Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Part I. Papers in representation theory

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Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Part I. Papers in representation theory

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THE WEIZMANN SCIENCE PRESS OF ISRAEL Publishers of the Fol/9wing Journals ISRAEL JOURNAL OF BOTANY; ISRAEL JOURNAL OF CHEMISTRY; ISRAEL JOURNAL OF EARTH-SCIENCES; ISRAEL JOURNAL OF MATHEMATICS; ISRAEL JOURNAL OF TECHNOLOGY; ISRAEL JOURNAL OF ZOOLOGY; JOURNAL o' ANALYSE (Sc1ENCE

FOR

(SCIENCE)

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Supporting Institutions THE NATIONAL COUNCIL FOR RESEARCH AND DEVELOPMENT; BAR-ILAN UNIVERSITY; BEN GURION UNIVERSITY OF T.HE NEGEV; TECHNION THE

HEBREW

ISRAEL INSTITUTE OF TECHNOLOGY; TEL

UNIVERSITY OF

JERUS7'.LEM;

THE

WEIZMANN

Aviv

UNIVERSITY;

INSTITUTE OF

SCIENCE

ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS Editorial Board

:Z.. Arad

Bar-llan University, Ramat-Gan, Israel

H. Furstenberg

Hebrew University, Jerusalem, Israel

S. Gelbart

Weizmann Institute, Rehovot, Israel

I. Gohberg

Tel-Aviv University, Ramat-Aviv, Israel

B. Pinchuk

Bar-llan University, Ramat-Gan, Israel

· L. Rowen

Bar-llan University, Ramat-Gan, Israel

L. Small

University of California, La Jolla, California

L. Zalcman

Bar-llan University, Ramat-Gan, Israel

Executive Editor: D. Greenberg

© 1990 THEWEIZMANN SCIENCE PRESS OF ISRAEL

Preface During the week of May 1989, a Workshop on "L-Functions, Number Theory,_and Harmonic Analysis" was held at Tel-Aviv University, to honor and review the impact of the work of Ilya I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Some of the papers in this Festschrift were presented originally at that Workshop, while others were solicited shortly afterwards; all were prepared especially for this collection, and are dedicated to Ilya with admiration, appreciation and affection._ Piatetski-Shapiro has been making major contributions to applied as well as theoretical mathematics for the past forty years. His works in cell biology, geophysics, automata, homogeneous networks, digital computers, _etc., are not adequately reflected in this Festschriff; however, the impact of his works in pure mathematics, ranging from trigonometrical series and analytic number theory to group algebraic geometry, and automorphic L-functions, is felt throughout these volumes. Ilya was born in Moscow on March 30, 1929. His pre-Uruversity schooling was completed during the lniddle 1940's, under extremely difficult, and depressing, wartime conditions. It is perhaps not surprising that Ilya emerged from these experiences with the tenacity and inner strength which today are his trademarks. But it is clearly a testimony to his unique character that he also emerged from this background with the special warmth, generosity, and humility which those of us who know him well so value. As Ilya passes his sixtieth birthday, it seems his mathematical powers are increasing rather than diminishing! While these volumes were being prepared for publication, we learned that Ilya will receive this year's Wolf Prize in Mathematics. We know that he will continue to work vigorously for many years to come, and we extend to him on this occasion our best wishes for future health and happiness.

The Editors Spring 1990

iii

Contents of Festschrift in honor of I.I. Piatetski-Shapiro Part 1: Papers in Representation Theory PREFACE

..... ...........................................................................................................

BIBLIOGRAPHY OF I. I. PIATETSKI-SHAPIRO

.................................................

iii

xi

PERSONAL REMINISCENCES I.I. Piatetski-Shapiro, My Advisor and Friend

1

Siegel Domains etc. S. Gindikin

5

L.N. Vaserstein

PAPERS IN REPRESENTATION THEORY Exceptional Representations and Shimura's .. . .... ........ ...... ... ........ ........ ... ... .. Integral for the Locaf Unitary -Croup U(3) Stephen S. Gelbart and Jonathan D. Rogawski

19

Quelques Remarques sur les Theoremes Reciproques Guy Henniart

77

Another Look at the Local II-correspondence for an Unramified Dual Pair Roger Howe

93

Rankin-Selberg Convolutions: Archimedean Theory Herve Jacquet and Joseph Shalika

125

The Smallest Representation of Simply Laced Groups D. Kazhdan and G. Savin

209

On Formal Dimensions,for Reductive p-adic Groups Marie-France Vigneras

225

Demonstration d'une Conjecture de Dualite de Howe dans le cas p-adique, p =fa 2 J.-L. Waldspurger

267

LIST OF CONTRIBUTORS, WITH ADDRESSES

vii

325

Part 2: Papers in Analysis, Number Theory and Automorphic .C-functions (contained in Vol. 3 of this series)

iii

PREFACE PAPERS IN ANALYSIS, NUMBER THEORY AND AUTOMORPHIC £--FUNCTIONS The Riesz-Raikov Theorem for Algebraic Numbers J. Bourgain

.......................................

The Exterior Square Automorphic L-functions'on GL(n) Daniel Bump and Solomon Friedberg

...........................

1 47

A Problem on Certain p-adic Zeta-functions Jun-ichi lgusa

67

Poles of Eisenstein Series and £-functions Stephen S. Kudla and Stephen Rallis

81

Rank-one Residues of Eisenstein Series Robert P. Langlands

111

Orbits of Group Actions and Values of Quadratic Forms at Integral Points G. A. Margulis

127

Spectrum of a Position of a Convex Body and Linear Duality Relations · V. Milman

151

On Ruelle's Zeta-function· S.J. Patterson

163

The Converse Theorem for GL(n) · Ilya Piatetski-ShapirQ

185

Factorization of Operator Valued Analytic Functions and Complex Interpolation Gilles Pisier

197

viii

Multiplicities of Galois Representations in Jacobians of Shimura Curves Kenneth A. Ribet

... . ...... .. .. ... ... .. .... . . .. ..... ... ... ... .. . .

221

On _Cusp For;ms II P. Sarnak

237

Remarks on the Distribution of Poles of· Eisenstein Series Atle Selberg

251

On Multiplicity of Local Factors Freydoon Shahidi

279

Automorphic Forms in GSp(4) David Soudry

291

Kuznetsov Formulas for Products of Groups - of R-rank One - N.R. Wallach and Miatello

305

R:

Hecke Operators and Periods of Modular Forms Don Zagier LIST OF CONTRIBUTORS, WITH ADDRESSES

..............................................

321 337

I.I. PIATETSKI-SHAPIRO

Bibliography (through 1989) 1. I. Piatetski-Shapiro

On an asymptotic formula for the number of abelian groups of order x or less (Russian) Mat. Sb. 26, 479-486 (1950). 2. I. Piatetski-Shapiro On principles of distribution of the fractional parts of the exponential function (Russian) Izv. Akad. Nauk SSSR, Ser. Math., 15, 47-52 (1951). 3. I. Piatetski-Shapiro On a variant of the Waring-Goldbach problem (Russian) Uspehi Mat. Nauk, 6:5 (45), 157-158 (1951). 4. I. Piatetski-Shapiro On the problem of uniqueness of the expansion of a function into a trigonometric series (Russian) M. Uchen. Zap Mose. Univ. 155, Math, 5, 54-72 (1952). 5. I. Piatetski-Shapiro On a variant of th.e Waring-Goldbach problem (Russian) Mat. Sb. 30 (1952). 6. I. Piatetski-Shapiro On a generalization of the concept of uniform distribution of fractional parts (Russian) Mat. Sb. 30, 669-676 (1952). 7. I. Piatetski-Shapiro On the problem of uniqueness of the expression of a function into a trigonometric series (Russian) Dokl. Akad. Nauk SSSR, 85, 497-500 (1952). 8. I. Piatetski-Shapiro On the distribution of prime numbers in the sequences [f(n)] (Russian) Mat. Sb. 33, 559-566 (1952). 9. I. Piatetski-Shapiro Fractional parts and some problems in the theory of trigonometrical series (Russian) Usp. Mat. Nauk 8:3, 167-170 (1953). 10. I. Piatetski-Shapiro An addition to the work "On the problem of uniqueness of the expansion of a function into a trigonometric series" (Russian) M. Uchen. Zap. Mose. Univ. 165:Mat. 7, 79-97 (1954). 11. I. Piatetski-Shapiro Abelian modular functions (Russian) Dok!. Akad. Nauk SSSR, 95, 221-224 (1954). xi

12. I. Piatetski-Shapiro An analogue of a Lefschetz' theorem (Russian) Dokl. Akad. Nauk SSSR 96, 917-920 (1954). 13. F.A. Berezin and I. Piatetski-Shapiro Homogeneous extensions of complex space (Russian) Dokl. Akad. Nauk SSSR 99, 889-892 (1954). 14. I. Piatetski-Shapiro A classification of modular groups (Russian) Dokl. Akad. Nauk SSSR 110, 19-22 (1956). 15. I. Piatetski-Shapiro Singular modular functions (Russian) lzv. Akad. Nauk SSSR, Ser. Math., 20, 53-98 (1956). 16. I. Piatetski-Shapiro On the theory of abelian modular functions (Russian) Dokl. Akad. Nauk SSSR 106, 973-976 (1956). 17. I. Piatetski-Shapiro and A. G. Postnikov Normal Bernoulli sequences of symbols (Russian) Izv. Akad. Nauk SSSR, Ser. Math. 21, 501-514 (1957}. 18. I. Piatetski-Shapiro On an estimate of the dimension of the space of automorphic forms for some types of disccmtinuous subgroups (Russian) Dokl. Akad. Nauk SSSR 113, 980-983 (1957). 19. I. Piatetski-Shapiro and A.G. Postnikov Normal Markov sequence of symbols and normal continued fractions (Russian) Izv. Akad. Nauk SSSR, Ser. Math. 21, 729-746 (1957). 20. I. Piatetski-Shapiro Some questions of harmonic analysis in homogeneous cones (Russian) Dokl. Akad. Nauk. SSSR 116, 181-184 (1957). (Math. Rev. Vol. 23 # A475). 21. I. Piatetski-Shapiro On the distribution of the fractional parts of the exponential function (Russian) Moskov. Gos. Ped. Inst. Uc. Zap. 18, 317-322 (1957). (Math. Rev. Vol. 22 # 4696). 22. I. Piatetski-Shapiro On a problem proposed by E. Cartan (Russian) Dokl. Akad. Nauk SSSR 124, 272-273 (1959). (Math. Rev. Vol. 21

#. 728). ·

23. I. Piatetski-Shapiro Discrete subgroups of the group of analytic automorphisms of the polycylinder and automorphic forms (Russian) Dokl. Akad. Nauk SSSR 124, 760-763 (1959). (Math. Rev. Vol. 21 # 3581). 24. I.M. Gelfand and I. Piatetski-Shapiro Representation Theory and Theory of Automorphic Functions Uspehi Mat. Nauk 14, No. 2(86), 171-194 (1959).

