes p (see [16] for the case 5 where 0(3, 2) is locally isomorphic to Sp 2(R)) . Notes. For a complete bibliography on "hyperboloid analysis", we refer the reader to [21]. Also for an adelic treatment of the Weil representation and auto morphic forms, see [4]. We use the terminology that a function vanishes at a cusp oo) = if = + d)-s has an expansion of the form ,E en e2" .J-H
s) k
D(s,f)
s)
a1(1 dln2)n-j (d,
dn2)n-J. k
dn2)n-j
·
(¢1r)(z) (cz
n;,;O
=
¢ c0
¢(r(z))
aj(l d l n2)
r( a
(here " is the ramification of the multiplier at a and N is the smallest positive integer so that
[ 1 N1 r-1 E rb J
r o
rl the arithmetic group in question) (see [18]).
BIBLIOGRAPHY 1. T. Asai, On the Doi-Naganuma lifting associated with imaginary quadratic fields, preprint. 2. A. Borel , Introduction aux groupes arithmetiques, Hermann, Paris, 1 969. 3. L. Ehrenpreis, The use of partial differential equations for the study of group representations,
Proc. Sympos. Pure Math. , vol. 26, Amer. Math. Soc. , Providence, R.I., 1 974. 4. S. Gelbart, Wei/'s representation and the spectrum of the metaplectic group, Lecture Notes in Math . , vol. 530, Springer-Verlag, Berlin and New York, 1 976. 5. Harish-Chandra, Automorphic forms on semisimp/e Lie groups, Lecture Notes i n Math., vol. 62, Springer-Verlag, Berlin and New York , 1 965. 6. S. Kudla, Theta-functions and Hilbert modular forms, preprint. 7. H. Maass , Automorphe Funktionen und indefinite quadratische Formen, Sitzungsberichte der Heidelberger Akademie der Wissenschaften, 1 949. 8. - - -, Verteilung der Punkte in Gittern mit indefiniter Metrik, Math. Ann. 138 (1 959), 286-3 1 5 . 9. Uber eine neue Art von nicht analytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann . 121 ( 1 949), 1 4 1 - 1 8 3 . 10. S. Niwa, Modular forms of half integral weight and the integral of certain theta functions, Nagoya Math . J. 56 ( 1 975), 1 47-163. 11. T. Oda, On modular forms associated with indefinite quadratic forms of signature (2, n - 2), preprint. �� ,
S . RALLIS
314
12. S. Rallis and G. Schiffman, Wei/ representation. I. Intertwining distributions and discrete spectrum, preprint. 13. , Automorphic forms constructed from the Wei/ representation : holomorphic case, Amer. J. Math . (to appear). 14. , Discrete spectrum of the Wei/ representation, Bull. Amer. Math . Soc. 83 (1 977), 267-270. 15. , Automorphic cusp forms constructed from the Wei/ representation, Bull . Amer. Math . Soc. 83 (1 977), 27 1 -275 . 16. , On a relation between st. cusp forms and cusp forms o n tube domains associated to orthogonal groups, preprint. 17. G . Shimura, Introduction to the arithmetic theory of automorphic functions, Iwanami Shoten ( 1 9 7 1 ) . 18. , On modular forms of half integral weight, Ann. of Math. (2) 9 7 (1 973), 440-4 8 1 . 19. T. Shintani, On construction ofholomorphic cusp forms ofhalf integral weight, Nagoya Math . J. 58 ( 1 975), 83-1 26. 20. C. L. Siegel, Lectures on quadratic forms, Tata Institute Lecture Notes, No. 7, 1 955. 21. R . Strichartz, Harmonic analysis on hyperboloids, J. Functional Analysis 1 2 ( 1 973), 341-3 8 3 . 22. A. Wei!, Sur certaines groupes d'operateurs unitaires, Acta Math. 1 1 1 (1 964), 1 43-2 1 1 . 23. D . Zagier, Modular forms associated to real quadratic fields, Invent. Math . 30 (1 975), 1 -4 8 . ---
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PRINCETON UNIVERSITY
Current address : Ohio State University
Proceedings of Symposia in Pure Mathematics Vol. 33 (1 979), part 1 , pp. 3 1 5-322
A COUNTEREXAMPLE TO THE "GENERALIZED RAMANUJAN CONJECTURE" FOR (QUASI-) SPLIT GROUPS
R. HOWE AND
1. 1.
