Geometry and Representation Theory of Real and p-adic groups

329 Pages·1996·7.47 MB·English

by  Juan Tirao, Dan Barbasch (auth.), David A. Vogan Jr., Joseph A. Wolf (eds.)

Progress in Mathematics
Volume 158
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
Geometry and
Representation Theory
of Real and p-adic groups
Juan Tirao
David A. Vogan, Jr.
Joseph A. Wolf
Editors
Birkhauser
Boston • Basel • Berlin
Editors:
Juan Tirao David A. Vogan, JT.
Facultad de Mathemcitica, Department of Mathematics
Astronomia y Fisica Massachusetts Institute of Technology
Universidad Nacional de C6rdoba Cambridge, MA 02139
Ciudad Universitaria
C6rdoba, Argentina 5000
Joseph A. Wolf
Department of Mathematics
University of California
Berkeley, CA 94720
Library of Congress Cataloging-in-Publication Data
Geometry and representation theory of real and p-adic groups / Juan
Tirao, David A. Vogan, Jr., Joseph A. Wolf, editors.
p. cm. -- (Progress in mathematics ; v. 158)
Papers from the Fifth Workshop on Representation Theory of Lie
Groups and Its Applications.
Includes bibliographical references
ISBN-13: 978-1-4612-8681-3
1. Lie groups. 2. Representations of groups. I. Tirao, Juan,
1942- . II. Vogan, David, A., 1954- III. Wolf, Joseph Albert,
1936- . IV. Workshop on Representation Theory of Lie Groups and
Its Applications (5th: 1995 : Universidad Nacional de Cordoba)
V. Series: Progress in mathematics (Boston, Mass.) ; vol. 158.
QA387.G46 1997 97-36150
512'.55-dc21 CIP
AMS Classifications: 22E25, 22E45, 22E46, llF70, 53C30, 17B30, 32MlO
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ISBN-13: 978-1-4612-8681-3 e-ISBN-13: 978-1-4612-4162-1
DOl: 10.1007/978-1-4612-4162-1
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CONTENTS
Preface . . . . . . . . . . . . vii
The Spherical Dual for p-adic Groups
Dan Barbasch ......... . 1
Finite Rank Homogeneous Holomorphic
Bundles in Flag Spaces
Tim Bratten . . . . . . . . . . . . . . . . . . . . . . 21
Etale Affine Representations of Lie Groups
Dietrich Burde . . . . . . . . . . . . · ... 35
Compatibility between a Geometric Character Formula
and the Induced Character Formula
Esther Galina · ... 45
An Action of the R-Group on the
Langlands Subrepresentations
Eugenio Garnica Vigil .... · ... 57
Geometric Quantization for Nilpotent Coadjoint Orbits
William Graham and David A. Vogan, Jr. 69
A Remark on Casselman's Comparison Theorem
Henryk Hecht and Joseph L. Taylor . . . . . . 139
Principal Covariants, Multiplicity-Free Actions,
and the K -Types of Holomorphic Discrete Series
Roger Howe and Hanspeter Kraft . . . . . . . . . . . 147
Whittaker Models for Carayol Representations of GLN(F)
Roberto Johnson ............... 163
Smooth Representations of Reductive p-adic Groups:
An Introduction to the theory of types
Philip C. K utzko .. . . . . . . . . . . . . . . 175
Regular Metabelian Lie Algebras
Fernando Levstein and Alejandro Tiraboschi 197
Equivariant Derived Categories, Zuckerman Functors
and Localization
Dragan Milicic and Pavle Pandiic . . . . . 209
A Comparison of Geometric Theta Functions
for Forms of Orthogonal Groups
Jiirgen Rohlfs and Birgit Speh ..... 243
Flag Manifolds and Representation Theory
Joseph A. Wolf . . . . . . . . . . . . 273
PREFACE
The representation theory of Lie groups plays a central role in both clas-
sical and recent developments in many parts of mathematics and physics. In
August , 1995, the Fifth Workshop on Representation Theory of Lie Groups
and its Applications took place at the Universidad Nacional de Cordoba in
Argentina. Organized by Joseph Wolf, Nolan Wallach, Roberto Miatello, Juan
Tirao, and Jorge Vargas, the workshop offered expository courses on current
research, and individual lectures on more specialized topics. The present vol-
ume reflects the dual character of the workshop. Many of the articles will be
accessible to graduate students and others entering the field. Here is a rough
outline of the mathematical content. (The editors beg the indulgence of the
readers for any lapses in this preface in the high standards of historical and
mathematical accuracy that were imposed on the authors of the articles.)
Connections between flag varieties and representation theory for real re-
ductive groups have been studied for almost fifty years, from the work of
Gelfand and Naimark on principal series representations to that of Beilinson
and Bernstein on localization. The article of Wolf provides a detailed introduc-
tion to the analytic side of these developments. He describes the construction
of standard tempered representations in terms of square-integrable partially
harmonic forms (on certain real group orbits on a flag variety), and outlines
the ingredients in the Plancherel formula. Finally, he describes recent work
on the complex geometry of real group orbits on partial flag varieties. This
is related to very active research on geometric realization of singular unitary
representations.