xii

25. I. Piatetski-Shapiro Geometry of Homogeneous Domains and Theory of Automorphic Functions. Solution of a problem of E. Cartan Uspehi Mat. Nauk 14, No. 3(87), 190-192 (1959): 26. l.M. Gelfand and I. Piatetski-Shapiro About a theorem of Poincare DAN USSR, Vol. 127 (3), 490-493 (1959). 27. I. Piatetski-Shapiro Theory of modular functions and associated problems from the theory of discrete groups (Romanian) Acad. R.P. Romme An. Romino-Soviet. Ser. Mat. Fiz. (3) 14, No. 4 (35), 11-48 (1960). (Math. Rev. Vol. 24 # A3313). 28. I. Piatetski-Shapiro Theory of modular functions and related problems in the theory of discrete groups (Russian) Uspehi Mat. Nauk 15, No. 1 (91), 99-136 (1960) Translated as Russian Math. Surveys 15, 97-128 (1960) (Math. Rev. Vol. 24 # A3312). 29. I. Piatetski-Shapiro Geometry of classical domains and theory of automorphic functions (Russian) Sovremennye Problemy Matematiki. Gosudarstv. lzdat. Fiz.-Mat. Lit., Moscow, 191 (1961). (Math. Rev. Vol.25 # 231). · 30. I. Piatetski-Shapiro of bounded homogeneous domains in an n-dimensional complex space (Russian) Dokl. Nauk SSSR 141, 316-319 (1961). (Math. Rev. Vol. 24 # 2681). 31. l.M. Gelfand and I. Piatetski-Shapiro Unitary representation in homogeneous spaces with discrete stationary groups DAN USSR, Vol. 147, No. 1, 17-20 (1962). 32. I. Piatetski-Shapiro On bounded homogeneous domains in an-dimensional complex space (Russian) lzv. Akad. Nauk SSSR Ser. Mat. 26, 107-124 (1962) .. (Math. Rev. Vol. 25 # 4552). 33. I. Piatetski-Shapiro The structure of j-algebras (Russian) lzv. Akad. Nauk SSSR, Ser. Mat. 26, 453-484 (1962). (Math. Rev. Vol. 25

#

4553).

34. I. Piatetski-Shapiro Domains of upper half-plane type in the theory of several complex variables (Russian) Proc. Internat. Congr. Mathematicians (Stockholm, 1962), 389-396 Inst. Mittag-Leffler, Djursholm (1963). (Math. Rev. Vol. 31 # 380). 35. l.M. Gelfand and I. Piatetski-Shapiro The Laplace, operator on the Riemann surfaces and the theory of representation Outlines Joint Sympos. Partial Differential Equations (Novosibirsk, 1963), 87-88. Acad. Sci. USSR Siberian Branch, Moscow (1963). (Math. Rev. Vol. 34 # 4729). 36. l.M. Gelfand, I. Piatetski-Shapiro and M.L. Zetlin Some classes of games and games of automata DAN USSR, Vol. 152, No 4, 845-848 (1963). xiii

37. E.B. Vinberg, S.G. Gindikin and I. Piatetski-Shapiro On classification and canonical realizations of complex homogeneous bounded domains Trudy Moskov. Mat. Obsc. 12, 359-388 (1963). 38. l.M;. Gelfand and I. Piatetski-Shapiro Automorphic functions and representation theory Trudy Moskov. Mat. Obsc. 12, 389-412 (1963). 39. l.M. Gelfand, Ju.G. Federov and I. Piatetski-Shapiro' Determination of crystal structure by search (Russian) Dok!. Akad. ·Nauk SSSR 152, 1045-1048 (1963) (Math. Rev. Vol. 31

#

2033).

40 .. V.I. Bryzgalov, I.M. Gelfand, I. Piatetski-Shapiro and M:L. Zetlin Homogeneous automata games and. their simulation on digital computers (Russian, English Summary) Avtomat. i Telemeh. 25, 1572-1580 (1964) (Math. Rev. Vol. 30 # 1897). /

41. l.M. Gelfand, M.I, Graev and I. Piatetski-Shapiro Representations of adele groups (Russian) Dok!. Akad. Nauk SSSR 156, 487-490 (1964) (Math. Rev. Vol. 29

#

2237).

42. V.C. Gurfinkel, A.N. Ivanova, Y.M. Koz, I. Piatetski-Shapiro and M.L. Shik CharacteriZation of the work of motion units in a stationary regime· Biophysics, Voi. 9, No. 5, 636-638 (1964). 43. I. Piatetski-Shapiro Arithmetic groups in complex domains (Russian) Uspehi Mat. Nauk 19, No. 6(120), 93-121 {1964). (Math. Rev. Vol. 32

#

7790).

44. S.P. Novikov, I. Piatetski-Shapiro and l.R. Shafarevich Fundamental directions in the development of algebraic topology and algebraic geometry (Russian) Uspehi Mat. Nauk 19, No. 6(120), 75-82 (1964). (Math. Rev. Vol. 30 # 2003). 45. I. Piatetski-Shapiro and M.L. Shik On spinal motion regulation Biophysics, Vol. 9; No. 4, 488"492 (1964). 46. V.I. Brizgalow, I. Piatetski-Shapiro and M.L .. Shik On a two-level model of interaction of automata . DAN USSR, Vol. 160, No. 5; 1039-1041 (1965}; 47. S.G. Gindikin and I. Piatetski-Shapiro On the algebraic structure of the field of Siegel modular functions (Russian) Dok!. Akad. Nauk SSSR 162, 1226-1229 (1965). (Math. Rev. Vol. 31 # 2218). 48. I. Piatetski-Shapiro The geometry and classification of bounded homogeneous domains (Russian) Uspehi Mat. Nauk 20, No. 2 (122), 3-51 (1965). Math .. Rev. Vol. 33 # 4323). 49. I. Piatetski-Shapiro Geometrie des domaines classiques et theorie des fonctions automophes. Traduit du Russe par A.W. Golovanoff. · Travaux et Recherches Mathematiques, No. 12, Dunod, Paris, 1v+160 pp. (1966). (Math. Rev. Vol. 33 # 5949). xiv

50. I. Piatetski-Shapiro and I.R. Shafarevich Galois theory of transcendental extensions and uniformization (Russian) . Contemporary Problems in Theory and Functions (Internat. Conf., Erevan, 1965), pp. 262-264 Izdat. "Nauka", Moscow (1966). (Math. Rev. Vol. 34 # 7511). 51. I.M. Gelfand, MJ. Graev and I. Piatetski-Shapiro Theory of representations and automorphic functions (Russian) Izdat "Nanka", Moscow, '512 pp. (1966). (L. de Branges Zbl i38, 72). (Math. Rev. Vol. 36 # 3725). 52. L.U. Vartanova, T.S. Gelankina, V.I. Kells-Borok, S, Mebel and I. Piatetski-Shapiro On the depth of the center of earthquakes Study of Seismic Observation on Computers, Nauka, Moscow, 10-30, (1966). 53. I. Piatetski-Shapiro and I.R. Shafarevich Galois theory of transcendental extensions and uniformization (Russian) Izv. Akad. SSSR Ser. Mat. 30, 671-704 (1966). (Math. Rev. Vol. 34 54. L.A. Gutnik and I. Piatetski-Shapiro Maximal discrete subgroups of a unimodular group (Russian) Trudy Moskov. Mat. Obsc. 15, 279-295 (1966). (Math. Rev. Vol. 35

#

#

2581).

261).

55. I. Piatetski-Shapiro, V.A. Volkonski, L.V. Levina and A. Pomanski An iterative method of solution of integer programming problems DAN USSR, Vol. 169, No. 6, 1289-1292 (1966). . 56. A.J. Freidenstein, L Piatetski-Shapiro and K.V. Petrakova Osteogenesis in transplants of bone marrow cells J. Embroyol. Exp. Morph., Vol. 16, 3, 381-390 (1966:). 57. N.B'. Vassilev and I. Piatetski-Shapiro Time of adjustment of automaton to an external surrounding Automatica and Telemechanica, Vol. 7, 119-123 (1967). 58. I. Piatetski-Shapiro and A.B. Sidlovskii Aleksandr Osipovic Gelfond (Russian). Uspehi Mat. Nauk 22, No. 3(135), 247-256 (1967). (Math. Rev. Vol. 35

#

2711).

59. I. Piatetski-Shapiro Automorphic functions and arithmetic groups (Russian) Proc. Internat. Congr. Math. (Moscow, 1966), pp. 232-274 Izdat. "Mir", Moscow (1968). (Math. Rev. Vol. 38 # 5719). 60. S.G. Gindikin, E.B. Vinberg and I. PiatetskicShapiro Homogeneous Kahler manifolds (English translation by Adam Koranyi) Geometry of Homogeneous Bounded Domains (C.I.M.E. 3rd Ciclo, Urbino, 1967), 3-87; Edizioni Cremonese, Rome,4 (1968). (Math. Rev.Vol. 38 # 6513). 61. I. Piatetski-Shapiro Redu

0,

Ep =Ip

which is now known as the Siegel disk. Pyatetski was interested in realizations of the half-plane type in connection with proplems in the theory of automorphic functions in symmetric domains.' It is only now that I realize that with the exception of his student of trigonometric series and trigonometric sums, and the 'later works on the theory of automations and applied mathematics (which were a result of certain peculiarities of Soviet math life), Pyatetski had faithfully studied automorphic functions in several variables his whole life. It is amazing that despite this fact, he does not encounter any problems when these studies take him into the realms which require a deep knowledge and precise understanding of algebraic geometry, arithmetic, theory of representations, or the theory of complex homogeneous manifolds. The case at hand required just such detours. Pyatetski wantedto prove that, in a reasonably general situation, fields qf automorphic functions are in fact (isomorphic to?) fields of ·algebraic functions in several variables. He was following the way outlined by Siegel. It was necessary to the dimensions of spaces of automorphic forms. To do this it was possible to use the Fourier decompositions of automorphic forms. The appropriate abelian subgroup of the discrete group, with respect to w.hich decomposition into Fourier series takes place, is constructed exactly in the realization of half-plane type. Siegel, of course was considering his halfcplane in connection to the theory of Siegel modular functions. Starting.from the Cartan classification, it is possible to see that in three of the four classical series of domains there exists a realization of the structure analogous to: 1Jv = Rn + iV, where V C Rn is a linearly homogeneous convex cone. Pyatetski called such domains Siegel domains of type one. ·The group of transformations· is of the type

Zl-'-+g(z)+a,

aERn,z

where g is the continuation of automorphisms of V on en. This group will be transitive on. 1Jv. In one of the series, as I already mentioned, Vis the cone of positive symmetric matrices; in the other it is the sphere (light) cone; and in the third it is the cone of positive Hermitian matrices over the quaternions. The domains connected to the cone of the complex positive Hermitian matrices are also symmetric. These are contained in the first series of the classical domains, but are not equal to whole of the series. In that series t,here are domains with a skeleton (Shilov boundary) which has a real dimension greater than n, ·while Siegel domains of the first type it always has the minimal dimension n. Therefore such domains cannot in principle be biholomorphically equivalent to the Siegel domains of the first type. The. most trivial example is the complex ball in en

lz11 2 + ·· · + lznl,2 < 1;

