The Admissible Dual of GL(N) via Compact Open Subgroups. (AM-129), Volume 129 9781400882496

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Annals of Mathematics Studies Number 129

The Admissible Dual of GL(N) via Compact Open Subgroups by

Colin J. Bushnell and Philip C. Kutzko

PRINCETON UNIV ERSITY PRESS

PRIN CETO N , NEW JERSEY 1993

Copyright © 1993 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, and Elias M. Stein

Princeton University Press books are printed on acid-free paper and meet the guidelines for permanence and durabil­ ity of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

Library of Congress Cataloging-in-Publication Data Bushnell, Colin J. (Colin John), 1947The admissible dual of GL(N) via compact open sub­ groups / by Colin J. Bushnell & Philip C. Kutzko. p.

cm .— (Annals of mathematics studies ; no. 129)

Includes bibliographical references and index. ISBN 0-691-03256-4— ISBN 0-691-02114-7 (pbk.) 1. Representations of groups. 2. Nonstandard mathemat­ ical analysis. I. Kutzko, Philip C ., 1946-

. II. Title.

III. Series. Q A171.B978 512'.2— dc20

1993 92-33614

To Lesley and David

C ontents In tro d u ctio n ....................................................................................................1 C o m m en ts for th e reader ................................................................. 17 1. E x a ctn ess and in tertw in in g .......................................................... 19 (1.1) Hereditary orders ................................................................... 19 (1.2) Hereditary orders relative to subfields ..............................24 (1.3) Tame corestriction ................................................................. 28 (1.4) Adjoint maps ...........................................................................35 (1.5) Simple strata and intertwining ...........................................43 (1.6) The simple intersection property ....................................... 46 2.

T h e stru ctu re o f sim p le s t r a t a .....................................................49 (2.1) Equivalence of pure strata ...................................................49 (2.2) Refinements of simple strata ...............................................51 (2.3) Split refin em en ts.....................................................................57 (2.4) Approximation of simple strata ......................................... 65 (2.5) Nonsplit fundamental strata ............................................... 76 (2.6) Intertwining and conjugacy ................................. 83

3.

T h e sim p le characters o f a sim p le stra tu m ...........................89 (3.1) The rings of a simple stratum .............................................89 (3.2) Characters and c o m m u ta to rs.............................................. 98 (3.3) Intertwining ...........................................................................104 (3.4) A nondegeneracy property .................................................114 (3.5) Intertwining and conjugacy ............................................... 116 (3.6) Change of r i n g s .....................................................................126

4.

In terlu d e w ith H ecke a lg e b r a s ...................................................143 (4.1) Induction and in te rtw in in g ................................................ 143 (4.2) Scalar Hecke algebras ..........................................................146 (4.3) Unitary structures ................................................................152

5.

S im p le ty p es .......................................................................................157 (5.1) Heisenberg representations ................................................ 158 (5.2) Extending to level zero .......................................................165 (5.3) A bound on in te rtw in in g .................................................... 176 (5.4) Affine Hecke algebras and Weyl groups ..........................177 (5.5) Intertwining and Weyl groups ...........................................181 (5.6) The Hecke algebra of a simple type ................................ 188 (5.7) Intertwining and conjugacy for simple types .................195

6.

M a x im a l ty p es ...................................................................................199 (6.1) Extension by a central character ......................................199 (6.2) Supercuspidal representations ...........................................202 vii

7.

T y p ica l (7.1) (7.2) (7.3) (7.4) (7.5) (7.6) (7.7)

rep resen tation s ............................... 207 Some Iwahori decompositions ...........................................208 Iwahori factorisation of a simple type ............................ 215 Main theorems ..................................................................... 222 Proof of the principal le m m a ............................................ 231 The strong intertwining property .....................................234 Jacquet functors and Hecke algebra m aps .................... 241 Discrete series and formal degree .....................................256

8.

A ty p ica l rep resen tation s ............................................................. 265 (8.1) Split t y p e s ............................... 266 (8.2) Jacquet module of a split type I .......................................275 (8.3) Jacquet module of a split type I I .....................................290 (8.4) The main theorems ............................................................. 297 (8.5) Classification .........................................................................298

R eferen ces ...................................................................................................307 In d e x o f n o ta tio n and term in ology ................................................ 311

viii

This work gives a full description of a method for analysing the admissi­ ble complex representations of the general linear group G = G L (N , F ) of a non-Archimedean local field F in terms of the structure of these repre­ sentations when they are restricted to certain compact open subgroups of G. We define a family of representations of these compact open sub­ groups, which we call simple types. The first example of a simple type, the “trivial type” , is the trivial character of an Iwahori subgroup of G. The irreducible representations containing a given simple type are then classified, via an isomorphism of Hecke algebras, by the irreducible rep­ resentations of some H = GL(M, K)> where K is some finite extension of F and M some divisor of N , which contain the trivial type in H . This leads to a complete classification of the irreducible sm ooth repre­ sentations of G, including an explicit description of the supercuspidal representations as induced representations.

