Table Of ContentLecture Notes in
Mathematics
Edited by A. Dold and B. Eckmann
728
Non-Commutative
Harmonic Analysis
Proceedings, Marseille-Luminy, France,
June 26 to 30, 1978
Actes du Colloque d'Analyse Harmonique
Non Commutative
Edited by
Jacques Carmona and Michele Vergne
Springer-Verlag
Berlin Heidelberg New York 1979
Editors
Jacques Carmona
Universite d'Aix-Marseilie II
U.E.R. Scientifique de Luminy
70. Route Leon Lachamp
F-13288 Marseille Cedex 2
Michele Vergne
Universite Paris VII
U.E.R. de Mathematiques
2, Place Jussieu
F-75221 Paris Cedex 05
AMS Subject Classifications (1970): 16A66, 17 BlO, 17 B35, 20G20,
22E30, 22E45, 32M15, 35ElO, 43A80, 43A90
ISBN 3-540-09516-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-09516-0 Springer-Verlag New York Heidelberg Berlin
Library of Congress Cataloging in Publication Data
Colloque d'analyse harmonique non commutative, 3d, Marseille, 1978,
Commutative harmonic analysIs, (Lecture notes in mathematics; 728) English or French.
Bibliography: p. Includes index. 1. Harmonic analysis··Congresses. 2. Lie algebras-
Congresses. 3. Lie groups--Congresses. I. Carmona, Jacques, 1934-II. Vergne, Michele.
III. Title. IV. Series: Lecture notes in mathematics (Berlin); 728.
OA3.L28 no. 728 [0A403] 510'.8s [515'.2433]
ISBN 0-387-09516-0 79-17579
This work IS subject to copYright. All rights are reserved, whether the whole or
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publisher, the amount of the fee to be determined by agreement with the publishel
f.) by Springer-Verlag Berlin Heidelberg 1979
2141/3140-543210
PREFACE
La troisieme rencontre diAna lyse Harmonique Non Commutative
sur les Groupes de Lie a eu lieu a Marsei IIe-Luminy, du 26 au 30 Juin 1978,
dans Ie cadre des activites du Centre International de Rencontres Mathema
tiques.
Ce volume contient Ie texte des Conferences presentees durant
Ie Colloque a Jlexception de celles qui concernaient des travaux deja publies
ou dont la publication etait prevue par ai IIeurs.
Outre les participants il cette rencontre, nous tenons a remerciel
JlU. E. R. Scientifique de Marsei IIe-Luminy et Ie Centre International de
Rencontres Mathematiques qui ont rendu possible la tenue de ce Colloque.
Jacques CARMONA
Michele VERGNE
TABLE DES MATIERES
Dan BARBASCH Fourier Transforms of some invariant
Distributions on a Semisimple LieGroup
and Li e Algebra... • • • • • •• ••• •• • •• • •• •• • • 1
WeitaCH.ANG Global Solvabi lity of Bi-invariantDiffe-
rentialOperators onSolvable Lie Groups 8
Laurent CLOZEL "Base Change" geometrique :
Relevement de la serie principale de
G L(n, C fIR) • • • • • • • • • • • • • • • • • • • • • • • • • • • • •• 1?
Jacques DIXMIER Sur la methode des orbi tes ••••••••••••••• 42
<C)........
Michel DUFLO Polyn8mes de Vogan pour SUn, 64
Mogens FLENSTED-JENSEN On a fundamental series of representations
related to an affine symmetric space ••••••• 77
Hans PlesnerJAKOBSEN Higher Order Tensor Products of Wave
Equations •••••••••••••••••••••••••••••• 97
Kenneth D. JOHNSON & GeneralizedHua Operators on Bounded
•
Adam KOR.ANYI Symmetric Domains •••••••••••••••••••••
Anthony JOSEPH W-ModuleStructure in the Primitive
Spectrum of the enveloping Algebra of a
Semisimple Lie Algebra •••••• 116
0 0 00 0 0 0.00.