SIEGEL DOMAINS

Vol. 2, 1990

7

here the skeleton coincides with the boundary. It is natural to suppose that it is possible to obtain the transitive group by considering the isotropy group of some point of the boundary. One could take a point of the boundary or of the skeleton, for example; and if, by using some analogue of the Cayley transformation, this point can be sent to infinity, then one will obtain the transitive group of affine automorphisms. This plan can be easily realized for the complex sphere, and the following domain results:

with the transitive group: Zj

H .Xz+ci;j > 1,.cj E e,.X ER+

z1H.X 2 z1+a+i2::)2.XzjCj + icji 2 ) j

Let us consider this question for the classical domains of type one in more detail. These domains lie in the Grassmanians. Let Gn,p be the Grassmanian manifold of p-dimensional hyperplanes of- en, p :::; n /2. Let us define the Stiefel coordinates (the matrix analogue of projective homogeneous coordinates) on Gn,p- Let us then consider the set Zn,p of p x n matrices with rank p:

Z

= {z;j},i Sop,j

Son

There is a projection Zn,p -+ Gn,p, where Zn,p is assigned the p-dimensional hyperplane spanned by the rows (z;,*). We now factorize Zn,p with respect to the following equivalence relation: ·

Z

gZ, g E GL(p; e);

thus we have frame bundle on Gn,p (the principal GL(p; e)- bundle). Let Z = (Zi, Z 2 ) where Zi, Z2 are of size p x p,p x (n - p) respectively. Let us consider subset G0 of Gn,p, where det Z 1 =f. 0. On G0 we have a local system of coordinates

The elements ( are the coordinates. The closure of this chart coincides with the whole of G. The collections of i 1 , .•. , ip of numbers of columns with nonzero minors provide other charts. The group GL(n; e) acts on Gn,p:

Z

H

Z 9 ,g E GL(n; e)

The symmetrical domain Up,n is Cartan dual to the complex space Gp,n and it is realized as a domain of Gn,p· It is necessary to take one of the open orbits of the group SU(p, n-p), which happens to be a real form on GL(n; e). Assume we have in en an Hermitian form with signature (p, n - p) which is given by the matrix H, i.e. zH z*, where points of en are regarded horizontal vectors. In en we consider the domain V+

zHz* > 0.

8

IMCP

S.GINDIKIN

Then Un,p consists of p-hyperplanes which belong to 'D+ away from 0. Thus we have ZHZ*

0, Z E Zn,p

in Stiefel coordinates (the Hermitian matrix is positive definite). The transformation H, which does not change its signature, is equivalent to the transformation from GL(n, C) to Gn,p· Let n = 2p, Z = (Z1, Z2), where the Zj are square matrices. Further, let

Then Un,p can be given by the following condition

This domain is completely contained in a coordinate neighborhood G 0 , and has the following form when expressed in the inhomogeneous coordinates (:

This is a disk type realization. If the form H is changed to :

iEP) 0 ' then we will obtain the domain

Except for the boundary, this domain will again belong to G 0 , thus in the coordinates ( we obtain an unbounded region i((* -() 0 of the half-plane type (this is a Siegel half-plane of type one corresponding to the cone of Hermitian positively defined matrices). The transition from H to H 1 induces a Cayley fractional linear transformation on(. Assume that now q = n-2p > 0, Z = (Zi, Z 2 , Z 3 ), where the Zj are matrices of dimension p x p, p x p, p x q. Accordingly,

iEp 0 0

-Eq

)

.

Then Un,p looks like i(Z1Z; - Z2Z;)- Z3z; in the inhomogeneous coordinates

0

.

Vol. 2, 1990

SIEGEL DOMAINS

9

We have

- (*) -TJTJ*

i

0.

Here ( is a p X p matrix, TJ is an p x q matrix. When p = 1, we obtain a domain equivalent to the complex ball, which was mentioned earlier. In this realization it is easy to find transitive group of affine transformations ( 1--t (

+ a,

T/

1--t

T/, a = a* ;

T/

1--t

T/ + b, (

1--t (

T/

1--t

9T/, (

g(g*, g E GL(p; C) .

1--t

+ iTJb* +

bis a p x q matrix,

Amazingly, Pyatetski was able to deduce the general construction of Siegel domains of type two from this example. Let V be a linearly homogeneous convex cone in Rw, and F is an Hermitian V positive transformation cm x cm --+ en. That is F( u, v) is linear with respect to u, anti-linear with respect to v, and F(u,v) = F(v,u). Let us consider a domain '.D(V, F), lmz - F(u,u) EV in C(;'.".:)'. Pyatetski called domains of this type Siegel domains of type two. These are always domains of holomorphy, biholomorphically equivalent to bounded domains. In '.D(V, F) there are automorphisms

(z, u)

1--t

(z +a+ 2iF(u, b) + iF(b, b), u + b) , a E Rk, u E cm .

Assume that F satisfies the added condition of homogeneity: that is, for g from the transitive group of automorphisms of the cone V there exist (complex) linear transformations g of the space cm such that

F(gu,gv)

= gF(u,v).

Therefore '.D( v, F) also has automorphisms

(z, u)

1--t

(gz,gu).

Altogether, we obtain a group of automorphisms of '.D(V, F). Siegel domains of type two turned out to be a convenient tool for the study of automorphic functions. Unfortunately automorphic forms cannot be decomposed into the regular Fourier decompositions, but have to be decomposed into Fourier series with a J acobian coefficient (the Fourier-Jacobi series). Nevertheless, this proved to be enough to obtain the necessary 'estimates of the relevant dimensions. I want to stress again that in the first article of Pyatetski, Siegel domains appear only as a tool for obtaining the above mentioned estimates. Everyone who had ever worked with semisimple groups and symlll.etric spaces knows th;i.t sooner or later comes that unpleasant time when it is no longer enough to simply concern oneself with the classical groups - for which everything is clear and obvious. It is then necessary to start thinking about exceptional groups (probably mathematicians who

10

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IMCP

concern themselves solely with exceptional groups are an exception themselves). Pyatetski conjectured that two exceptional symmetric domains admit a realization in the form of Siegel domains of the second type. To check this was not going to be easy. While there are the two exceptional domains, there is only one exceptional cone, the cone of positive octave matrices of third order. The domain that corresponds to this cone is the 27-dimensional Siegel domain of type one. The other special domain seemed even more difficult, it did not even have an appropriate cone. This seemed ominous. Nevertheless, it turned out that it does have a construction through classical objects. It was realized as_a Siegel domain of type two for a spherical cone. I remember well the emotions and surprises connected with this. · Now it is clear that there was a natural way to clear everything up. It was necessary to see that the realization of-symmetric domains as Siegel domains is equivalent to expressing the universal structure of the roots of ,a symmetric space with a bounded system of roots. These have the form >.;, >.; :f >.;, 2>.;. In the irreducible case the invariants are the rank and the total dimension a of the subspaces spanned by the_roots >.; :i= >.;, and the total dimension -b of the subspace!J spanned by 2>.;. If b = 0, then we have a Siegel domain of type one. What was happening at the time, however, was an exhaustive 11tudy of V-positive Hermitian forms for classical cones (a set of extremely beautiful problems in linear algebra). This activity woUld unavoidably expose - at· some point or another - that there exist "too many" Siegel domains and that there have to exist homogeneous nonsymmetrical domains as well. The symmetrical classical domains are all connected-with only the cone of Hermitian matrices, after all. This fact is especially easy to see by looking at the dimensions of the root subspaces. · But, of course, it was first necessary to allow th.e existence of sud\ nonsymnietric domains! Yet, at the time, the consensus was negative. I remember one Jxchange occurred during one of Pyatetski's lectures on the domains. Pyatetski turned to Onishik, who was following the latest literature about complex manifolds more carefully then the other participants of the seminar; and asked, "Has it been proved already that complex homogeneous domains are symmetric?" "Almost" came-the answer. The picture was the following: H. Cartan used direct methods to show that all homogeneous domains in C 2 and C3 _jU"e symmetric. It was quite natural to assume that this holds for larger dimensions. This was slowly being proved with weaker and weaker assumptions about the automorphism group. A.Borel and Koszul assumed that the group was simple, and Hano used the assumption the group was unimodular (the last result was published in 1957, not long before this conversation took place). Everyone was completely sure that the restrictions placed on the nature of the group were quite nat,ural. Who would have thought that in the general situation the group is solvable! It is interesting to note that later, we had carefully reread Carta.n's paper and saw that he did not make the conjecture that all bounded homogeneous domains are symmetric. He simply thought that a construction of nonsymmetric domains woUld require some extremely fresh ideas. How right he was! As we all know, there do not exist symmetric Siegel domains of type two which are connected to the cone of symmetric matrices. The easiest example can be constructed in din;i-ension 4 (the lowest dimension not considered by Cartan). The domain of C4 which is

Vol. 2, 1990

SIEGEL DOMAINS

11

nonsymmetric is the the domain with the coordinates z1, z2, za, w given by the following· conditions: where.y; is the imaginary part of z;. It is funny to remember now, how suspiciously we listened for the first time to the proof that this domain is nonsymmetric. This was done · in 1959, so Pyatetski's sixtieth jubilee coincides with the thirtieth jubilee of this example. I would riot be surprised if Pyatetski had arrived at this result not as a result of looking at the examples and trying to understand them, but rather in a more natural way. At the time, he was interested in the structure of the boundary of symmetric domains in connection with the theory of automorphic functions. In cases compact manifolds (Siegel) and even compact nvrmal spaces (Remmert) the field of meromorphic" functions is-actually equal to the field of algebraic.functions ·if there are enough of them. Following an iqea of Siegel once again, one could attempt to try and compactify the fundamental domain of the discrete subgroup in such a way as to make sure that all automorphic functions could be extended to this compactification. This approach has a number of nice points since it does not rely_on the assumption that all automorphic functions are ratios of automorphic forms. Gradually, this approach was becoming the main one for Pyatetski. But this did require an understanding of the analytic components of the boundary· of the symmetric domain in question, and most domains have a fairly complicated,stratification. In fact, oilly the complex ball has a smooth one. In all the other cases, the Shilov boundary splits into isolated points, but it possesses holomorphic components of large dimension as well. _The boundary of the six-dimensional domain U2,s, for example, contains 'a five dimensional component which is but not symmetric. This is also related to an'other important notion. One can consider a realization of a symmetric domain under which a certain fixed component of the boundary is "sent" out to infinity. If the component is a point, then one will obtain a Siegel domain of type two. In the general case, however, the con_struction gives a wider class of domains which Pyatetski called Siegel domains of type three. These are bundles over bounded domains with affine-homogeneous fibers. This technique turned out to be·quite fruitful. It became possible to show, for example, that C7 contains a continuum of nonequivalent homogeneous domains. -It then became clear that symmetric domains are, in a certain sense, an exception among the homogeneous domains. The situation was becoming more and more intriguing and Pyatetski. devoted more and more of his attention to homogeneous domains (still not forgetting about automorphic functions). . In 1961, the theorem about realizations of symmetric domains in the form of Siegel domains of typ.e two was proved, the components of the boundaries and the appropriate realizations in the form of Siegel domains of type three were researched, and the Siegel · domains of type two were studied in great detail. It seemed that the canonical appearance of type two domains was described, and that there were a ·finite number of such domains in each dimension. There was still no known examples of non-symmetric domains of Siegel type one (I will not speculate as to whether Pyatetski itlready suspected their existence). This stage is recorded in the book written by Pyatetski- Shapiro called' "Geometry of Classical Domains and_ the Theory of Automorphic Functions," which was published in Russian in 1961.

