Table Of ContentProgress in Mathematics
Volume 220
Series Editors
Hyman Bass
Joseph Oesterle
Alan Weinstein
N oncommutative
Harmonic Analysis
In Honor of Jacques Carmona
Patrick Delorme
Michele Vergne
Editors
Springer Science+Business Media, LLC
Patrick Delorme Michele Vergne
Institut de Mathematiques de Luminy Centre de Mathematiques
UPR 9016 CNRS Ecole Polytechnique
13288 Marseille Cedex 9 91128 Palaiseau Cedex
France France
Library of Congress Cataloging-in-Publication Data
Noncommutative harmonic analysis : in honor of Jacques Carmona I Patrick Oelorme and
Michele Vergne, editors.
p. cm. - (Progress in mathematics ; v. 220)
ISBN 978-1-4612-6489-7 ISBN 978-0-8176-8204-0 (eBook)
DOI 10.1007/978-0-8176-8204-0
1. Harmonic analysis. 2. Lie groups. I. Carmona, Jacques, 1934- II. Oelorme, Patrick.
III. Vergne, Michele. IV. Progress in mathematics (Boston, Mass.); v. 220.
QA403.N66 2003
515'.2433-dc21 2003052075
CIP
AMS Subject Classifications: llF70, 11S40, l1S90, 22E35, 22E45, 22E46, 53055
ISBN 978-1-4612-6489-7 Printed on acid-free paper.
©2004 Springer Sciencet-Business Media New York
Origina1ly published by Birkhliuser Boston in 2004
Softcover reprint ofthe hardcover lst edition 2004
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987654321 SPIN 10916445
www.birkhasuer-science.com
From the ICM Conference in Nice (1970).
Left to right: Rene Carmona, Jacques Carmona, Michele Vergne and Michel Duflo
Working afternoon in Luminy at the end of the 1990s.
Left to right: Mogens Flensted-Jensen, Patrick Delorme, and Jacques Carmona
Contents
Preface ..................................................................... ix
Jacques Carmona ......................................................... xiii
Morris identities and the total residue for a system of type Ar
Velleda Baldoni-Silva and Michele Vergne ............................... 1
A reduction theorem for the unitary dual of U (p, q)
Dan Barbasch ....................................................... 21
Symmetric spaces and star representations III. The Poincare disc
P. Bieliavsky and M. Pevzner .......................................... 61
Local zeta functions for a class of symmetric spaces
Nicole Bopp and Hubert Rubenthaler ................................. 79
Quelques remarques sur les distributions invariantes
dans les algebres de Lie reductives
Abderrazak Bouaziz ................................................ 119
Espace des coefficients de representations admissibles
d'un groupe reductif p-adique
Patrick Delorme .................................................... 131
Dualite entre G / G]R et Ie groupe renverse - G]R
P. Harinck and M.-N. Panichi ........................................ 177
Sur certains espaces d'homologie relative d'algebres de Lie:
cas des polarisations positives
Annie Hersant ...................................................... 201
La formule de Plancherel pour les groupes de Lie
presque algebrique reels
M. S. Khalgui and P. Torasso ........................................ 213
Analytic continuation of nonholomorphic discrete series
for classical groups
A. W. Knapp ....................................................... 253
viii Contents
A branching law for subgroups fixed by an involution
and a noncompact analogue of the Borel-Weil theorem
Bertram Kostant .................................................... 291
Representations of S L2 and the distribution of points in JP>n
J. Kuttler and N. R. Wallach ......................................... 355
A localization argument for characters of reductive Lie groups:
an introduction and examples
Matvei Libine ....................................................... 375
Intertwining ladder representations for SU(p, q) into Dolbeault cohomology
John D. Lorch, Lisa A. Mantini, and Jodie D. Novak ................... 395
Summation formulas, from Poisson and Voronoi to the present
Stephen D. Miller and Wilfried Schmid ............................... 419
McKay's correspondence and characters of finite subgroups of SU(2)
W Rossmann ....................................................... 441
Methodes de Kashiwara-Vergne-Rouviere pour les espaces symetriques
C. To ross ian ....................................................... 459
Einstein integrals and induction of relations
E. P. van den Ban ................................................... 487
Preface
The articles in this volume, honoring the work of Jacques Carmona, show the diversity,
vitality, and importance of the area of noncommutative harmonic analysis.