Masaki KASHIWARA Functions on the Shi lov Boundary of the
Michele VERGNE Generalized Half Plane 136
00000.000000000000
Masaki KASHIWARA K-types and SingularSpectrum 177
Michele VERGNE
2
Henri tv1OSCOVICI Avanishing Theorem for L - cohomology
in the ni (potent case.................... 201
•
Niels Vigand PEDERSEN Semicharacters of Solvable Lie Groups
00
Stephen RALLIS The Eichler Commutation Relation and the
Continuous Spectrum of the Wei I Represen-
tation 211
00000000000000000.0 ••••• 00.00...
• to appear elsewhere
FOURIER TRANSFORMS OF SOME INVARIANT DISTRIBUTION
ON SEMI SIMPLE LIE GROUPS AND LIE ALGEBRAS
Dan Barbasch'~
Institute for Advanced Study
Princeton, NJ 08540
Let G be a connected semisimple Lie group and u EGa unipotent element.
Then the orbit of u by the adjoint action of G defines an invariant tempered
distribution which we denote by Tu' In this paper we propose to investigate the
explicit relation between the distribution T and the measure s as soc ia ted to
u
semisimple orbits, more precisely the distribution defined by Harish-Chandra.
These results make it possible to calculate the Fourier transform of T in the
u
sense of Harish-Chandra, i.e. to expand T in terms of distributions coming from
u
characters of irreducible unitary representations. As motivation for this problem
we consider the following situation. Let ~ ~ G be a discrete subgroup such that
2
G/r is compact. Let n be the left regular representation of G on L (GII).
Then the well known Selberg trace formula states that
(1) tr n(f)
where f E Coo(G) and C is the set of conjugacy classes of elements in by
c
the adjoint action of G. An important problem for harmonic analysis and number
theory is to find the multiplicities of irreducible representations occuring in
2
L (GII). This can be done by observing that
(2) tr n(f): l: mw tr w(f)
wE C(G)
where e(G) is the set of equivalence classes of irreducible unitary represent-
ations and the multiplicities. If we can find the Fourier transform in the
sense of Harish-Chandra of each i,e. find an explicit formula
(3) tr w(f)d\-l(W)
then we can calculate the explicitly.
Due to the work of Arthur ~ 1J, Warner : 2 1 and Warner and Osborne r 4 J on
the Selberg trace formula for vo 1 (G IT') < 00, distributions of the form with
Y unipotent occur. Formulas of type (3) were obtained for y semisimple by
Sally & Warner I 3 J, Herb ~ 5 J and Chao [ 6 J. The corre sponding formulas for Y
*
Supported in part by a National Science Foundation Grant.
2
unipotent are simple consequences of these formulas and the relations in theorems
2 and 3.
The measures Ty have obvious analogues for the Lie algebra. Let B be
the Cartan-Killing form. Then we can consider the Euclidean Fourier transform
for any x E ?-, with respect to B. Then is an invariant eigendistribution
for G-invariant constant coefficient differential operators. By using the results
of Rossmann [ 9 J it is then possible to obtain formulas for when x is nil potent
These questions are completely answered in the case real rank ~ in
Barbasch :8J. The general<i.zations proved here are inspired by the results in that
paper.
Before we state the main results we prove a general lemma.
Lemma 1: Let ~ = ~+ ~ + ~ be a parabolic subalgebra with its corresponding
Langlands decomposition and P = MAN the corresponding parabolic subgroup. Assume
that there is X E n. such that p.x is open and dense in ~. Let H E (It be
such that cent"j H = 7rI. + O"l. and r = dim cent";} H. Then
(4) lim t(n-r)/2T (f) TI(H) -lcXTx (f)
t~ 0+ tH
where TI(H) Idetn(ad H)I and C depends only on the normalization of the
x
measures.
Rr££i: This is well known. We sketch a proof for completeness. Let
be a Cartan decomposition and K the group corresponding to 1.