12

S.GINDIKIN

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. At the same time, Vinberg and I also began studying complex homogeneous domains. It is necessary to add that Pyatetski had a strong effect not just on the younger participants of the seminar. Karpelevich, for example, also began studying the boundaries of symmetric ·spaces under the influence of Pyatetski's lectures. Viriberg and I had soon applied a "division of labor" to our works. Vinberg successfully began studying the "real". part of the problem - homogeneous convex cones. He quickly understood everything that was known about the self-adjoint homogeneous cones, and explained to us the connection with Jordan Algebras. Very soon, he also constructed an example of a homogeneous non-selfadjoint cone in R5 • This was vital for the project as a whole, since because of it C5 contained a non-symmetric Siegel domain of type one (and C 11 contained a continuum of nonequivalent ones, but this was a later result). I, on the other hand, began by computing the Bergman and Szego kernels for Siegel domains of type two connected with self-adjoint cones. Hua's book, in which these kernels were computed for the classical symmetric .domains, was quite popular in Moscow at the time, having just been translated .into Russian. These very beautiful and very clear computations were done on a certain compact subgroup. Therefqre, the question about non-symmetric domains remained unclear. It turned out that working with the transitive solvable group was quite ·convenient. It was somewhat surprising to find out that it was possible to compute the normalizing multipliers, which Hua computed in the classical domains through the invariant volumes of the analogue of the spheres. This was made possible by generalizing Siegel integrals (the analogue of Euler integrals for cones). In the process of my computations I had discovered that the description of Siegel domains connected with classical cones incomplete. Moreover, I saw that there is a continuum of holomorphically nonequivalent Siegel domains of type two. I remember well the immediate reaction Pya.tetski had to my news: "This must i:nean that any homogeneous domain admits a realization in the form of a Siegel domain of type two!" I would never have imagined this . . . · Quite soon he proved his conjecture that a solvable, simply-transitive split Lie group acts on this domain. The most important part of this work was the complete restatement of the problem into algebraic terms. Pyatetski introduced a concept of j-algebra, a Lie algebra which has a linear operator j and a linear form w with the following axioms satisfied: j2:::::: -E;

[.ix,jy] = j[jx, y] + j[x, jy] + [x, y] ; w([jx, jy]) = w([x, y]), w([jx, x]) > 0

for x

=/. 0 .

Obviously •the Lie algebra is the Lie algebra of the group of automorphisms, j is the operator of the almost complex structure and one ofthe axioms is the condition of the integrability of j. Furthermore, he considered the Bergman metric which li.appens to be the invariant Kahler metric. The corresponding symplectic form is exal:t, as was noticed by Kozul. Thus he obtained a one-form w. This what was left of the condition of homogeneous bounded domain of en, a condition which is relatively hard to work with. Technically the result consists of exhibiting a certain structure in any split solvable j-algebra which corresponds to a Siegel domain of type two. No one had any doubts that the result is correct in the general· case, so the three

Vol. 2, 1990

SIEGEL DOMAINS

13

of us united in order to obtain it. By the end of 1962 we had proved the existence of a realizatfon of any complex homogeneous domain in the form of a Siegel domain of type two. The proof was complicated and unwieldy, so despite the fact that the result became well known, and many mathematicians used it; it is probable that few ·understood the proof (which is still complicated even today). After that, Pyatetski stopped his systematic research of complex homogeneous domains. Vinberg and I kept going for a while. We had studied the root structure of homogeneous domains and applied what we had learned. He applied it to constructions of algebraic computations connected to the homogeneous cone, while I applied it to the construction of the theory of special functions in homogeneous cones and integral formulas in complex homogeneous domains. . In 1963 Vinberg and I decided to study generic homogeneous Kahler manifolds, i.e. manifolds which have a transitive group of holomorphic automorphisms which preserves the Kahler metric. Therefore, we gave up the assumption tl}.at· the corresponding symplectic form was exact. We made a conjecture that as manifolds, complex Kahler manifolds are direct 1 products of homogeneous bounded domains of locally flat and sililply connected compact Kahler manifolds. The description of the last two types is known. It is respectively the factorization of Hermitian spaces over lattices and flag-manifolds.· The metric is' different from that of the·direct product. Thus, from that point of view, one only obtains a bundle over a bounded manifold, with fibers which are a direct product of manifolds of the other two types. · This problem can also be reduced to a purely algebraic question about the so-called Kahler Lie algebras, a natural generalization of j-algebras. Still it was very tempting to use some geometric or analytic differences of these three types of manifolds for .obtaining the splitting. These plans were not destined to be realized - we were only able to find an algebraic proof of the conjecture for split solvable groups (years of 1964-65, published in 1967). We made some more progress, then interrupted the work on the complete proof for a while, or so we thought. As it turned out we never returned to it, but more then twenty years later, in 1988, our conjecture was proved (Dorfmeister and Nakajima). This seems like a good place to end my story about the past. I had wanted to tell the story as I had remembered it without consulting with either Pyatetski-Shapiro or Vinberg about the I hope there are no dramatic differences in- our memories. But we all know the fable about comparing statements of several eyewitnesses. Nowadays, I have again- started working on homogeneous complex manifolds after a long break, and I would like to discuss a few results which I believe have relevance to those stated above.

Pseudohermetian symmetric domains. Let X =Gu/Ube a flag mauifold (simply connected, compact, homogeneous Kahler manifold), where Gu is a compact Lie group, Uthe centralizer of the torus. A flag domain V is an open orbit in. X of some real form G of the complexification Ge of the group Gu, [1]. In particular, if Xis a compact Hermitian symmetric space, then a flag domainis the Cartan dual noncompact Hermitian symmetriC space. But even in this case there exist other domains - the pseudohermitian symmetric spaces with non compact stationary subgroup .. A typical example: the domains 'Dp,q with p + q = n - 1 in cpn which are of

14

S. GINDIKIN

IMCP

the from With q = 0 we have a complex ball. In general, among the flag domains only the Hermitian symmetric ones are Stein manifolds. The rest are q-pseudoconcave, where q is the dimension of the maximal complex homogeneous submanifolds which are contained in them. Consequently, in V the spaces of q dimensional 8 cohomology classes are infinitely dimensional regardless of which coefficients are chosen. It would be interesting to build some sort of analysis on these cohomology classes . The Radon-Penrose transformations must play an important role here, in integration of cohomology classes over compact complex submanifolds. One would also want an explicit formula for inversion of this transformation, as well as integral formulas which reconstruct cohomology classes from the boundary values. For 'Dpq these formulas have been found, and the integral formulas have kernels which are holomorphic on a certain Stein manifold which is a bundle over 'Dpq (2,3]. Therefore, it is possible to apply holomorphic techniques to the leading 8 cohomologies. Probably, the analogues of these formulas exist for flag manifolds in great generality. It would also be interesting to understand whether the flag domains or pseudohermitian domains can be included into some broader class of homogeneous manifolds, similar to the way that the symmetric domains can be included into the class of complex homogeneous domains or homogeneous Kahler manifolds. This, of course, requires full information about the structure of that class. It is probable that the class of pseudo-Kahler homogeneous manifolds is too large. Geometric structures on compact Hermitian manifolds. Our ability to realize the Hermitian symmetric manifolds in the form of bounded domains of en is a relatively accidental occurrence. They "live" in the dual Hermitian symmetric compact manifold, just as the Siegel domains which are equivalent to them. The automorphism of these manifolds, as well as the generalized Cayley transformations between them, are contained in the group of automorphisms of a compact manifold. Geometry of this manifold significantly affects the geometry of the included domains. This geometry is especially rich in the cases with rarik greater than one. On the other side of this richness are the different facts about the "rigidity" in the case of rank greater than one. Recently, a new fact was discovered in this vein. It has been shown, that besides the Riemanian structure, there also exists a different infinitesimal structure which X which lies in Ge/Kc which is invariant with respect to G and the isotropic subgroups of all the points are compact. We hope that xc is a Stein manifold although there are problems with the proof in the general situation. The spaces of holomorphic functions or of sections of complex bundles are tightly connected with the analysis on the group G, since its orbits are the the sets of uniqueness for holomorphic functions. It is possible that they are connected with discrete series representations (nonholomorphic ones!). It seems that it is also possible to consider the holomorphic automorphic forms on xc for r c G. A similar situation was considered by Gelfand and me (7]. For groups G which act in the Hermitian symmetric domains, inside Ge there were constructed semigroups G+ which are Stein manifolds invariant under left and right multiplication by G, and G is their Shilov boundary. The representations of holomorphic discrete series can be continued to G+. These examples suggest that complex Stein manifolds invariant under the action by G for which the orbits are the sets of uniqueness may be useful for analysis. It is probable that one additional axioms. The fact that this action is proper can be justified in a number of cases. Infinitesimal structures on the horosphere manifolds. There is a generalization of the symmetric space not connected with groups, but stated in purely geometric terms [8]. This arose in connection with problems in integral geometry. It goes back to the classical idea of Plucker to characterize geometry of a

16

S.GINDIKIN

IMCP

manifold by families of submanifolds and the way these submanifolds intersected (Plucker considered lines in three dimensional projective space). It turns out that in symmetric spaces, horospheres intersect in the maximally simple and a maximally singular way. For · simplicity let us consider the case of a complex semisimple Lie group G (this is a pseudosymmetric space with the automorphism group G x G). Let S be the set of horospheres of Ge (the two-sided shifts of the maximal unipotent subgroup), and let dim Ge= p. Let us consider the dual family of submanifolds 3 9 , g E G of S. 3 9 , g E G consists of for which g E Ge. For Ep let us look at the p-parameteric family Ep of p-hyperplanes tangent to Sg, g E G at the point E Sg. Then Ep for all groups and symmetric spaces admits the following inductive description. For a (j - 1)-subspace u and a (j + 1) subspace 7r let us call the sheaf {u, 7r} the combination of j-subspaces p such that u C p C 7r. Then I:1 consists of one-subspaces which lie in some two- subspace 7r and

e

e, e

u;

=

LJ. {u ,7r (u)}, 1

u+E;-1

On the p-th step we obtain EP (for some operator u 1-+ 7r(u)). This structure takes place for all groups, the difference coming only from the choice of 7r(u), that is, the sheaf containing the (j -1)-hyperpl¥1es of I:;-1. If, for some n-dimensional manifold X, we have an n-parametric family of p-dimensional {xe, ES} so that the family of the of the tangent planes Ep of the dual submanifolds S., will have an analogous inductive structure; then there exists a local formula of inversion Jor the integral transformation of integration .on Xe. This formula is analogous to the inversion formula for the horosphere transformation. Thus in this situtation (which has no groups!), it is possible to build a harmonic analysis. The inductive character of the construction allows us to pass to the infinite-dimensional case. At the end, let me return to the beginning. In the last year and a half I have begun meeting with people who had seen Pyatetski. I had asked every one of them, "How is Ilya?" Normally they would answer ··"He is still working all the time". This answer always calmed me. I know him well enough to know that if he is working all the time then everything is O.K. with him. I hope to be hearing such an answer for many years to come. References

e

0

1.