PIATETSKI-SHAPIRO
1. Introduction. In [Sat], Satake explains how the Ramanujan and Ramanujan Peterson conjectures concerning the coefficients of cuspidal modular forms can be formulated group theoretically. Briefly, the interpretation is that the local consti tuents (see [F]) of the automorphic representation associated to a classical cusp form should be tempered in the sense of Harish-Chandra [KZ] . (See also [GGP].) Temperedness is a technical condition on the asymptotic behavior of matrix co efficients that has proved crucial in Barish-Chandra's work on the Plancherel formula. Since it makes sense for any reductive group, the question as to whether the cuspidal automorphic representations of any global algebraic group might not be tempered came to be known as the generalized Ramanujan conjecture. Examples of automorphic cusp forms whose component at oo was nontempered have been constructed by several authors (see [Ga], [HW], (M], and the remarks in [B]). However these examples were for anisotropic groups, and the question remained open for quasi-split groups. Here we will construct examples of cusp forms for Sp4 which are nontempered almost everywhere. 1 Our construction uses the oscillator representation. Regarding Sp4 as one mem ber of the dual reductive pair (Sp4 , 02) (see [H]) yields an injection of the auto morphic forms on 02 into the automorphic forms on Sp4 • This construction is parallel to those given by Heeke, Maass, and Shalika-Tanaka for cusp forms on Sp2 = Sl2 corresponding to grossencharaktere of quadratic fields. Indeed these are automorphic forms on Sl2 corresponding to the dual pair (Sp2 , 02 ). We remark that although every automorphic representation of 02 gives rise to a representation of Sp4 , not all such representations of 0 2 are involved in the correspondence with Sp2 • It turns out that it is precisely the 02 representations which were missing in the cor respondence with Sp2 that give rise to cuspidal representations of Sp4 • Locally, at a place where the binary quadratic form is anisotropic, there is precisely one repre sentation of 0 2 which is missing from the pairing with Sp2 , namely the nontrivial character which is trivial on S0 2 • Thus the nonconnectedness of 02 plays an imAMS (MOS) subject classifications ( 1 970) . Primary 22E55 . 'Our first examples were for U,, , ; however the Sp, example i s easier to present, and i n some sense is better, since Sp, is not merely quasi-split but split. © 1979, American Mathematical Society
315
R . HOWE AND I . I . PIATETSKI-SHAPIRO
316
portant role in this construction. The local (Sp 4 , 02) correspondence was analyzed by C. Asmuth in his thesis [A]. The cusp forms that we construct are peculiar in some respects. In particular, they fail to have Whittaker models [P 1] . Thus the possibility remains open that the generalized Ramanujan conjecture is true for cuspidal automorphic representations with Whittaker models. Probably the best known example of a cuspidal represen tation without Whittaker model is the representation Or o of Sp4 over a finite field, discovered by Srinivasan [Sr] . The cusp forms constructed here are closely related to Oro. Indeed, let k be a non-Archimedean local field with integers (9 and residue class field F. Then Oro is a representation of SpiF) and may be pulled back to a re presentation of Sp4 ((9) via the canonical surjection It is not hard to see by means of the Cartan decomposition that the induced re presentation ind Oro = ind §�! i�l O ro will be irreducible and supercuspidal. It is essentially this ind 8 1 0 which will be the local constituent at certain finite places of the automorphic representations we construct. We note that Oro is one of the simplest cuspidal representations of the type Lusztig [L] calls unipotent. The examples for U2 , 1 also involve a cuspidal unipotent representation of that group. This suggests that the failure of the generalized Ramanujan conjecture is related to the existence of these cuspidal unipotent representations. Lusztig has classified these representations for the classical groups over finite fields and has shown that although they are rare, they exist for groups of arbitrarily high rank. The question of computing £-functions attached to the automorphic forms we construct is of some interest. Using the definition of £-functions for GSp 4 in [P 2], [NPS] we made some preliminary computations. The feeling is that the £-function for the local cuspidal representation which comes from the oscillator representation has a pole. This is consistent with the expectation that after an ap propriate lifting our cuspidal nontempered automorphic representation corres ponds to a noncuspidal automorphic form. In this way perhaps our example can be shown to fit consistently in the pattern of Langlands' philosophy. Detailed compu tations will be given elsewhere. 2. The construction. [W 2] ) . We take char
of the form
where A, (2. 1 )
B, C,
Let k be a global field (an A-field in the terminology of Wei! k =F 2. Let Spik) be the set of matrices , with entries in k,
and D are 2
x
2
matrices satisfying
where is the 2 x 2 identity matrix and A 1 is the usual transpose of Let K b e a quadratic extension o f k . The norm form from K to k i s a quadratic
I
A.