The "singular" tempered representations neglected in Wolf's article take
center stage in Garnica's. The basic problem he considers is a very old one: to
understand the reducibility of representations unitarily induced from discrete
series representations. Knapp and Stein showed that for connected G, the
irreducible constituents of such an induced representation all have multiplicity
one, and are parametrized by the characters of an "R-group," which is a
product of copies of Z/2Z. Garnica extends the work of Knapp and Stein to
disconnected groups. He finds that the irreducible constituents all have the
same multiplicity (now a power of two), and that the number of constituents
is also a power of two. Both of these numbers are described in terms of an
R-group.
Galina's article explores further the connection between the geometric
constructions that Wolf describes and formulas for the characters of repre-
sentations. Essentially she finds the geometric content in an old formula of
Harish-Chandra, Hirai, and Wolf for characters of induced representations.
The coefficients in character formulas can often be expressed as dimen-
sions of certain Lie algebra homology groups. (This idea goes back at least
viii PREFACE
to Kostant's work on the Bott-Borel-Weil theorem in 1961. One recent in-
carnation is a conjecture of Osborne proved by Hecht and Schmid.) Because
the character of a representation is independent of the particular globalization
chosen, the Lie algebra homology should not depend strongly on the globa-
blization. Hecht and Taylor prove an unpublished theorem of Casselman of
this nature: that the Lie algebra homology groups of a maximal nilpotent
subalgebra with coefficients in a Harish-Chandra module or its smooth glob-
alization coincide.
The problem of geometric realization of representations on flag varieties
is taken up again in Bratten's article. A fundamental problem in the field
is that representations on Dolbeault cohomology are almost always too large
to carry G-invariant Hilbert space structures. (Wong has shown that they
always carry the "maximal globalization" of the underlying Harish-Chandra
module.) One would like to have an equally compelling geometric realization
of the minimal globalization, since that space can carry a G-invariant inner
product. Bratten proves that the compactly supported cohomology groups
(with coefficients still in sheaves of germs of sections of holomorphic vector
bundles) provide such realizations.
The connection between flag varieties and representations has an alge-
braic side as well. Two cornerstones are Zuckerman's "cohomological induc-
tion" construction of representations, and the Beilinson-Bernstein algebraic
localization theory. The article of MiliCic and Pandzic addresses the founda-
tions of both of these constructions. Their definitions clarify and simplify a
number of basic results, including the "duality theorem" of Hecht, Milicic,
Schmid, and Wolf.
Another very active research area is the theory of reductive dual pairs.
Weil found in the 1960s a group-theoretic framework for the theory of theta
functions, and made it into a powerful construction of automorphic forms.
Howe abstracted from Weil's work the following setting. One has a large
reductive group H, a very small representation p of H, and a pair of reductive
subgroups G1 and G2 , each of which is the centralizer of the other in H.
Essentially one studies the restriction of p to G1 X G2 • Howe has made this
setting into a tool for studying group representations. His paper with Kraft
is a beautiful introduction to these ideas. The goal is to understand the
restriction to a maximal compact subgroup of a holomorphic discrete series
representation. The authors explain why this is problem is interesting, why
it is a problem in invariant theory, and then why the theory of reductive dual
pairs has something to say about it. Finally they prove some powerful new
results.
The article of Graham and Vogan is first of all an introduction to the
Kirillov-Kostant philosophy of coadjoint orbits. This philosophy provides a
geometric "classical analogue" of unitary representation theory, in the same
sense that classical mechanics is an analogue of quantum mechanics. The
simplest technique for constructing representations from coadjoint orbits relies
on the existence of a G-invariant Lagrangian foliation of the orbit. Graham
PREFACE ix
and Vogan consider nilpotent coadjoint orbits, for which such foliations usually
do not exist. They find instead C-invariant Lagrangian coverings: larger
families of Lagrangian subspaces, with a compact family passing through each
point. They use such coverings to describe representations of C, but are not
able to find unitary structures.
Of course there is much more to Lie theory than the theory of reductive
groups. The article of Levstein and Tiraboschi concerns a class of two-step
nilpotent (or "metabelian") real Lie groups C . The "regular" condition is that
every non-trivial irreducible unitary representation of C is square-integrable
modulo the center. This condition has a direct algebraic description, but is
not easy to check; roughly speaking, one needs to verify that every non-zero
element in a linear space of skew-symmetric bilinear forms is non-degenerate.
The authors analyze regular metabelian Lie algebras, and classify those with
two-dimensional center. They find some unexpected new examples.
