The Orbit Method in Representation Theory: Proceedings of a Conference Held in Copenhagen, August to September 1988

233 Pages·1990·3.269 MB·English

by  M. Duflo, M. Vergne, N. V. Pedersen (auth.)

II
Progress in Mathematics
Volume 82
Series Editors
J. Oesterle
A. Weinstein
M. Duflo N . V. Pedersen M. Vergne
The Orbit Method in
Representation Theory
Proceedings of a Conference Held in
Copenhagen, August to September 1988
With 23 Figures
1990 Birkhauser
Boston . Basel . Berlin
M. Duflo N.V. Pedersen
University of Paris-VII Mathematics Department
75251 Paris Cedex 05 University of Copenhagen
France 2100 Copenhagen
Denmark
M. Vergne
eNRS
DMI
45 rue d'Ulm
75005 Paris
France
Library of Congress Cataloging·in·Publication Data
The Orbit method in representation theory: proceedings of a
conference held in Copenhagen. August to September 1988JM. Duno.
N.V. Pedersen. M. Vergne. editors.
p. cm.- (Progress in m~lhemalics: v. 82)
··Held at the University of Copenhagen from August 29 to September
2. 1988. . in honor of L. Pul;ans~ky··-Prcf.
Includes bibliographical references.
ISBN·13: 978-1-4612-8840·4 e-ISBN-13:978-1-4612·4486-8
DOl: 10.1007/978-1-4612-4486·8
I. Orbit mctOOd- Congresses. 2 Lie groups- Congresses.
3. Representations of groups-Congn:sses. 4. Lie algebras
Congresses. 5. Representations of algebras-Congn:s!oCs.
6. Pul;anszky. L.-Congresses. J. Duno. Michel. II. Pedersen. N.V.
(Niels Vigandl III. Vcrgne, Michtlc. IV. Pukanszky. L.V.
K0benhavns UniversiteL VI. Series: Progre~ in mathematics
(Boston. Mass.): vol. 82.
QA387.013 1990
512".55--dc20
89·1&439
Printed on acid· free paper.
o Birkhiiuser Boston. 1990
Soficover reprint of the hardcover 1st edition 1990
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ISBN·13: 978·0·8176-3474·2 (Hardcover)
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Preface
The present volume contains the proceedings of the conference "The Or
bit Method in Representation Theory" held at the University of Copen
hagen from August 29 to September 2, 1988. Ever since its introduction
around 1960 by Kirillov, the Orbit Method has played a major role in
representation theory of Lie groups and Lie algebras. We therefore felt
it was desirable to devote a conference fully to this topic. As one of
the main contributors to the orbit method, L. Pukanszky, celebrated his
sixtieth birthday in November 1988, we decided to hold the conference
in honor of him.
At the conference were given 22 invited lectures of which 9 are pub
lished in these proceedings. Furthermore, L. Pukanszky, who unfor
tunately became ill just prior to the conference, has submitted a pa
per for the proceedings. The conference was organized by M. Dufio,
M. Flensted-Jensen, H. Plesner Jakobsen, N. Vigand Pedersen, and M.
Vergne. These proceedings are edited by M. Dufio, N. Vigand Pedersen,
and M. Vergne.
Most of the manuscripts were typeset (in 'lEX) at the Mathematics
Department, University of Copenhagen, and the final manuscript was
prepared there.
We heartily thank the following institutions for generous support:
Carlsberg Fondet, The Danish Natural Science Research Council, Den
Danske Bank, The Danish Mathematical Society, Julius Skrikes Stiftelse,
Knud H(3jgards Fond, The Danish Ministry of Education, Otto M(3nsteds
Fond and Thborg Fondet.
Copenhagen NIELS VIGAND PEDERSEN
September 1989
v
Contents
Preface ............................................................. v
List of Participants ................................................. ix
Towards Harmonic Analysis on Homogeneous Spaces
of Nilpotent Lie Groups ............................................. 1
LAWRENCE CORWIN
Orbites Coadjointes et Cohomologie Equivariante ................... 11
MICHEL DUFLO AND MICHELE VERGNE
Representations Monomiales des Groupes
de Lie Resolubles Exponentiels .................................... 61
HIDENORI FUJIWARA
The Surjectivity Theorem, Characteristic Polynomials
and Induced Ideals ................................................ 85
ANTHONY JOSEPH
A Formula of Gauss-Kummer and the
Trace of Certain Intertwining Operators ............................ 99