COUNTEREXAMPLE TO GENERALIZED RAMANUJAN CONJECTURE
317
form, labeled � . on K regarded as a 2-dimensional vector space over k . Denote the isometry group of this form by 0 2 (k). Let -c denote the nontrivial element of the Galois group Gal of K over k. Then -c is in 0 2 (k), and �(x, x) = x-c(x) for x in K. Set U = {x E K : �(x, x) = 1 } .
(2.2)
Evidently U is a subgroup of K x , the multiplicative group of K, and so acts on K by multiplication. This action identifies U with S0 2 , and we have an exact sequence 1 -+ U -+ 02 -+ Gal -+ 1 .
(2. 3)
Let A be the adele ring of k. Let v be a typical place of k, and k. the completion of v. Then Sp4 (A ) is the group of matrices with entries in A satisfying (2. 1 ), K. = K ®k k. is the completion of K at 1', and �. is the form induced by � on K•• and so forth. Following Weil [W 1] and Saito [So] we know there is a unitary representation w of SplA ) on L2 (A 2 ® k K) � £2 (A 2 ® A KA ) � £ 2(K1). From these authors' work we can glean that this representation will have the following properties. (i) w preserves the Schwartz-Bruhat space .!I'(Kl) . Denote by w"' the restriction of w to .!/'. (ii) Just as .!I'(Kj) is the (restricted) tensor product of the local Schwartz-Bruhat spaces .!I'(K';), so also w"' is the restricted tensor product of representations w'; of Sp 4(k.) acting on .!I'(K;), and the w';' are restrictions of unitary representations w. on £2(K;). (iii) The action of the subgroup of Sp 4 of elements of the form k at
is essentially induced by the linear action on A2 , modified to provide unitarity. Precisely, choose a basis e � o e2 for K over k. Then for f E .!I'(A 2 ® A KA) , w e have (2.4)
w
(g
�H ) {f) ( v1 ® e 1 + v2 ® e2)
=
s(A) I det A l- 1 /(A- 1 (v 1 ) ® e1
+
A - 1 ( v2) ® e2) .
In (2.4), I I is the standard "module" (cf. [W 2]) or absolute value on A, v 1 , v2 are in AZ, and s(A) is a character on GL 2(A ) of order 2. We note that (2.4) breaks up in an obvious way into local actions. (iv) The action p of 0 2 (A) on .!I'(Kl) induced by the linear action of 02 (A) on KA commutes with w(Sp4(A ) ) . Note this action of 0 2(A) also breaks up into a pro duct of actions of local 0 2( K.) ' s . (v) For f E .!I'(Kj) define (2. 5)
e( f) = I: f( x) . Xl:;K2
Then e is a distribution on .!I'(K2) , and e is invariant by w "' (Sp4 (k) ) . That is, for g in Sp4(k) and / E .!I'(Kl), we have (2.6)
e(w(g)(f)) = e(J) .
318
R . HOWE AND I . I . PIATETSKI-SHAPIRO
It is obvious that 8 is also invariant by p(02(k)) . Therefore, if we define, for f in .9"(K.n, a function 01 on SpiA) x 02 (A) by the formula (2.7)
Oj(g, h) = B(w(g)p(h)f)
then () 1 factors to a function on the quotient space Sp4(k)
x
02 (k)\Sp4 (A)
x
02( A ) .