A connection between Lie theory and differential geometry is explored in
Burde's article. An affine structure on a manifold M is a maximal collection
of charts with the property that any change of coordinate map between two
charts is locally an affine map x 1-+ Ax + b on IRn. Burde characterizes left-
invariant affine structures on a Lie group C in terms of what he calls etale
affine representations: homomorphisms Q of C into the affine group Aff(E) of
a vector space E, with the property that some vEE has a discrete stabilizer
in C and an open orbit in E. Semisimple groups never admit such representa-
tions, and nilpotent groups often do; indeed Milnor conjectured in 1977 that
every nilpotent Lie group admits an etale affine representation. Burde makes
a thorough study of such representations. He finds very general conditions for
their existence, but is also able to construct nilpotent groups for which they
do not exist.
In the theory of p-adic reductive groups, one of the fundamental results
is Borel and Casselman's description of the representations with an Iwahori-
fixed vector. Briefly, suppose C is a split group over a p-adic field, and J
is an Iwahori subgroup. Borel showed that a smooth irreducible representa-
tion 7r of C admits a J-fixed vector if and only if 7r is a subquotient of an
unramified principal series representation of C. It follows that such represen-
tations may be studied by means of the Iwahori Hecke algebra of compactly
supported J-bi-invariant functions on C. This Hecke algebra has a relatively
simple structure (found by Matsumoto) and its representations were explicitly
parametrized by Kazhdan and Lusztig. In the case of C L( n) over fields of large
residual characteristic, Howe and Moy found a way to study arbitrary repre-
sentations, replacing J and the trival representation of J by smaller compact
open subgroups and representations of them. They constructed isomorphisms
between the corresponding Hecke algebras and Iwahori Hecke algebras for
smaller groups over extension fields. Finally Bushnell and Kutzko removed
the restrictions on the residual characteristic. Kutzko's article here provides
a careful and elementary introduction to his work with Bushnell, as well as to
ideas for extending their work to groups other than CL(n).
x PREFACE
The ideas described in Kutzko's article reduce many problems in the rep-
resentation theory of p-adic groups to the case of unramified representations.
Unramified representations are the subject of Barbasch's article on his work
with Moy, especially the unitary spherical representations of classical groups.
Barbasch describes large families of particularly interesting unitary represen-
tations, and introduces the technology (particularly Lusztig's theory of graded
Hecke algebras) needed to study them.
The article of Johnson provides another kind of introduction to the rep-
resentation theory of GL(n) over a p-adic field F. Johnson considers super-
cuspidal representations of a special kind first constructed by Carayol. By
an old result of Gelfand and Kazhdan, every supercuspidal representation of
GL(n) admits a Whittaker model; Johnson constructs these models explicitly
for Carayol's representations.
One of the basic reasons for existence of representation theory for reduc-
tive groups over real and p-adic fields (that is, local fields) is the representation-
theoretic theory of automorphic forms. Those ideas are represented here by
the article of Rohlfs and Speh. One of Langlands' central conjectures is that
the theory of automorphic forms is controlled by dual groups, and therefore
that automorphic forms on different inner forms of the same group are closely
related. Rohlfs and Speh investigate this idea in the case of inner forms of
orthogonal groups, and the automorphic forms contributing to the analytic
torsion of locally symmetric spaces. (Changing the inner form of the orthog-
onal group in this case amounts to modifying the signature of the underlying
quadratic form.) Using ideas of Kottwitz about stabilizing the trace formula,
and a careful study of certain special representations of real groups, they ob-
tain very precise information about analytic torsion on different inner forms.
Juan Tirao
David A. Vogan, Jr.
Joseph A. Wolf
Editors
THE SPHERICAL DUAL FOR p-ADIC GROUPS
DAN BARBASCH
o. INTRODUCTION
The local Langlands conjectures have played a very significant role in the
study of the representation theory of reductive algebraic groups. Roughly
they say that the parametrization of equivalence classes of irreducible rep-
resentations should be given in terms of conjugacy classes of continuous ho-
momorphisms of the Weil group WIF into the dual group LC. When C is
split, the local Langlands conjectures say basically that the parametrization
of equivalence classes of irreducible representations should be given in terms of
conjugacy classes of homomorphisms of the Wei I group WIF into the dual group
LC. More precisely, let v C be the connected complex group with root data
dual to the root data of C. Then consider v C conjugacy classes of continuous
homomorphisms
(0.1)
such that the image consists of semisimple elements. In the real case, these
conjectures were crucial for the classification of admissible irreducible (g, K)
modules in the work of Langlands, Shelstad, Knapp-Zuckerman and Vogan.
In the p-adic case they playa significant role in the work of Kazhdan-Lusztig
and Lusztig. There is a technical modification in that one considers maps of
the Weil-Deligne-Langlands group,
(0.2) cP : WIF x SL(2) ~ LC.
This parametrization is not very well suited for describing the unitary dual
(except for the tempered part). However, motivated by global considerations,
Arthur proposed that in (0.1) and (0.2) one ought to replace WIF (WIF x SL(2)
respectively) by its product with another SL(2). Denoting the group in (0.1),
(0.2) by WIF , the parametrizing space would be conjugacy classes of maps
(0.3)
such that the restriction to WIF represents a tempered parameter (see Section
1.4 for details about the parametrization of tempered representations). We