R.O. Wells, Jr., J.A. Wolf, Poincare series and automorphic cohomology on flag domains, Ann. of Math. 105 (1977), 397-448. 2.. S.G. Gindikin, G.M. Henkin, Integral geometry for 8-cohomology in q-linearly concave domains in Cpn, Functional Anal. Appl. 12, no. 4 (1978), 6-23 (in Russian). 3. S;G. Gindikin, Integral formulas and integral geometry for 8-cohomology in cpw, Functional Anal. Appl. 18, no. 2 (1984), 26-33. 4. G.B. Goncharore. Generalised conformal structures on manifolds, Selecta Math. Sovetica 6?, no. 4 (1984), . 5. S.G. Gindikin, Generalized conformal structures, preprint, Weierstrass-institut, Berlin, P-Math.-18/89, Berlin, 1989, 1-28. 6. D.N. Ahiezer, S.G. Gindikin, On Stein extensions of real symmetric spaces, Math. Ann. (to appear).

Vol. 2, 1990

7. 8.

SIEGEL OOMAlNS

17

l.M. Gelfand, S.G. Gindikin, Complex manifolds whose skeletons are real semisimple Lie groups and holomorphic discrete series of representations, Function. Anal. Appl. 11, no. 4 (1977), 19-27. S.G. Gindikin, Horospheres and twistors, Leet. Notes in Phys. 313 (1988), 3-10.

Lab. Build. "A" Moscow University GSP-234, Moscow 119899, U.S.S.R.

ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS, Vol. 2, 1990

Exceptional Representations and Shimura's Integral for the Local Unitary Group U(3) by

Stephen S. Gelbart* Department of Mathematics The Weizmann Institute of Science Rehovot 76100,

ISRAEL

and Jonathan D. Rogawski* Department of Mathematics University of California, Los Angeles Los Angeles, California 90024, USA To Ilya Piatetski-Shapiro on the occasion of his sixtieth birthday

Table of Contents Introduction 1.

Preliminaries

2. 3. 4.

The local Howe pairing (U(l), U(3)). Heisenberg models Kirillov models

5.

Exceptional representations

6.

The Shimura integral

7. 8.

The ramified case The unramified case

* Both authors are supported by a grant from the U.S. Israel Bi-National Science Foundation. The second-named author is also partially supported by a Grant from the NSF and a Sloan fellowship.

19

20

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S.S. GELBART AND J.D. ROGAWSKI

Introduction The purpose of this paper is two-fold. Firstly, we treat various aspects of Howe's correspondence for the dual reductive pair (U(3), U(l)) over a local field. denotes a quasi-split unitary group in n variables.

Here U(n)

Secondly, we lay the groundwork for

global applications by developing the theory of integrals of Shimura type attached to an irreducible admissible representation

7l'

of U(3), these integrals being defined in terms of

the dual reductive pair just mentioned. One of our applications is a characterization of the globally exceptional representations of U(3). In a forthcoming paper [GeR1], we shall discuss the equivalence o:f the following conditions on a cuspidal representation

7l' :

(1)

7l'

is globally exceptional

(2)

7l'

is a Weil representation

(3)

7l'

is the endoscopic transfer ([Ro]) of a one-dimensional automorphic representa-

tion of U(2) x U(l) (4)

the standard Lcfunction L(s,71' 0 'f/) has pole on the line Re(s) = 3/2 for some unitary Hecke character 'f/ of E.

This type of result illustrates a compatibility between the endoscopic transfer for U(3) and the theory of theta series for U(3).

Together with results contained in [Ro], it implies

that the cuspidal representations that contribute to H 1 (r, t(b)db .

The function cp, which belongs to the Kirillov model of

7r

(§4), is a function on B with values in r; · similarly, '!? belongs to the Kirillov :model of the Weil representation associated tor* 0 tR (cf. §6). For unramified data, SH(cp,i?,t) is computed in terms of the standard L-functions L( s, 7r 0 T/) defined in §1. To carry out this computation, 'l?(b) is expressed in terms of a certain fixed element B .in the space of T.

We then.compute the inner product of cp(b) with translates of Busing the recursion

relations imposed by the Hecke operators. This is a local version of the theory of primitive theta functions developed in [Sh] and [GlR]. To the best of our knowledge, it is also the first time an unramified local zeta-integral has been explicitly computed when neither the familiar theory of Whittaker functions, nor the well-known GL(n) theory, is applicable. Throughout the_paper, we include results for GL(3) as well as U(). This is necessary for global applications since a global unitary group in three variables is locally isomorphic to GL(3) at half of the places. The unramified calculations are in fact considerably more involved in the GL(3) case. The representation of an L-function in terms of an integral of the product of a cusp form, Eisenstein series, and theta function was first investigated by Shimura in the case of the metaplectic cover of GL(2)

([Shim]).

The general underlying representation-

theoretic mechanism was explained by· Piatetski-Shapiro in [P2], where it was to as "Shimura's method".

Piatetski-Shapiro also carried out parts of this program for

Vol. 2, 1990

U(3) ([P1]).

23

SHIMURA'S INTEGRAL FOR LOCAL U(3)

In particular, the theory of Heisenberg and Kirillov models (§3.1, 4.1-2)

is due to him, and we would like to thank him for allowing us to include here some of his unpublished results (from [P1]).

We note also that the exceptionality of local Weil

representations was asserted (without proof) in [P2]. At around the same time, Shintani inde);)endently studied a global Shimura integral in the case of holomorphic automorphic forms of level one on U(3) and obtained complete results in that case ([Sh]).

Shintani

studied the Fourier-Jacobi coefficients of an automorphic form, which in the holomorphic case are theta functions, and examined the recursion relations they satisfy when the form is an eigenfunction of the ring of Hecke operators. In §8, we develop this theory in a local, _representation-theoretic setting and use it to evaluate SH( r.p, {), e) for unramified data with_ the help of some difficult calculations due to Shintani. The significance of a Shimura-type global integral is that it gives an explicit integral representation for L( s, 7f ® e) even when 7f

has no Whittaker model;

cf. [GP] for background discussion.

It seems particularly fitting to contribute this paper to a Festschrift in Piatetski-

Shapiro's honor, and it is with pleasure and affection that we do so.

The second-named

author would also like to thank the Weizmann Institute of Science for hospitality and excellent working conditions on several occasions during the preparation of this work. We are grateful to Miriam Abraham for her professional typing of the manuscript, and Laure Barthel for her careful reading of Sections 1 through 5.

24

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S.S. GELBART AND J.D. ROGAWSKI

§1.

Preliminaries

1.1.

In this section, we fix some notation which will be used throughout the paper.

Let F denote a fixed local field of characteristic zero and let E / F be a quadratic extension. Conjugation with respect to E/ F will be denoted by a bar, or a. If Fis non-archimedean, we let Op be the ring of integers in F, pa prime element of F, and q

= Card(Op/(p)).

Let w EI F be the character of F* of order two associated to E / F by class field theory. Let U( iI>) be the unitary group of a skew-Hermitian form in three variables:

ii>=(: : :) -1 where

eEE*,

TrE/F(e)

0

0

=0.. We have U(iI>) ={g E GL3(E): tgiI>g =iI>}.

Throughout the paper, G will denote either U(iI>) or GL3(F) and Z will denote the center of G.

In both cases, B and M will denote, respectively, the Borel subgroup of

upper-triangular matrices in G and the subgroup of diagonal matrices. In. case F is nonarchimedean, we denote a special maximal compact subgroup of G by K. If G we set K

= GL 3 (0p ).

If E/ Fis unramified and p =/= 2, we assume

= GL3(F),

eis a unit and take K

to be the subgroup of integral matrices in G. Recall that if (W, A) is a symplectic space over F, then the Heisenberg group H(W) is the central extension of W consisting of pairs [w, t], with w E W, t E F, and group law defined by:

[w, t][w',t'J

=

[w + w1 ,t + t'

1

+ 2 A(w,w

The center of H(W) is {[O, t] : t E F} and is isomorphic to F.

1

)]

25

SHIMURA'S INTEGRAL FOR LOCAL U(3)

Vol. 2, 1990

The unipotent radical N of B is the Heisenberg group of a two-dimensional vector space. If G

= U(cJ/),

N consists of matrices of the form

1

0 where w E W and t E F. TrE/F(exfi):

It is isomorphic to H(W) with W

= E,

where A(x, y)

=

If G = GL3(F), then N is the strictly upper-triangular group, whiCh we

·can describe as the set of matrices of. the form 1

a

+t)

[w,t] = ( 0

1

b

0

0

1

where w = (a,b) E F 2 , t E F.

In this case, N is isomorphic to H(W) with W = F 2 ,

where A((a, b), (a', ,8')) =all - a'b.

In both cases, we denote the center of N by U and

define R to be the centralizer of U in B. Let d( a, b, c) be the diagonal matrix with entries a, b, c. _Let diag( x) be the scalar .matrix with entry x. Let S' be the torus of elements of the form d(l, ,8, 1), where ,8 E E 1 if G

= U(cJ/)

and ,8 E F* if G

= GL3(F),

and set S =-S'Z.

If G

= U(cJ/),

d(a, 1, a- 1 ) for a E E* and let T be the torus of elements of the form d(a). If G

let T be the torus of elements of the form d(a, 1, c). Then R

=S ·N

and M

set d(a)

=

= GL3(F),

= Z · T.

Finally, (7r, V) will denote an irreducible admissible representation of G. 1.2.

Local factors.

We describe the local factors ii:J. the (non-archimedean)

unramified case. · Let A be an unramifi.ed character of E* and let 7r(A) be the principal series representation of U(3) induced from the character d(a)diag(x) for the Langlands class in LU(3)

=

--+

A(a) of M.

A

GL3( C::)X1Wp associated to 7r(A) is.

26

IMCP

S.S. GELBART AND J.D. ROGAWSKI

given by 1

where Fr is a Frobenius element in Wp.' If 'f/ is an unramified character of E*, we define

We remark that L(s,7r(A)0'f])

= L(s,1l"E0'f]), where 1l"E, the base change of 7r to GL3{E),

is the representation of GL3(E) with Langlands class

Since we are interested in global applications, we also have to consider the group GL3. Suppose that E / F is a global quadratic extension and G is the in three variables with respect to E / F.

unitary group

Let v be a place of F which splits in E and let

wand w 1 be the places of E dividing v. Then Gv is canonically isomorphic to GL3(Ew), which we identify with GL3(Fv), by means of the isomorphism Fv

-+

Ew. Let

7r

be the

unramified representation of Gv corresponding to the Langlands class

in GLa( 1 be a non-degenerate skew-Hermitian form on Vi and let

The Howe pairing for unitary groups.

dimension nj for j =

Let

'1>2 be a non-degenerate Hermitian form on V2. Then '1>1 0 '1>2 is a skew-Hermitian form on V

= Vi ®E V2

and A

= TrE/F(1

0 '1>2) is an F-linear alternating form on V.

Gj be the unitary group associated to j.

Let

Let W be the F-vector space underlying V.

There is a natural map of the product G1 x G2 into Sp(W) whose kernel is the central subgroup {(z,z- 1 ): z E E 1 }. Furthermore, (Gi,G2) is a dual reductive pair. If we take n 2

=

1, then G 2 is the unitary group U(l) in one variable, which is

isomorphic to E 1 , and the image of G2 in Sp(W) coinci.des with the image of the center of G1. Fo.r a proof of the following theorem, see (MVW], pg. 51.

Vol. 2, 1990

SIIlMURA'S INTEGRAL FOR LOCAL U(3)

Theorem 2.4.1:

In the above notation, suppose that

nz =

29

1.