Then we may assume that f(k.Z) = feZ) for any k E K. Since G = K·P we can
write
J
(5 ) TX(f) f (p·X)di-l(p) f(Z)dZ
P n
and
.r
(6) TtH(f) f(tn·H) TI(tH)-l f(tH + Z)dZ
N n
n-r
Since dim n = 2' the proof is complete.
As a consequence we have the following theorem.
Theorem 2: Let 1-= sl(n, 11.) and [X, H, Y} be a Lie triple.
a) If ad H has odd eigenvalues then there is a parabolic ~X such that
X satisfies the conditions in Lemma 1.
b) If ad H has even eigenvalues only, then
. (n-r) /2 _
(7) 1l.IJl t T (X-Y) (f) - cXTX (f)
t~O+ t
~: In view of Lemma 1 we have to determine which X E ~ nilpotent
satisfy the condition in Lemma 1. Of course part b) is more difficult because
X-Y is an elliptic element. The existence of such a formula is motivated by the
3
fact that the conjugacy class of X (or equivalently [X, H, Y}) with respect to
G is determined by the G-conjugacy class of X-Yo (A proof of this can be found
in Barbasch [ J. It was first observed by R. Rao and then proved by B. Kostant.)
We first consider conjugacy classes of nilpotents with respect to GL(n).
Representatives are given by the Jordan canonical form. This consists of boxes of
the type
(where nj could be one). Then the conjugacy classes of nilpotents are para
metrized by partitions
On the other hand let dP be a parabolic sub algebra such that
m+at. where ml + .•. + ~ = n
Then it is straightforward to see that the nilpotent with the open orbit in n
is given by ~ boxes of size k, (~-l -~) boxes of size k-l, etc. Since
this exhausts all possible conjugacy classes of nilpotents, we can apply Lemma 1
in this case. As a consequence we also get the fact that any nilpotent
X E sl (n, R) has an open orbit in some n for a parabolic.
The conjugacy classes of nilpotents with respect to SL(n) are the same as
for GL(n) except for the case when all the n '5 are even. In that case X
j
and -X are not conjugate with respect to SL(n). In this case, since H is
formed of diagonal boxes
n.-l
J
and it becomes clear that ad H has only even eigenvalues. (This is relevant for
example for the fact that dim Cent H = dim Cent X = dim Cent (X-Y) as can be
seen from a calculation of the representations of 51(2) ~ [X, H, Y} occuring
in ,}.)
We now recall that Harish-Chandra has done a very deep analysis of the
distribution
4
Here L is a Cartan subgroup, bEL' an element in the regular set and ilL (b)
a certain determinant function. A convenient reference for basic results about
Ff, for the group as well as their analog for ~, can be found in Varadarajan
J. In particular \Oe need thee [ae t that, for any Z semisimple, Ttz(f)
has an asymptotic expansion in t. In particular the first nonzero term in this
expansion is a measure. We have to show that TX is the only distribution
occuring in the first nonzero term. He note that, if X corresponds to the
partition n 2 n 2 ... 2 n then X-Y can be conjugated into an element of
l 2 k
the type
• .•n 1-l
-(nl-l).· •
etc.
Since the centralizer of this element is contained in the Levi component of a
parabolic defined by the size of the boxes involved, we can write down the
measure corresponding to the orbit of t(X-Y) explicitly (in terms of the
decomposition G = K· P) and then calculate the limit.
This completely solves the problem of calculating Fourier transforms of
cr
nilpotent orbits in the case = sl(n, R).
The case of a general semisimple Lie algebra is much more complicated. In
the previous formulas the semisimple orbit had the same dimension as the nilpotent
orbit. As can be seen from Barbasch ~8J this cannot always be done, even when
'1
real rank = 1.
The next theorem, treating some special nilpotent" in ";1- sp(n) illustrates
what the difficulties are.