A c k n o w le d g e m e n ts: The first-named author was supported in p art by SERC grant GR/E47650. The second-named author was supported in part by NSF grant DMS-8704194 and by SERC grant G R/F73366. Both authors wish to thank the Institute for Advanced Study for its hospitality during their visit in the Academic Year 1988-89. This visit was supported in part by NSF grant DMS-8610730. They also wish to acknowledge the hospitality of the Center for Advanced Studies of the University of Iowa. This document was typeset by

The Admissible Dual of GL(N) via Compact Open Subgroups

IN T R O D U C T IO N In this book, we give a new and effectively complete classification of the irreducible sm ooth complex representations of the general linear group G = G L (N , F ) over a non-Archimedean local field F. The study of the representation theory of G and other reductive groups over a local field arose from several sources, notably general questions of harmonic analysis on locally compact groups, and connections with arithm etic and autom orphic forms:— see, among others, the influential accounts [G G P -S], [HC], [JL], [S], The subject has been largely driven by a wide-ranging family of conjectures of Langlands unified in the “Principle of Functoriality” . This was first enunciated in [L ai] and subsequently refined:— see [La2] and the expositions in [BoCs]. Progress with these conjectures will presumably require a clear description of the irreducible representations of these reductive groups, of which GL(N) is the most accessible example. The methods of this work have been developed in the hope th a t they will prove amenable both to generalisation and ap­ plication in this program. W hen discussing the representation theory of a reductive algebraic group, in virtually any context, the one technique which immediately suggests itself is th at of induction from parabolic subgroups. If G is a reductive algebraic group (identified here with its group of points over some fixed field of definition) and P is a proper parabolic subgroup of G with unipotent radical U and Levi decomposition P = M U , then M = P / U is a reductive group with smaller semisimple rank than G, and therefore presumably of simpler structure. An irreducible representation of M can be inflated to P and then induced to G, the composition factors of this representation yielding irreducible representations of G. (Such induced representations always seem to have finite composition length, so this procedure can, in principle, be carried out.) The representation theory of G can thus be approached via a clear strategy: (a) classify the irreducible representations o f Levi factors o f proper parabolic subgroups o f G; (b) describe the decomposition o f representations o f G induced from proper parabolic subgroups; (c) describe the representations o f G which cannot be obtained from steps (a) and (b). In some circumstances, for example in appropriate categories of repre­ sentations for groups over connected fields, step (c) is effectively empty. 1

In t r o d u c t i o n

For sm ooth representations of reductive groups over a non-Archime­ dean local field, considerable progress has been made along these lines. In this context, it is known that a representation of G induced from an irreducible representation of a parabolic subgroup has finite compo­ sition length [Cs]. Representations which do not occur as factors of such “parabolically induced” representations are here called supercuspidal. Given an irreducible smooth representation 7r of G which is not supercuspidal, there is a proper parabolic subgroup P of G, with unipotent radical 17, and an irreducible supercuspidal representation cr of P / U such th at 7r occurs as a factor of the representation Ind( 0. To see this in the case n > 1, we note th at the normaliser of U n (21) is ju st {x £ G : x ^ x " 1 = ^3n }. Any x which n o r m a l i s e s m u s t also 21

1. E x a c t n e s s a n d i n t e r t w i n i n g

normalise the order {a E A : tyn a C ^3n }. However, since is an invertible fractional ideal of 51, we have *p- n 0, where [x] denotes the greatest integer < #, for x £ M. We then have a canonical isomorphism L/r+1(Q t)/t/n+1(2l)

2 , - • • , vjv} of V, we can define a lattice chain C = C(V) as the set of all OF-lattices of the form Pf (Of V1 -f 0f^2 -f - - - -h OF^r + pF^r-fl “f - - -pF^iv), where 1 < r < N and m E Z. Then e{C) = JV, and any lattice chain C which has V as a basis is contained in C. In term s of the associated hereditary orders, we have 5l(£) C 51(£). Thus a choice of basis V of F specifies a minimal hereditary OF-order in A, namely 5l(£(V)). (1 .1 .9 ) P ro p o s itio n : Let C1, C2 be oF-lattice chains in V , and p u t 51i = E n d °(£ 2), i — 1,2. The following conditions are equivalent: (i) t h e oF - l a t t i c e c hains Cl h a v e a common OF-basis; (ii) the sets C 1 fl C2, C l U C? o f oF -lattices are oF -lattice chains; (iii) there exist hereditary Of -orders 5(3 , 5I4 in A such that 5I3 D 51; D 5l4l i = 1,2. Proof: Suppose first th at (i) holds. Let V = { t q , . . . , v n } be a common basis of the lattice chains Cl . If L denotes the OF-span of the t;*, then L E C 1 H C2 and the hereditary (indeed maximal) OF-order E nd0jP(L) 23