Then the covering

7r: Mp(W)---+ Sp(W) splits over the image of G1 in Sp(W).

The choice of lifting G1 in the ker( 7r)

=

- < v*,w >for

v, wEX, v*,.w*EX*. Let G = GLn(F) and identify .G with the stabilizer of X and X* in Sp(W).

The

SchrOdinger model of r,µ is realized on S(Fn) and any lifting of G to Mp(W) deterniines, . by restriction of a metaplectic representation, an oscillator representation of G. There is a lifting of G to Mp(W) uniquely determined by the requirement that g E G act on the Schrodinger model by the formula:

· We call the representation given by. this formula the standard oscillator representation of

GLn(F). All other oscillator

differ from it by a twist. In particular, we

see that the oscillator representations are independent of t/J. Let P be the standard parabolic subgroup

{g E GLn(F): tgen =.\en .for some A E F*} where .en is the unit vector to GLn-i(F) x GL1(F).

· · · , 0, 1). The standard Levi subgroup M of Pis isomorphic Let Po be the stabilizer of en, and let uo be the determinant

character of Po. It is clear that w,µ is isomorphic to the ind(G, Po, the function 'Pf(g) = Let

(compactly) induced

of GLn(F). An isomorphism is given by mappingf(x) to

ldet(g)i 1/ 2 f(tgen)

in the space of Ind(GLn(F), Po,

x be a unitary character of Z(G)

·

= F.* and let w,µ(X) be the largest quotient of

w,µ on which Z acts via X· To determine w,µ(x), define

Vol. 2, 1990

for

f

31

SHIMURA'S INTEGRAL FOR LOCAL U(3)

E S(Fn), x E Fn. This integral clearly converges, and for z E Z, (w.p(z)c,o)x(g) Let

x(z)c,ox(g).

and

u1

denote the character JdetJ on GL1 and GLn-1, respectively.

u2

The functional c,o-+ 'Px(g) transforms under P by the character map c,o-+ 'P.x defines an isomorphism of w.p(x) with Ind(G, P, unitary induction from P by ip.

Observe that Cp

= GL3(F).

®xu1(n-l)/ 2 and the ® xu1(n-l)/ 2 ). Denote

= u2 ® u!(n-l)

and hence w.p(x) is

the trivial representation of GLn-1·

isomorphic to ip(x ® triv.), where triv. Now let G

=

A representation

'Tr

of G will be called a Weil representation

if it is o(the form w.p(x) ® x' for some unitary character x' of G. Proposition 2.6.1:

(a) w.p(x) is a non-zero irreducible representation of G. It is

isomorphic to ip(X ® triv.). (b)

x' are distinct characters of Z, then w.p(x) and w.p(x') are inequivalent

If x

as representations of SL3(F). · Proof: 2. 7. that W

W1

=

The irreducibility of ip(X ® triv.) is well-known ([BZ]) and (b) is clear. Mixed model,

= W1 ffi W2

X1 ffi Xi

We return for a moment to the notation of §2.1. Suppose

and that A

= Ai ffi A2, where A; is a symplectic form on W;.

a polarization of W1 and realize

on S(Xi)). Let F be any model for

Then

rJ ®

rJ

Let

(the 1/1-representation of H(W1) acts on S(Xi) ® F. We ·identify -

S(Xi)®F with S(Xi ,F), the space of Schwartz functions on Xi with values in :F.. There is a natural surjection

with central kernel through which

rJ @

factors and we thus obtain a realization of T ,P

on S(Xi, .F). This realization is called a mixed model.

32

S.S. GELBART AND J.D. ROGAWSKI

Now let G = U(w) and let V

= E 3.

For 1 5 j

IMCP

3, let

Let W; denote the F-vector space underlying Vj. Set W13 = W1 EB Wa and let Aia denote _ the restriction of A= TrE/F(w) to W13. with respect to V = W13 EB W2.

a mixed model for -r.p of H_(W)

We

The decomposition W13 =' W1 EB Wa is a complete

. polarization with.respect to A13 and the ¢-representation of H(W1a) can be realized on

S(Wa). Identify H(W2) with H(E) and let ( -rJ, :F) be a ¢-representation for H(E). Then we obtain a mixed-model realization of -r.p on S(Wa, :F). We identify Wa with E and write

S(E,:F) for S(W1,:F). Now let w.p be- a mixed-model realization of the oscillator representation on S(E,:F). Then w.p is determined only up to twisting by a character of G. We can fix w.p uniquely by fixing an oscillator representation

of U(l) on :F and requiring that elements s ES'

(cf. §1.1) act by the formula: _(1)

w.p(s)w(w)

for

wE S(E,:F).

The following formulas also hold:

(2)

=

w) for a EE*, with 1(a) a unitary char-

acter of E* which depends on the choice of

(3) - w.p([c,t])w(w) =

and if [c,t] EN.

Note that since we identify N with H(E), -rJ([cw,O]) iS defined (cf. §1). Note also that since the metaplectic covering defined with 't'v is a B-map from V to Ind(r,B,R). Fix a character

eof M

whose restriction to S is unitary.

character of B and denote its restriction to R by eR· Let w

We also regard

= w( r* 0

the space of w, then there exists a non-zero R-map L' : W--> X*.

'l?(b)

e as a

eR)· If W denotes Fix w E Wand set

= L'(w(b)w). Let

< ' > denote the Hermitian eR-pairing between T and r* 0 eR,. i.e.,

< r·x, r·y >= eR(r)- 1 < x, y >. In the following sections we will study the local Shimura integral sn(rp,i?,e)=

r

}R\B

e(b)db=

-

r

ls\T

), then S(O) = S). the subgroup of integral matrices in N. Af+ = {t E M : l8(t)I ::::; l}.

Let R(O) = S(O) · N(O), where N(O) is

Define 8(t) = la/cl fort= d(a,b,c), and let

We will call (r,X) in R('I/;) unramified if the space of

R(O)-fixed vectors in X is non-zero. There exist unramified representations (r,X) E R('I/;).

Lemma 8.1.1:

Moreover.

if r is unramified, then (a)

The space of R(O)-fixed vectors in Xis one-dimensional, and

(b)

Fort E M, the space of tR(O)t- 1 invariants of r is non-zero if and only if t E M+.

Proof:

Assume first that G = U(iI>). In particular, we assume E/F is unramified

and (is a unit (cf. §1.1). Therefore OE is self-dual with respect to the form TrE/F((xy), and we can realize r.p in a lattice model with respect to OE ((MVW), page 42). Let S be the space oflocally constant functions f on E such that f(w+a) ='I/;(! TrE/F((wa))f(w) for a E OE. Then N acts by the formula

r.p([u, t])J(w) = 'l/;(t

1

.

+ 2 TrE/F((wu))J(w + u)

For d(l, ,8, 1) ES' and z E Z, the formula r(zd(l, ,8, l))J(w) = J(,Bw) defines an extension

S,S. GELBART AND J.D. ROGAWSKI

50

of T.;; to a representation To of R.

IMCP

Let Jo be the characteristic function of OE.

scalars, Jo is the Unique N(O)-fixed vector for T.p, and it is also fixed under S(O).

Up to This

shows that To is unramified. However, To is the unique unramified element in R( 'ljJ), since

R(t/J) consists of the twists T1 =To® v,

vis non-trivial character of R/N, and no

non-trivial twist is unramified. This proves (a), and (b) follows easily. The GL3(F) case follows similarly using the. Schrodinger model.

D

Lemma 8.1.2: Let T E R( 'ljJ ). Then Tis unramified if and only if w( T) is unramified. Proof:

Assume first that G

R-map L: w-+

T

= U(

O if m=O if

m>O

if m=O

By Lemma 8.3.1,

and by the above relations,

= 2)a(m + I,n -1) + 60,mx:1a(m,n)]£(pn,p-m)B f(p- 1, l)cp = 2)a(m, n + 1) + 60,nx:2a(m + 1, n)]f(pn ,p-m)B f(p,p)cpc

where Oij is the Kronecker delta. Arguing similarly with .A2, we obtain the following two recursion relations:

.A1a(m,n) = a(m -1,n)

+ qx(p)[a(m + 1,n -1) + 60,m11:1a(m,n)J

Vol. 2, 1990

63

SIDMURA'S INTEGRAL FOR LOCAL U(3)

A2a(m, n)

= x(p)a(m, n -

1) + q[a(m - 1, n + 1) + c50,n11:2a(m, n)]

+q2x(p)[a(m + 1, n) + c50,m11:1a(m, n + 1)] . It is easily checked by induction on m+n that if a(O, 0)

Hence, if i,o(l)

=

=

0, then L('ir(b)v)

= 0, then a(m, n) = 0 for all m,

n.

0 for all b E B and then L is identically zero, sini;e

G =BK. This proves the first assertion.

From now on, we assume that a(O, 0)

AlF(X, Y)

(I)

= 1.

This gives the relations

= XF(X, Y) + qx(p)[F(X, Y) - F(O, Y)]Y x- 1 + qx(p)11:1F(O, Y) +

F(X,O)]Y- 1 + q211:2[F(X,O) - F(O, O)]x- 1

q2 [F(X, Y) -

A2F(X, Y) ::;:: x(p)Y F(X, Y) + q[F(X, Y) - F(X, O)]xy-l + q11:2F(X, 0)

(II)

+ q2x(p)[F(X, Y) - F(), Y)]x- 1 + q2x(p)11:1[F(O, Y) - F(O,O)]Y- 1

To write F(X, Y) as a rational function, we need to write F(X, 0) and F(O, Y) as rational functions. Set Y.

(Iy) (IIy)

= 0 in (I) and (II).

[A1 - X]F(X, 0)

Since F(O, 0)

1, this gives

= qx(p)11:1+q 211:2x-1[F(X,0) -1] +

q2Fy(X, 0)

[A2 - q11:2]F(X,O) = q2x(p)X- 1{F(X,O) -1] + qXFy(X,O) + q2x(p)11:1Fy(O,O)

where Fx, Fy denote the partial derivatives with respect to X and Y, respectively. If we set X = 0 in (Iy) and (IIy ), we obtain a pair of linear equations for a(O, 1) and a(l, 0):

A1- qx(p)11:1

= q211:2a(l,O) +

A2.- q11:2 = q2x(p)a(l, O)

q 2 a(O, 1)

+ q2x(p)11:1a(o; 1) .

This proves (a). Set a= a(l, 0) and f3=a(O,1). Next we solve for F(X, 0) by eliminating I



Fy(X, 0) from (Iy) and (IIy ). Multiply (Iy) by X and (Hy) by q to obtain

.

64

IMCP

S.S. GELBART AND J.D. ROGAWSKI

Subtracting the' equations and multiplying by X gives ·

A similar argument gives an expression for F(O, Y).

This proves (b).

Now (I) and (II)

can be re-written as

(I)

(II)

[>.1 -

x - q 2Y- 1 -

= [qx(p)1'1

qx(P )Yx- 1]F(X, Y)

- qx(p)Yx- 1]F(O, Y)

+ [q2 "2 -

q2 Y- 1]F(X, 0) - q2 ,.2x- 1

[>.2 - x(p)Y - x- 1q2 x(p) - qXY- 1]F(X, Y)

= [q1'2 -

qXY- 1]F(X, 0)

+ [q2x(P)"l

- x- 1q2 x(p)]F(O, Y) - q2 x(p)1'1Y- 1 .