BERTRAM KOSTANT
The Penney-Fujiwara Plancherel Formula
for Symmetric Spaces ............................................. 135
RONALD L. LIPSMAN
Embeddings of Discrete Series into Principal Series ................ 147
TOSHIHIKO MATSUKI AND TOSHIO OSHIMA
Vll
Is There an Orbit Method for Affine Symmetric Spaces? ........... 177
GESTUR 'OLAFSSON AND BENT 0RSTED
On a Property of the Quantization Map for the
Coadjoint Orbits of Connected Lie Groups ........................ 187
L. PUKANSZKY
The Poisson-Plancherel Formula for a Quasi-Algebraic Group
with Abelian Radical and Reductive Generic Stabilizer ............ 213
PIERRE TORASSO
Vlll
List of Participants
G. Almquist, Lund E. Kehlet, Copenhagen
H. Haahr Andersen, Arhus M.S. Khalgui, Thnis
M. Andler, Paris A.A. Kirillov, Moscow
E. van den Ban, Utrecht S. Kleiman, Cambridge (USA)
J. Bang-Jensen, Odense N .J. Kokholm, Copenhagen
Y. Benoist, Paris T. Koornwinder, Amsterdam
C. Berg, Copenhagen A. Koranyi, New York
T. Branson, Copenhagen and B. Kostant, Cambridge (USA)
Iowa City K. Kumahara, Tottori
C.J.B. Brookes, Cambridge H. Leptin, Bielefeld
(England) R.L. Lipsman, College Park
J.-Y. Charbonnel, Paris W. Lisiecki, Warsaw
E. Christensen, Copenhagen J. Ludwig, Luxemburg
L. Corwin, New Brunswick L.-E. Lundberg, Copenhagen
B. Currey, Saint Louis B. Magneron, Paris
V.K. Dobrev, Trieste A. Melin, Lund
F. Du Cloux, Palaiseau I. Mladenov, Sofia
J .-Y. Ducloux, Paris T. Moons, Diepenbeek
M. Dufio, Paris O.A. Nielsen, Kingston
E.G. Dunne, Oxford A. Ocneanu, University Park
G. Elliott, Copenhagen G. 'Olafsson, Gottingen
R. Felix, Eichstatt D. Olesen, Roskilde
A. Fialovsky, Bonn J. B(lSriing Olsson, Copenhagen
M. Flensted-Jensen, Copenhagen B. 0rsted, Odense
H. Fujiwara, Kinki T. Oshima, Tokyo
R. Goodman, New Brunswick G. Kjrergard Pedersen,
F. Greenleaf, New York Copenhagen
U. Haagerup, Odense N. Vigand Pedersen, Copenhagen
G. Heckman, Leiden R. Penney, Purdue
J. Hilgert, Erlangen M. Poel, Groningen
J. Jacobsen, Arhus D. Poguntke, Bielefeld
H. Plesner Jakobsen, Copenhagen H. Prado, Iowa City
J .C. Jantzen, Hamburg M. Reimann, Bern
S. J(lSndrup, Copenhagen R. Rentschler, Orsay
A. Joseph, Paris and Rehovot H. Schlichtkrull, Copenhagen
IX
W. Soergel, Hamburg M. Vergne, Cambridge (USA)
H. Stetkrer, Arhus and Paris
T. Sund, Oslo N. Wildberger, Toronto
E. Thieleker, Tampa H. Yamada, Japan
P. Torasso, Poitiers S. Yamagami, Tohoku
x
Towards Harmonic Analysis on Homogeneous Spaces
of Nilpotent Lie Groups
LAWRENCE CORWIN1
The work described here is a joint project with Fred Greenleaf.
Let G be a simply connected nilpotent Lie group, with Lie algebra
(!S, and let K be a connected Lie subgroup, with Lie algebra Ji. We
would like to do harmonic analysis on K\G. More generally, let X be a
character of K; we would like to consider questions of harmonic analysis
for the unitary representation T induced from x. The first task facing us
is to understand what that means.
In order to begin to do harmonic analysis, we must solve a problem in
group representation theory: determining the direct integral decompo
sition of the representation T on 1fT = {f : G ---> C I f(kx) = x(k)f(x)
if k E K, f measurable, and If I E .c2(K \ Gn. That problem was
answered in [4] (see also [6]). Let £' E (!S* be such that for Y E Ji,
X( exp Y) = e2,..il' (Y), and set OT = £' + Ji.l C (!S*. The Kirillov corre
spondence gives a bijection between Gil and Ad* (G)-orbits in (!S*. The
decomposition of T can be described as follows:
(1)
where J.l is the push-forward of (a finite measure equivalent to ) Lebesgue
measure on OT to (!S* / Ad*(G) and thence to Gil, and n,.. is the number
of Ad*(K)-orbits in 0,.. nOT. (Here and in the future, 0,.. is the Ad*
orbit in (!S* corresponding to 71".) Furthermore, n,.. is either J.l-a.e. finite
or J.l-a.e. infinite; the former occurs iff for generic £ E Ji.l,
dim G . £ = 2 dim K . l.
(In fact, n,.. is then either J.l-a.e. even or J.l-a.e. odd; see [1].) We can
also decompose p as
r
(2) T e:: 7I"odv(O) ,
lOr! Ad· K
lSupported by NSF grant DMS-86-03169
1