Furthermore, () 1 will have moderate growth at oo , so that () 1 is an automorphic form on Sp4(A) x 02( A ). Our strategy now is the usual one in the theory of {}-series. The coset space 02 (k)\02 (A) is compact. We will see shortly that for an appropriate choice of f, the restriction of () 1 to 02(A) is an arbitrary Schwartz-Bruhat function on 02 (k)\02 (A). Thus in particular, we can arrange that ()1 will transform under some given irreducible representation a of 0 2 (A). From a study of the local representa tions Wv, we can then conclude that if av is chosen properly at certain places v, then () 1 will transform under Sp4(kv) according to a certain supercuspidal representa tion. z It follows that 01 !Sp4( A) will be a cusp form. Further examination will show that at almost all places v, the cusp form () 1 will transform according to a certain irreducible nontempered representation a;, of Sp4(kv), determined by av. There fore, any irreducible component in the space of cusp forms on Sp4(A) generated by () 1 will transform at almost all places by these same a;, and our goal will be achieved. We proceed to details. PROPOSITION 1. The map f -> () 11 02(A) is surjective from .9"(K�) to .9"( Oz (k) \ 02( A ) ) . PROOF. Let (xb x2) be a point in K2 such that
e: g
----->
x1
and x2 span K over k. The map
(gxb gx2)
of 02(A) into K1 is a smooth injection. It also has closed image. This can be seen i n various ways, but perhaps most easily b y noting that the image o f 02(k) consists of rational points and is therefore discrete, and 02(k)\02(A) is compact. Since the image of e is closed , the pull-back e* : C�(K1) ----> Coo( Oz (A))
is surjective onto C�( 02(A )) . Let/ E C� (K1) vanish at all points of K 2 not in the image of e. Then by Witt's Theorem [J] it is easy to compute that (2.8)
01(g) =
L: e*(f)(hg).
k 02 (k)
Since it is well known that all f in Coo(02(k)\ 02(A)) are of the form given on the right-hand side of (2.8), the proposition follows. REMARK. The same result with the same proof holds for an arbitrary irreducible type I dual reductive pair (G, G') (see when the space with form attached to G has an isotropic subspace of dimension at least equal to the dimension of the formed space attached to G'.
[H])
'Actually it can be shown that 01 will transform under Sp.(A) according to an irreducible repre sentation a' determined by a , but this finer result is unnecessary for the present purpose and would unduly lengthen the paper.
COUNTEREXAMPLE TO GENERALIZED RAMANUJAN CONJECTURE
319
I f K. i s a field, i.e., i f K does not split a t then will b e anisotropic and will be compact. Therefore SI'(K�) will break up into a discrete direct sum of spaces transforming under (k.) according to its (one or two dimensional) irreducible representations. Asmuth has established the following facts [A]. (See also [H].)
v, {3.
02
PROPOSITION 2. If K
02(k.)
sum decomposition does not split over at v, then there is an orthogonal direct over ch actspmodule. saceby theTherestmappi ricitsioninrreduci ogf btole underalthl eirjoireduci nandt tacthbusleiohasnrepresent ofthe02form (k.at)ioandns of 0as2(k.0()whi. Each 2 2(k.) into the (irreducible admissible deunifitnesarizanablein)jectirepresent on of athetionsrepresent a t i o ns o f 0 ofl on S02, (the siigsnumthe nontri vialat!-diion)mtensi onalis super repre whi c h i s t r i v i a represent h en sentati o n of 0 2 cuspidal. 02 J,f' 02(k.) ( (r t r-1 �t-1 ) ) f t l rt l -1 f' (O, J f(xl > x2) dx1 dx2. l r 4 t 2 1 ., {3. 02 k
SI'(K;)
1: SI'(K�, O")
�
a
SI'(K;, O" )
Sp4(k.)
SI'(K�, O") )
O"
®
0"
1
x
w.