Theorem 3: Let 1= sp Cn, R) •
]
a) Let X p,q -_00 Ao-j- and Zp ,q -r -A0 A0-] where A [~p -~ and
q
p + q = n. Then
(8) lim tn(n+l)/2 T (f) (f)
tZ
t~ 0+ p,q
b) Let rO A-I where A 0. + 3 n - 1. Then
'0 0
n-l
(9) 1 d I tn(n+l)/2 L: (-1) Y-;3T . Cf) TX ~ (f)
'2 dt t=O+ y=n-S tZY+l,n_Y_l cr.,1:),0
5
Proof: The proof of these formulas follows by finding a transverse to the
semisimple orbit and expressing the measure with respect to that transverse. The
simplest case is when A is positive definite. Then the KAK decomposition is
sufficient. Rather than go through the proof we give a different one for
'} = sp(2, R) which illustrates the role played by the geometry of the orbits.
In this case the conjugacy classes of nilpotents are given by
+r 00]
~o 0 0] 0 0 1} 0
_ + 0 f0i 0 2 [~ 1 0 0 o 0
1) X-- OOOO ' H = 0 -1 y" - 0 0 o -{3
o 0 -(3 0 0 0 -3 0 2 o 0
2) x =
where A = +1, A = -I
3)
where A =+-[01 00 J •
The cases listed in 2 and 3 with + and are not conjugate. We
consider case 3 with +.
Let Zy = cent'] Y. Then Q = x + Zy is, according to some unpublished
results of R. Rao, a transverse to the adjoint action of G in particular to
any orbit intersecting it. Let ~ E Coo(G X U). Then there is a well defined
c -
correspondence (integration on fibers) ~ ~ f~ where f~ E c:(G'Q) which is
onto. Then for Z E Q,
(9) f f (xZx -1 )dx f f 13 (x, u)dw(u)dx
G/Gz 13 G Q n o(Z)
Let r = dim cent" X = dim Q. Then
(10) f~(tv) = t-(n-r)/2 fl3 (v)
t
where and Y = exp ( -110 g tH) and
t 2
ut = X + tyt·uy•
We now calculate the intersection O(X - Y) n Q. X - Y satisfies the
2
equation x(x + 1) = O. We get two orbits satisfying this equation:
6
[!
0 1 0 1
0 0 ~] [~ a 0 Jl 2
d E.It and a + bc + 1 0
d 0 0 0
-d2 0 c 0
The first one is o (X - Y) r, Q., the second is O(ZO) n Q. where Z0 = [-Io OI J·
The action of the centralizer of [X, H, Y} ("" sp (1» leaves Q invariant.
Then identifying sp(l) with sl(2, 11.), the first orbit is the orbit of
which has centralizer the same as in sl(2, It) and the second orbit
corresponds to
which corresponds to [ 0 lJ in sl (2, I) •
-1 0
Since O(X) n!:!,= (X}, the measure corresponding to O(X)
on this transverse is dx ® Ox where dx is the Haar measure and Ox the delta
function at X.
Thus by using formulas (8) and (9) we get
3 f
t fj3(tx' (X-Y»dx = t
G/Gx_y G
f
Since Xo is nilpotent we can never get j3(x, X)dx in the limit no matter what
G
differential operator we use.
On the other hand, +or Zo we get
t3 f fj3(tx'ZO)dx = t f u j3(x, X_t2y + tg'ZO)dgdx
G/G G SL(2) /G '
Zo Z0
But then the formula stated in the theorem reduces to the well known statement that
FBfC O; w) = d(O), c 1- 0
for the Lie algebra sl(2, It). In order to have a complete proof we need to do
1-
some more work since G'Q is open in but not equal to it. We omit the details.
Finally, we remark that certain invariant distributions with support in the
cone of nilpotent elements are associated to irreducible representations. The
relationship involves the euclidean transforms of semisimple and nilpotent orbits
and limits measures that are considered in this paper. We refer to Kashiwara &