Several contributions lead to a new perspective on classical questions.
The article of B. Kostant is concerned with the study of the restriction of finite
dimensional representations of a complex semisimple algebra to the subalgebra of
fixed points of an involution. Several applications are given: the structure of M, the
generalization of the Cartan-Helgason theorem, and a noncompact version of the
Borel-Weil Theorem.
1. Kuder and N. Wallach examine Nagata's counterexample to the Hilbert 14th prob
lem, and deduced new results from revisiting his algorithm in a more representation
theorical framework.
The article of S. Miller and W. Schmid gives a proof of a formula of Voronoi which
generalizes the Poisson formula. The summation involves a particular weight coming
from a coefficient of the L-function of a cuspidal automorphic form of GL(3, IR)/
GL(3, Z). Applications are given.
W. Rossmann shows that, in the McKay correspondence, between irreducible char
acters of finite subgroups of SU (2) and extended Coxeter-Dynkin graphs of type ADE,
characters are themselves given by a uniform formula.
Two contributions concern the description of unitary representations of real semisim
pIe Lie groups.
A. Knapp's article is devoted to the unitarity of the analytic continuation of discrete
series in a Borel-de Siebenthal chamber for a number of cases of classical groups.
D. Barbasch studies the spherical unitary dual of groups U (p, q). He gives in par
ticular the complete determination of the spherical unitary dual of the groups U (n, n)
and U(n + 1, n).
Two contributions are related to the orbit method.
In the case of representations of nilpotent Lie groups, A. Hersant examines some
spaces of relative homology with respect to positive polarizations.
M.S. Khalgui and P. Torasso give a proof for the Plancherel formula of real algebraic
groups, deduced from the philosophy of the orbit method.
x Preface
Two contributions concern invariant distributions on reductive Lie groups or Lie
algebras.
A. Bouaziz proves two results on invariant distributions on the Lie algebra of a
reductive Lie group in the Harish-Chandra class. If such a distribution is supported on
the closure of the orbit of a compact set, Harish-Chandra showed that this distribution
is tempered. Bouaziz shows that its Fourier transform is a locally L I-function. He also
shows the equivalence between two notions of invariant-tempered stable distributions.
M. Libine describes his recent argument of localization. This allows for a geometric
proof of the Schmid-Vilonen result relating the two formulas of characters as an inte
gral over a cycle due to Rossmann, and a fixed point formula. Key ideas are given for
the example of SL(2, lR).
P. Delorme proves an analog for p-adic groups of results of J. Franke, who, in his
proof of a conjecture of A. Borel, introduced a filtration on spaces of -automorphic
forms and described the corresponding graded space.
Y. Baldoni-Silva and M. Vergne find explicit formulae for the total residue of certain
rational functions with poles on hyperplanes determined by the roots of a system of
type An.
Two contributions focus on reductive symmetric spaces.
E. van den Ban gives an introduction to his joint work with H. Schlichtkrull on the
Paley-Wiener Theorem and Plancherel formula for reductive symmetric spaces.
P. Harinck and M.N. Panichi introduce, for a complex semisimple simply -connected
Lie group G a duality between G / G]R and a real form of G called a reversed group.
A correspondence between invariant stable distributions is obtained, generalizing the
correspondence obtained by A. Bouaziz when the real form G]R has a compact Cartan
subgroup.
Two contributions are devoted to deformation quantization.
C. Torossian shows, using Kontsevich graphs, how to deform the Campbell-Hauss
dorff formula for symmetric spaces. This allows him to reobtain Rouviere's result in
the solvable case on the convolution of invariant distributions, and to generalize the
result to a special case of symmetric cases.
P. Bieliavsky and M. Pevzner describe an intertwining map between the regular
representation of SL(2, lR) in the space of COO-functions on the Poincare disk with
a subrepresentation of S L(2, lR) in the space of Coo -functions on lR2 possessing the
Weyl-Moyal star-product. They suggest how to extend the result to the case of hermi
tian symmetric spaces of tube-type.
N. Bopp and H. Rubenthaler describe an explicit functional equation for Zeta distri
butions associated to prehomogeneous spaces for a real reductive group G (lR), in the
sense of Mikio Sato or Godement for Mn (lR). Open orbits are symmetric spaces. Zeta
functions are attached to minimal principal series and to distribution vectors invariant
under isotropy groups of this open orbits.