0" -+
Sp4
Sp4(k.). If O"
O"
0"1
'
If O" is any representation of other than the signum, the corresponding O"' is not even tempered [KZ]. We will show this is true when O" ' is the trivial representa tion, which will suffice for the present purpose. The point is that there are functions e SI'(K;) which are invariant under and which are positive and do not vanish at zero. We compute the matrix coefficient
(2.9)
0 0 0 0 J, w. 0 0 0 0 0
(/')
Since has compact support, we see that for r and sufficiently large, (2.9) becomes
0)
±
Kv
Since the a-function of Sp4 is we see the matrix coefficients (2.9) decay too slowly at oo for O" ' to be tempered. If K. is not a field, then is a hyperbolic plane whose isotropic lines are the two places of K lying above v. The joint action of and Sp4 on SI'(K�) can be analyzed in a manner directly analogous to Proposition 2, but the work is slightly longer since the decomposition is continuous rather than discrete. Since we cannot refer to the literature, we prove only what we need. We consider only non-Archimedean places. Let C and C' be the subgroups of integer matrices in (k and Sp4(k.). Let SI'(K;)C' be the C'-fixed vectors in SI'(K;).
There is a linear isomorphism ( 02 The map is an 02(k.) intertwining map.
02 .)
PROPOSITION 3.
a
a : CC:
(k.) / C)
_..,
SI'(K�)c' .
PROOF. By performing a partial Fourier transform ( see [RS] for the analogy in
R. HOWE AND I. I. PIATETSKI-SHAPIRO
320
the (02 , Sl2) pair) we find w': is equivalent to the representation of Sp4(k.) on .9'(k�) induced by the natural linear action of Splk.) on k�. In this realization of w';', the action p';' of S02(k.) becomes the (scalar) dilations of kt normalized for unitarity. To get 0 2 , you add Fourier transform on kt with respect to the symplec tic form
of which Sp4(k.) is the isometry group. It is easy to check that the orbits of C' in kt are the sets 11:k((J� - 11:0t) , together with 0. From this one sees that .9'(k�)c' has a basis composed of the functions (h. where f/Jk is the characteristic function of 11:k8t. All the f/Jk are invariant by dilations of x: by elements of �- Assuming the Fourier transform 1\ on k� to be normal ized properly, as we may, it will be seen that (2. 10) In particular ¢0 is invariant under lV� and under A, that is, under C. Define a map by
pf¢
a (/) = .( )( 0) for
f
e
C{ 0 2(k.)/C) .
Since the dilations by powers of 11: are a set of coset representatives for p.(02(k.))fp.(C), it is easy to see that a is a linear isomorphism. It is also trans parently an 02 (k.) intertwining map, so the proposition follows. We will say an irreducible admissible representation of Sp4(k.) is if it contains a C'-fixed vector. Similar terminology applies to 02(k.).
C'-spherical an (irreduci ble admissibinletertwi ) C'-nspheri cal represent a t.9'io(nK�)of to the spLetaceSuppose Y11of• be thThen eis kernel of all i n g operat o rs from the joint actionwhereof is an irreducible C-spheri on cal (K�)/ isonirreduci .9'representati b l e , and of t h e form of ationsTheofcorrespondence from the asettioonsfallof C-spherical represent into the set of iC's an-spinherijecticalonrepresent a
PROPOSITION 4. Sp4(k.). Ya'
'
Sp4(k.)
' a .
a
0 2(k.).
02 (k.)
®
' a , a
-+
a
'
a
02 (k.)
x
Sp4(k.)
Qa '
=
Sp4(k.).