Using (b) and multiplying by XY, we re-write again as

(I) (II)

[>.1XY - X 2 Y -

f

q2 - qx(p)Y 2]F(X,

y)

[>.2 - x(p)Y- q2x(p)x- 1 - qXY- 1]F(X, Y)

= Now (I) gives Qi(X)Q2(Y)F(X, Y) as the quotient of

and >. 1XY - x2y - Xq2 - qx(p)Y2. equal to P1(X)P2(Y)

is equal to

+ CXY.

To prove (c) we must show: that this quotient is

So we must check that

Vol. 2, 1990

SIIlMURA 'S INTEGRAL FOR LOCAL U(3)

65

This is a lengthy caluclation which, however, can be verified easily by a computer program for algebraic manipulations (such as MAPLE).

Proposition 8.5.2:

Qi(X) =

Proof:

TI

For

7r

D

as in'§8.3, 1UJ.d Qi, Q2 as in Theorem 8.5.1,

(1-q- 1zj 1X), and Q2(Y) =

TI

(1-q- 1 zjY).

We have x(p) = z1z2z3 and

Hence

The formula for Q2(Y) follows and the caiculation of Qi(X) is similar.

Proposition

Proof:

Suppose that

The equations for a,

f3

in Theorem 8.5.l(a) become

D

66

IMCP

S.S. GELBART AND J.D. ROGAWSKI

Using that 11:111:2 = q- 1, we get a = x(p)- 111:2,

/3

The formula for C ill

= x(p)11:1.

Theorem 8.5.1 gives

By Theorem 8.5.1 and Proposition 8.5.2, the formulas for P1(X) and Ql(X) become.

+ q- 1x(p)- 111:2]X + q- 211:1X2)

P1(X) = (1 -

= (1 - q- 1x(p):-- 111:2X)(1 -iJ- 1x(p)11:; 111:1X) Ql(X) = (1- q- 1x(p)- 111:2X)(l -q- 1x(p)11:2 111:1X)(l - q- 1x(p)- 111:1 1x) and the expression for P1(X)/Q1(X) follows.

The expression for P2(X)/Q2(X) follows

similarly.

o

Local Fourier coefficient of an unramified Weil representation.

8.6.

Let

w.p be an oscillator representation for GLa(F). We consider the mixed model realization of w.p on the space S(F 2 ) © :F as in §2.8.

It depends on the choice of

turn determines the character 'Y of F*.

x

Lo :

S(F 2 )

Let

which in

be· a character of Z and define a map

© :F---+ :F by

Lo(g © 0) = {

iF•

x- 1 (a)g(a, a- 1 )w 1 (a)Oda

x) (cf. §3.2).

where g E S(F 2 ) and 0 E :F. Let .r = Lemma 8.6.1: Proof:

Lo(w.p(r)g © 0) = r(r)Lo(g © 0) for all r ER.

If r = d(a, a, a)d(l, b, 1), then

Lo(w.p(r)g © 0) ;="

=

£.

l.

x- 1(a)g(a,

= r(r)Lo(g © 0) .

67

SIIlMURA'S INTEGRAL FOR LOCAL U(3)

Vol. 2, 1990

Similarly, Lo(w.p([w, t])g ® 8)

=

t])Lo(g ® 8) for r

= [w, t] EN.

D

Lemma 8.6.1 implies that Lo factors through w(r) and hence we obtain a non-zero Heisenberg functional L : w( r) -+ :F. (Cf. Sections 2.6, 2. 7.) As before, let 8 be a mixed vector in

f which is fixed by N(O).

Note that

T

is

unram:ified if and only if 8 is fixed by R(O). Let go(x,y) be the characteristic function of

OF Ee OF. Lemma 8.6.2:

Assume that Tis unramified. Then the vector go® 8 E S(F 2 ) ® :F

is fixed by K. Proof:

Let vo = go ® 8. Suppose that r =

x). Then 'Y and

x are unramified

by Proposition 8.1.2. Let w =

Then K is generated by

w, M(O),

0 ( 0

0

1 1 - 0

and N(O). Note that go® 8 is

in 0.- Hence,

if u E U(OF ), w.p(u)vo = t/J(uxy)go(x, y) ® 8 = vo. Similarly, if n = [(ci, c2), OJ E N(O), then.

since x, 'If E OF and r(c1x,c2y)8

= 8 if go(x,y)-/= 0.

This shows that vo fixed by N(O).

On the other hand, if t = d(t1, t2, ta), with t; E Oj;., then

since 'Y is

and

fixed 8. Finally, by a standard formula (page 41 of [MVW],

cf. the notation of our §2.8), there is a constant c: such that

IMCP

S.S. GELBART AND J.D. ROGAWSKI

where dz is the self-dual measure on Y.

This is equal to c:vo since the kernel of 1jJ is Op.

We conclude that the line generated by vo is fixed by K. But then the intersection Ka of K with the special unitary group SU() fixes vo, and hence c:

Set i?(t)

= L(w..p(t)go 0

0).

= 1 since w

E Ka.

O

Then{) is the element in the Kirillov model for w(r)

corresponding to the K-fixed vector w, where w is the image of go 0 0 in w(r). It is clear from the next proposition that i9(t) is non-zero (since i9(1)

= 0).

Thus we set

We also regard 0 as an element of X(l,r) by setting O(r)= r(r)O. Proposition 8.6.3:

For a, b ;:=:: 0, we have

L

i9(pa,p-b) =r(Pa-b)q-(a+b)/2.

f(pn,p-m)O(pa,p-b).

n-a=b-m m,n2:_0

Proof:

We have

= 'Y(Pa-b)q-(a+b)/2

L

[l(pt)wl(pt)o .

Observe that

since d(p-t, 1, p-t)

= d(p-t, p-t)d(l, pt, 1),

i9(pa ,p-b)

= r(Pa-b)q-(a+b)/2

and hence

L

f(pa+t,p-(b-t))O(pa ,p-b)

SHIMURA'S INTEGRAL FOR LOCAL U(3)

Vol. 2, 1990

69

n=a+t,m=b-'C [J

8. 7.

The local Shimura integral in the split case.

is the central character of ;.,( T), then w( T) corresponds to

By the results of §2.6, if x Langlands class of the

q-1/2/(P)

g(w(r))

=

(

Comparing Proposition 8.5.3 with Proposition 8.6.3, we conclude that q-1/21(P) Recall e

= x(p)1t1, x(p)l(p)-2 = x(P)-11t!11t2,

= q1/ 2r;,1 = x(p)- 17(p), e;-"{c).

form E(d(a, b, c))::::::

and let

\

and q1/21(p)

= x(p)tt21

ebe an unramified character of M

of the

Motivated by global considerations (cf. [GeR1]), we define

the following three characters of T = M / S':

XE(d(a,b,c))

.

= x(ac), tt(d(a,b,c)) =

i1d(a,b,c)i1 8

= jac- 1 j

and the following three £-factors 3

L(s, 71" ® e) =

II [(1 -

elz;q- 8 )(1 - e2zj 1.q- 8 ) ] - 1

j=l

Lp(s, eF) = (1 - 6e2q-s)-l LE(s,exE") = [ll - x(p)e6q- 8 )(1- x(Pr 1e- 16q- 8 ) ] - 1 We now use the notation of §8.1 and §8.6. In particular, -0 is as in §8.1.

8

,

70

S.S. GELBART AND J.D. ROGAWSKI

Let fo = faE11:11 · ll 8 -l/ 2 .

Theorem 8.7.1:

IMCP

Then for Re(s) large,

SH(1.p,i'J,fo}= and

< 1.p(t),i'J(t) > =
o

and this can be written as

L

m,n;;::o

a(m, n)(q-:-n/2en Al"+n + q-(n-1)/2g(n-1) Al+n-1 A2

+ ...

ranges over

72

S.S. GELBART ANDJ.D. ROGAWSKI

IMCP

We now follow the calculations in [Sh, §4]. We write SH(cp, tJfo, e) as

L a(m, n)Af A2[q-n/2en(A1/A2r + q-(n-1)/2e(n-1)(A1/A2r-1 + ...

Set C1

= q- 112eAifA2,

C2

= q- 112e- 1A 2/A 1.

Then SH(cp,tJeo,e) is equal to SH(l,e),

where SH(X, e) denotes the power series

L

m,n;:::o

a(m, n)Af A2'[Cf

Recalling that (f(X,Y)

+· .. + C1+1+c:..._2 + ... + c21xm+n

= L:a(m,n)xmyn, we see that SH(X,e) is equal to. C1 Ci _ l (f(A1X, C1A2X) - f(A1X, A2X))

1

+ 02 -

Lemma 8.7.2: (b)

(a)

.

l (C2f(C2A1X,A2X)-f(A1X,A2X))

f(X, "'1X)

= (1 -

q- 1x(p)- 1,.,2X)/Q1(X) .

f(,,,2X,X) = (1- q- 1x(P)"'1X)/Q2(X).

Proof:

T.his can be checked by direct calculation using Theorem 8.5.l(c).

Lemma 8.7.2, SH(X,e) is equal to

So by

Vol. 2, 1990

73

SHIMURA'S INTEGRAL FOR WCAL U(3)

Then a direct (and long) calculation, using Theorem 8.5.1, shows that this is equal to (1- q- 1 x(p)- 111:2A1X)(l - q- 1x(p)11:1A2X)(l - q- 2A1A2X 2 ) Qi(A1X)Q2(A2X) Therefore SH(l, e) is equal to (1 - q- 1x(P)- 111:26q-s)(1 - q- 1x(p)11:16q-s)(1 - q- 2eie2q- 2s) Qi(6q-s)Q2(6q-s) Note the numerator is equal to Lp(2s .

+ 2,ep)LE(s + 3/2,eE11:), -

since 11:1 = q- 112 c: and

11: 2 = q-1/ 2c:- 1. Theorem 8.7.l_nowfollows, since SH(r.p,i!,e) = SH(l,e).

74

IMCP

S.S. GELBART AND J.D: ROGAWSKI

References

[BZ]

Representations of the group GL(n,F), where

J. Bernstein and A. Zelevinsky: F is a local non-archimedean field.

[Cas]

W. Casselman, Some General Results in the Theory of Admissible Representations of p-adic Reductive Groups,

[CS]

mimiographed notes, 1973.

W. Casselman and J. Shakila: II:

[GP]

Russian Math. Surveys 31, 3, 5-70 (1976).

The Whittaker function.

The unramified principal series of p-adic groups Compositio Math. 41, 2, (1980) 207-231.

S. Gelbart and I. Piatetski-Shapiro: unitary group.

In:

Automorphic forms and L-functions for the

Lie group representations IL SLN 1041, Springer, Berlin-

Heidelberg-New York (1984). [GeR1] S. Gelbart and J. Rogawski:

On L-functions and Fourier-Jacobi coefficients of

automorphic forms on U(3). In preparation. [GeR2] S. Gelbart and J. Rogawski: [GlR]

Heisenberg models for U(3).

In preparation.

G. Glauberman and J. Rogawski: On theta functions with complex multiplication. J. Reine u. Angew. Math. (Crelles J) 395, 68-101 (1989).