1. E x a c t n e s s a n d i n t e r t w i n i n g

contains both 211 and 2I2 . On the other hand, the 21* both contain 2l(£(V)), so (i)=>(iii). Assuming (Iii) holds, write Cl for the lattice chain defining 2 1 1 < i < 4. Each L G C 1 U C 2 is, in particular, an SU-lattice, so C 1 U C 2 C C4. Therefore the set C 1 UC2 of lattices is linearly ordered by inclusion. It is surely stable under multiplication by F x , so C 1 U C2 is indeed a lattice chain. On the other hand, the set C l fl C2 is linearly ordered and i n ­ stable. However, any 2l3-lattice in V is an 2h-lattice, for i = 1,2. Thus C 1 P l£ 2 D £ 3. In particular, C1 C\C2 is nonempty, and so a lattice chain. This proves (iii)=>(ii). Finally, assume th at (ii) holds. Fix L E C1 f l £ 2, and choose a basis of the lattice chain C1 U C2 whose Oj^-span is L. This is the common basis required for (i). ■ R e m a rk s : (i) In (1.1.9), we can replace (ii) by the superficially weaker statem ent th at C 1 O C2 ^ $ and the set C l U C2 is linearly ordered by inclusion. Indeed, this is what we used in the proof. (ii) If C 1, C2 satisfy the conditions of (1.1.9), we get various inci­ dence relations connecting the orders 21* and their radicals. For example, set = rad(21*), 1 < i < 4, and let L be an ^ - l a tt i c e in V . W rite C l = {L j : j £ Z}. We can then find j £ 7L such th at L) D L D L )+1. We have ty iL C ^3iL) = L )+1 C L , hence ty iL C L for all ^ - la ttic e s L. It follows th at C 2I4 C 2(2. By symmetry, we also get ^ 2 C 2 ti. There are many relations of this sort.

(1.2) H ereditary orders relative to subfields Now suppose that, in addition to our hereditary order 21 = End°F (£) in A, we have a subfield E / F of A. Thus we may view V as an E -vector space, via the given inclusion E —» A. (1 .2 .1 ) P ro p o s itio n : Let 21 be a hereditary order in A, with 21 = E nd°F(£ ), for some OF-l&ttice chain C = {£;} in V. Let E / F be some subfield o f A. The following conditions are equivalent: (i) E x C A(21) (i.e. E normalises 21); (ii) each Li is an Oe -lattice and there is an integer e' such that Pe L{ = Li+ei for all i (i.e. C is an o^-iattice chain in the E-vector space V). Proof: Suppose th at E x C £(21). We show first th at each Li is an o^-m odule, hence o#-lattice. The group 17°(21) is the unique maxim al compact subgroup of £(21), so 0^ C E7°(21). In particular, each Li is a module for the ring R = 0f [0e ] generated over by 0^ . It is therefore enough to show that R = o#. First, we have R D U 1( oe ), so pE C R . Further, R contains a complete set of representatives of Oe mod pE, namely 0 together with the group of roots of unity in E of 24

1. E x a c t n e s s a n d i n t e r t w i n i n g

order relatively prime to the residual characteristic of E . The assertion follows. Now take a prime element tte of E . Then we E £(21), and it there­ fore acts as an autom orphism of C. T hat is to say, there is an integer e' such th at Li+e>= — Pe L{ for all i. This proves (i)=>(ii). The converse is similar. ■ Now suppose th at the equivalent conditions of (1.2.1) hold. Define (1.2.2) (1 .2.3)

B = E n d # (F ) = the A-centraliser o f E , Q =

Q3 = 0 l n £ ,

B.

(1 .2 .4 ) P ro p osition : With the notation above, 93 is a hereditary oe~ order in B, and Q is the Jacobson radical o f 03. Indeed, we have Q n = ^3n fl B for all n E Z. Further, the integer e' o f (1.2.1 )(ii) is given by «' = . ( * ) = . ( B M

=

Proof: Immediately from the definitions we have 03 = End®F(/2) fl B , which is exactly EndpB(£). Further, the intersection of B with End”F (£) is End™ E(C) for any n , and all the assertions follow. ■ We now forget about our order 01 for the moment, and concentrate on the pair A, E. Let W be an E-subspace of V such th at the canonical m ap E ® f W -*V , induced by the inclusion of W in V and the given action of E on V is an isomorphism (i.e. W is the E-span of an E-basis of V). This induces an isomorphism of E-algebras A — Endjp(E) 0_p Endir(W ). It is convenient to abbreviate A (E ) = Endjp(E ). Thus the choice of W induces an embedding of algebras (1.2.5)

tw • A (E ) —►A,

which extends the given embedding E —> A, viewing E as canonically embedded in ^4(E). We view A (E ) as a (E , E)-bimodule, and this gives us the trivial identity A (E ) = A (E ) 25

E.

1. E x a c t n e s s a n d i n t e r t w i n i n g

On the other hand, the isomorphism E