PROOF. Since C' is compact, the natural projection map .9'(K�)C ' -+ Q�; is sur jective. The space Q�: will be invariant by 02(k.), and since there is precisely one C' fixed vector in a ' [C), Q... will be irreducible under 02 x Sp4 if and only if Q�; is irreducible under 02 • According to Proposition 3, there is a surjective 02 inter twining map
a' : C';"(0 2/C) --+ Q�: . Therefore the irreducible 0 2 representations in Q�: will correspond to the C-fixed vectors in Q�;. So to show irreducibility of Q .. under 02 x Sp4, it will suffice to show that Q�' xc is one-dimensional. To this end, consider the action of the Heeke algebras [C] C';"(02 //C) and C';"(Sp4//C') on .9'(kt)C x C', as described in Proposi-
COUNTEREXAMPLE TO GENERALIZED RAMANUJAN CONJECTURE
32 1
tion 3. (We will pass freely between .?(k!) and .?(K�).) If the ¢k are as in (2. 10), then it is easy to see that the set of functions qZk¢k + q-Zk ¢ -k where q = l 1rl;;-1 form a basis for .?(k�)Cxc' . Let T1 be the Heeke operator on 0 2 defined by T1 (f)(x) = q-Z /(1rx) + q2J( 7r - 1 x) for f e .?(k�)cxc' . Then T1 generates the Heeke algebra C';'(02 //C). Let Ti. and T� be the Heeke operators on Sp4 , each of total mass 1 and supported on the C' double cosets with respective coset representatives
[� ! � g] [� � � g ]· and
-1
0 0 0
1
-1
0 0 0
1t' - 1
These operators generate C';'(Sp4 //C'). Direct computation yields the equations T1 (¢o) = q- z f/J - 1 + q2¢ � o (2. 1 1 )
Ti( ¢o) =
1 ((q - 1 ) ¢ - 1 + (q4 - 2q + 1 ) ¢0 + q4(q - 1 ) ¢1 ), q( q 4 _ I )
T2(¢o) =
1 ((q2 - 1 ) ¢- 1 + (q4 - 2 q2 + 1 ) ¢o + q4(q2 - 1) ¢1 ) · q2(q4 _ 1)
From Proposition 3, we know that .?(k�)cxc' is isomorphic as C':'(02ffC) module to the regular action of C':'(0 2//C) on itself. In particular ¢ 0 is a cyclic vector for .?(k�)cxc ' under the action of T1 . Since T1 and the Tj commute with each other we have, for any fin .?(k�)cxC' , (2. 1 2)
q(q4 - 1 ) Ti_(f) = ( q4 - 2 q + 1)/ + qZ(q - 1 ) T1 (f), q2 (q4 - 1 ) T2(f) = (q4 - 2qZ + 1 ) / + q2(q2 - 1 ) T1 (f).
Now return to Q�,xc' . It will be the quotient of .?(K;)cxc' . Therefore if ip0 is the image of ¢0 in Q�,xc' , then ifio is a cyclic vector for T1 acting on Q�,xc' . On the other hand, since Q11, is a sum of copies of (}' 1 as Sp4 module, we see ifi o will be an eigen vector for Ti and T?_. Hence by (2. 1 1), we see ip0 will be an eigenvector for Tr . so Q�,xc ' is one-dimensional as desired. Hence Q11, � (}' ® (}' 1 • Furthermore (2. 1 2) shows that the eigenvalue of T1 acting on ip0 determines those of T{ and T2 , and vice versa. Since these eigenvalues determine in turn (}' and (}' 1 [C], we see also that 1 (}' -+ (}' is injective, so the proposition is proved. We note that, from the realization of w';: on .?(k�). it is obvious that no C' spherical quotient of w';: can be tempered, because all such quotients will admit a distribution which is invariant by the isotropy group in Sp4 of a point in k�, and existence of such a distribution precludes temperedness. We may now proceed as indicated above. Select/ in .?(K1) such that 81 trans forms under 0 2(A) according to an irreducible representation whose local com ponent is the signum representation at at least one place where K does not split over Such representations obviously exist. Then under SplA), we know from Propositions 2 and 4 that 81 will transform according to an irreducible representa-
k.
322
R. HOWE AND
I.
I. PIATETSKI-SHAPIRO
tion 17;, determined by 17 v• of Sp4(kv) for all places v where K does not split over k, or where K does split and 81 i s fixed by c;. Together, these account for almost all places. Since 81 transforms by a supercuspidal representation at at least one place, it will be a cusp form. Therefore the (£2 -closure of) the SplA) translates of 81 will decompose into a direct sum of cuspidal automorphic representations. At all the places mentioned above, the local components of these automorphic representa tions will have to be the 17; determined by the 17v· Therefore these automorphic representations will be nontempered at almost all places. This completes the con struction. To conclude, we note that if k Q and K is an imaginary quadratic extension, then some of the forms we construct will be ordinary Siegel modular forms of weight Also, since automorphic forms on 02 obviously fail to satisfy strong m ultiplicity one, the corresponding forms on Sp4 will also contradict strong multiplicity one. =
I.
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