[Ka]

D. Kazhdan:

Connection of the dual space of a group with the structure

its

closed subgroups. Funk. Analysis and Its Appl. 1, 63-65 (1967). [Ke]

D. Keys:

Principal series of special unitary groups over local fields, Compositio

Math. 51, 115-130 (1984). [MVW] C. Moeglin, M.-F. Vigneras, J.-L. Waldspurger:

Correspondances de Howe sur

un corps p-adique. SLN 1291. Springer, Berlin-Heidelberg-New York (1987). [Mo]

C. Moen:

The dual pair (U(3), U(l)) over a p-adic field. Pacific J. Math. 127,

No. 1 (1987) 141-154.

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SIIlMURA'S INTEGRAL FOR LOCAL U(3),

75

[P1J

I. Piatetski-Shapiro:

Lecture Notes (by J. Cogdell), Yale University, 1977.

[P2]

I. Piatetski-Shapiro:

Tate theory for reductive groups and dis,tinguished repre-

sentations. Proc. Int. Cong. Math., Helsinki (1978) 585-590. [Ro]

J .D. Rogawski:

A utomorphic representations of un_ita.ry groups in three variables.

To appear: Annals of Math. ·Studies, Princeton U. Press. [Shim] G. Shimura:

On modular forms of half-integral weight.

Annals of Ma.th. Vol.

97 (1973), 440-481. . (Sh]

T. Shintani:

On automorphic forms on-unitary groups of order 3..- (Unpublished,

1979). · [Wa]

J.-L. Waldspurger:

Seminaire Bourbaki 1986/1987. Asterisque 152-153, Soc.

Math. de France.

[W] A. Weil: Sur certains groupes d'operateurs unitaires. Acta Math. 111, 143-211 - (1964).

ISRAEL MATHEMATICAL CONFERENCE PROCEEDINGS, Vol. 2, 1990

Quelques remarques sur les théorèmes réciproques Guy HENNIART

Classification AMS : 22 E 50 Mots clés-Key words : Automorphic and Galois representations, L and e factors,· converse theorems. Résumé

Dans une première partie, nous examinerons comment il est possible de caractériser les représentations admissibles irréductibles de GL(n) sur un corps local non archimédien par des facteurs L et e. Divers exemples montrent qu'on ne peut guère améliorer, en général, les résultats positifs de Jacquet, Piatetski-Shapiro et Shalika. Une seconde pàrtie traite du cas global; l'on y souligne des problèmes peu explorés et en particulier on examine comment pourraient se généraliser les énoncés de Piatetski-Shapiro pour les théorèmes réciproques pour GLn.

Abstract

In the first part of this paper, wè investigate possible characterizations of irreducible admissible representations of G L( n) over a non archimedean local fi.eld by Lande factors. Examples are given, showing that the positive results of Jacquet, Piatetski-Shapiro and Shalika cannot be much improVt;!d upon in general. In the second part we examine the global case, painting out some interesting open problems and wondering how to generalize Piatetski-Shapiro's statements on converse theorems form G Ln.

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Introduction

1.1. Dans cet article, nous voulons énoncer quelques remarques et questions sur ce qu'il est convenu d'appeler les théorèmes réciproques voire théorèmes inverses (converse theorems en anglais) pour GLn. Il s'agit en fait de questions de nature assez différente, selon que l'on se trouve dans le cas local ou le cas global. Nous décrivons brièvement ci-après le contenu de l'article, dont le plan suivra grosso modo cette description. 1.2. Prenons d'abord pour F un corps local non archimédien et fixons-en un caractère additif non trivial 'lj;. A tout couple ( 7r, 7r 1) de représentations admissiblès irréductibles de GLn(F) et GLn1(F') respectivement, Jacquet, Piatetski-Shapiro et Shalika [JPSS2,3] ont associé des facteurs L( 7r X 7r 1) et e:( 7T x 7r 1 , '1jJ ), et énoncé que toute représentation admissible irréductible générique 7T de GLn(F) (pour n 2) est caractérisée, à équivalence près, par les facteurs L( 7T x ?T 1 ), L( 7T v x ?Trv) et e:( 7T x 7r 1 , '1jJ), 7r 1 parcourant les représentations admissibles irréductibles supercuspidales de GL; (F), l'indice i variant de 1 à sup (1, n - 2) [JPSS2, p. 61, 1. 1-2]. D'après Jacquet [Ja], on peut se contenter de faire varier i de 1 à [n/2], la partie entière de n/2. Nous construisons des exemples qui montrent qu'en général on ne peut se contenter de i = 1, ni même de i < [n/2], même si l'on suppose 7r supercuspidale. Pour n = 4,. on montre qu'il ne suffirait même pas des facteurs L( (X o det) ® r( 7T) ), où r est une quelconque représentation polynomiale de GL 4 (C), disons de degré m, et où r(7r) est la représentation admissible irréductible de GLm(F) attachée à 7T par fonctorialité de Langlands. 1.3. Les exemples mentionnés sont en fait construits du côté galoisien. On utilise pour cela la conjecture de Langlands qui prédit, pour chaque entier n 1, l'existence d'une bijection 7r(a) entre l'ensemble QF(n) des classes d'équivalence de représentations complexes continues irréductibles de degré n du groupe de Weil de F et l'ensemble A'F( n) des classes d'6quivalence de représentations admissibles irréductibles supercuspidales de GLn(F), de sorte qu'on ait

L(a ®a')= L(7r(a)

X

7r(a'))

(et de même pour les contragrédientes) ainsi que

. pour a E QF(n), a' E QF(n'). Cette conjecture de Langlands est loin d'être démontrée pour n > 3, mais nous montrerons au § 4 que, pour n = 4, nos exemples galoisiens peuvent être transférés du côté de AF(4).

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THÉORÈ.MES RÉCIPROQUES

1.4. Prenons maintenant pour F un corps global. Soient 7r et 7r 1 deux représentations automorphes de GLn(Ap) et GLn1(Ap) respectivement. J Piatetski-Shapiro et Shalika (JPSS3, JS, PS] ont défini des fonctions L( 7r X 7r 1 ) qui sont des fonctions méromorphes d'un paramètre complexe s, sont données par le produit sur les places de F des facteurs locaux correspondants (produit qui converge pour s de partie réelle assez grande), sont des fonctions rationnelles de q-s si F est un corps de fonctions en une variable sur un corps fini à q éléments, sont bornées dans les bandes verticales si F est un corps de nombres, et vérifient l'équation fonctionnelle

L(7r x 7r 1)(s) = t:(7r

X

7r 1 )(s) L(7rv x 7r 1v)(l - s)

où e:( 7r x 7r 1) est aussi le prod1fü sur les places de F des facteurs locaux correspondants (voir pour une preuve publiée de cette équation fonctionnelle (AC p. 69] qui outre les références susdites utilise [Sh 1, 2, 3]). 1.5. Le problème des théorèmes réciproques est le suivant. On se donne pour chaque place v de F une. classe ?Tv de représentations admissibles irréductibles de GLn(Fv), non ramifiées pour presque toute place v, et on se demande quelles conditions imposer, sur des fonctions L associées à 7r = 0 1rv, pour que 7r soit une représentation automorphe de GLn(Ap). Plus précisément, on suppose que pour 7r 1 dans un certain ensemble de représentations automorphes (de GL;(Ap) pour certains entiers i), le produit n L( 7r v X converge pour s de partie réelle assez grande et définit V

une fonction L( 7r x 7r 1 ) holomorphe et vérifiant les propriétés énoncées en 1.4. Le problème est de déterminer un ensemble, si possible optimal, de représentations automorphes 7r 1 , pour que les hypothèses impliquent que 7r soit automorphe. 1.6. De tels théorèmes réciproques sont connus pour n = 2 [JL, Wel, We2] et n = 3 [JPSSl]·et un manuscrit de Piatetski-Shapiro [PS] circule depuis une quinzaine d'années environ, abordant le cas général pour les corps globaux de caractéristique non nulle. Nous formulons quelques problèmes qui nous semblent devoir être explorés dans cette direction, problèmes qui sont motivés par la conjecture de Langlands mentionnée plus haut et sa contrepartie globale. Enfin, nous nous demandons s'il ne suffit d'imposer conditions susdites pour 7r 1 parcourant les caractères des classes d'idèles de F, non pour impliquer que 7r est automorphe, mais pour qu'il existe une représentation automorphe T de GLn(AF) ne différant de 7r qu'à un nombre fini de places. C'est sur cette question optimiste que s'achèvera l'article. 1. 7. On a vu que plusieurs travaux de Piatetski-Shapiro ont été source d'inspiration du présent article .. C'est un plaisir et un honneur pour moi que de dédier cet article à LI. Piatetski-Shapiro, homme et mathématicien.

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Je tiens à remercier aussi les participants du Séminairè sur les représentations des groupes réductifs et les formes automorphes de Paris (et particulièrement G. Laumon), où certains des exemples et problèmes abordés ici ont été exposés. Enfin je remercie de grand cœur les organisateurs de la conférence en l'honneur de LI. Piatetski-Shapiro, qui m'ont invité à ·Y apporter ma contribution, sous la forme d'un exposé et du présent article.

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THÉORÈMES RÉCIPROQUES

Le cas local galoisien

2.1. Nous fixons un corps local non archimédien P, et un caractère additif non trivial 'ljJ de P. Nous fixons une clôture séparable algébrique P de P et notons Wp le groupe de Weil de P sur P. Pour chaque entier n 2: 1, nous notons Q}( n) l'ensemble des classes d'équivalence de représentations continues irréductibles de W F dans un espace vectoriel complexe de dimension n. Le premier but de ce paragraphe 2 est de donner des exemples de représentations Œ et Œ1 dans QF( n ), distinctes mais telles que pour toute représentation 7 E QF( i), i < n/2, on ait

c(Œ 0 7,,,P) = c(Œ 1 07,,,P). (Remarquons qu'on a toujours L(Œ 0 7) = L(Œ 1 0 7) = 1 puisque Œ, Œ1 et 7 sont irréductibles et qu'on a i < n; de même on a :

2.2. Dans nos exemples, nous prendrons toujours Œ1 = w 0 Œ, où w est le caractère non ramifié d'ordre 2 de px, vu comme caractère de Wp par l'isomorphisme de réciprocité de la théorie locale du corps de classes (nous noterons de la même façon un caractère de px et celui de Wp qui lui correspond). En effet; on dispose alors de formules simples pour les facteurs e:, que nous explicitons ci-après. Pour p une représentation complexe continue de W F on note a(p) son exposant, i.e. l'exposant de son conducteur d' Artin, et on note a(p) le plus grand indice a de ramification (en numérotation supérieure) telle que p soit non triviale sur le sous-groupe WF. On sait que si p est irréductible on a a(p) = 0 si p est un caractère non ramifié et a(p) = dim(p )(1 + a(p)) sinon [Hel, Thm. 3.5]. Pour IJ' et 7 comme plus haut on a, pour tout composant irréductible p de IJ'@ 7, . a(p):::; sup(a(Œ),a(7)) avec égalité si a(Œ) 3.6.1], qu'on a

(*)

-:f

a(7), et on déduit immédiatement [cf. Hel, .Cor.

a(Œ 0 7):::; sup(ia(Œ), na(7))

avec égalité si ia(Œ) -:f na(7). D'autre part, c'est une propriété très élémentaire des facteurs e; que

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2.3. _Prenons pour n un entier pair, n = 2m et supposons donnée