Table Of ContentLecture Notes ni
Mathematics
Edited by .A Dold and .B Eckmann
664
evitatummoC-noN
cinomraH sisylanA
Actes ud Colloque d' Analyse Harmonique
noN ,evitatummoC ,ynimuL-elliesraM
1 ua 5 Juillet 1974
Edited yb .I. ,anomraC .I. Dixmier
dna .M Vergne
Springer-Verlag
Berlin. Heidelberg (cid:12)9 New York 1975
Editors
.forP Jacques Carmona
Universite d' Aix-Marseille
tnemtrap6D de Math6matiques
70 Route L~on Lachamp
13288 Marseille Cedex 2/France
Prof..laques Dixmier
Universit6 Paris VI
.R.E.U d' Analyse
~tilaborP et applications
4 Place Jussieu
75230 Paris Cedex 05/France
.forP Mich~le Vergne
Universit6 Paris VII
.R.E.U de seuqitam6htaM
2 Place Jussieu
75221 Paris Cedex 05/France
Library of Congress Cataloging in Publication Data
Colloque d' analyse hammonique non convnutative,
Marseille, 1974.
Non commutative harmonic analysis.
(Lecture notes in mathematics (Berlin) ; 466)
English or French.
.i Harmonic analysis - -CongTess es. .2 Lic
algebmas--Congmesses. 3. Locally compact groups--
CongTesses. .I Carmona, Jacques, 1934-
II. Dixmie~, Jacques. III. Vergne, Mieh~le.
IV. Title. V. Series.
QA3.L28 no.~66 QA405 515'.785 75-19252
AMS Subject Classifications (1970): 16A66, 17 B20, 17 B35, 17 B45,
20G05, 22D10, 22D12, 22E45, 22E50, 31A10, 35P15, 43A05,
43A65, 82A15
ISBN 3-540-07183-0 Springer-Verlag Berlin (cid:12)9 Heidelberg (cid:12)9 New York
ISBN 0-387-07183-0 Springer-Verlag New York" Heidelberg (cid:12)9 Berlin
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ECAFERP
Un colloque cl~Analyse Harmonique Non Commutative a eu lieu
Marseille-Luminy, clu 1 au 5 julllet 1974~ clans le caclre cles activit~s au Centre
international ae Rencontres Math~matiques avec le soutien cle I~LI. E. R. cle
Luminy (Unlversit~ cl=Alx-Marseille).
Le present volume contient le texte aes e(cid:141) que les conf~ren-
clefs invites ont bien voulu nous faire parvenir. La liste aes articles ne
corncicle pas exactement avec cel le cles exposes pr~sent~s durant l e Colloque.
Clest le cas en partlculier aes conferences cle Kostant, Rai"sp WallaCho
Outre les participants b cette rencontre, nous tenons b remercier
IILIo Eo R~ cle I_uminy et le Centre International cle Rencontres Math~matlques qui
ont renclu possible la tenue ae ce colloque~ ainsi que le secretariat au D~parte-
ment de Math~matique-lnformatlque oe Luminy qui a assur~ la preparation mat~-
rielle ae ce volume.
Jacques CARMO NA
Jacques DIXMIF_.R
Mich~le VE RG NIE
TABLE DES MATIERES
Robert J. BLATTNER Intertwining operators and the half-
density Pairlng . ......... . .......... ... 1
Jonathan BREZIN Geometry end the Method of Kirillov ..... 31
Jacques CARMONA Sup les fonctions C de Harish-Chandra 62
w
Nicole CONZE-BERLINE Sur terrains quotients de I lalg~bre
enveloppante dlune alg~bre de Lie semi-
simple 13
~ 1 7 6 1 7 6 1 7 6
Jacques DIXMIER Id~aux Primitifs compl~tement premiers
dens Ilalg~bre enveloppante de s/3, CJ 83
Michel DUFLO Semigroups of complex measures on a
locally compact group ,,.oooo~ 65
Mogens F LENSTED-JENSEN Spherical Functions and Discrete Series 56
Paul GERARDIN Groupes r~ductifs et GPoupes r~solubles 97
Daniel KAS TLER Stability and Equilibrium in Quantum
Statistical Mechanics o~176176176176176176176176 68
Bertram KC~TANT VePma Modules and the Existence of Quasi- ~'~
InvaPiant Differential Operators 00.~176176 101
No01 LOHOLIE SuP la racine carrie du noyau de Poisson
dans les espaces sym~triques 921
o o o o o o o o o , .
Marie-Paul~ MALLIAVlN & Dlagonallsation du syst~me de de Rham-
Paul MALLIAVIN Hodge au dessus dlun espace Riemannlen
homog~ne ....e.....~ oo.eo..~ 531
VI
Mustapha RAIS Action de cer-tains groupes dans les espaces de
fonctions C oo ............. 147
(cid:12)9 oeeoeooooooeoo,
Francjois RODIER ModUle de WhTttaker- et car'act~r-es de
PepP~sentations ........................... 151
Wi Ifried SCHMID Some Remarks about the discrete series
char'actePs of Sp (n t )R .................... 172
David J. SIMMS An application of polar'isations and half- form 195
Hugo ROSSI ~E Continuation analytique de la s~Ple discr'~te
Mich~le VERGNE holomorphe .............................. 891
Notan Ro WALLACH On the Unitarizability of Representations
with Highest Weights oooooo ..... ~ ......... 226
INTERTWINING OPERATORS AND THE HALF-DENSITY PAIRING
Robert J. Blattner
.i Introduction.
These notes give the details of a result stated inaprevious paper
(2, Section 7(b)) which asserts that the Knapp-Stein intertwining
operator 3 linking the principal series representations of SL(2,~)
coming from characters of a split Borel subgroup differing by a Weyl
reflection can be constructed from the so-called half-density pairin$
(2, p. 152), due to Kostant and Sternberg, of geometric quantization.
Section 2 of the present paper recalls a construction of 4,
which manufactures a hermitian line bundle with connection (L,V) given
an orbit X of the eoadjoint representation of a simply connected Lie
group G and a point p e X. We also derive some results relating non-
vanishing sections of L and certain 1-forms they define. The
principal result here is Proposition 2.6. Section 3 is devoted to
working out our example. G is the universal covering group of
SL(2,~) and X is a hyperboloid of one sheet. Whereas Knapp and Stein
fix a Borel subgroup of G and let the Weyl group act on the inducing
representation, the geometry of our situation leads us to use the
Weyl group to move the Borel subgroup while holding the inducing
representation fixed. Section 4 comments on our result and raises
questions that need to be settled concerning the half-density pairing.
In what follows we shall assume the reader to be familiar with
Kostant's fundamental paper 4 on geometric pre-quantization and
also with the half-density pairing as set forth in 2.
This work was supported in part by NSF Grant GP-43376.
2. Remarks on the orbit method.
Let G be a connected, simply connected Lie group with Lie algebra
g. Let g* be the dual of g and let ad* denoted the coadjoint repre-
sentation of G on g*: ad* x = (t adx) -I. One constructs representa-
tions of G by the orbit method by choosing an orbit X of ad* in g*,
choosing an ad*-invariant polarization F of X, and constructing a
Hilbert space upon which G acts from these data. For details, see
Auslander and Kostant (i, Section 1.5).
Suppose F is real. Then the construction in (2, Section )3 is
equivalent to that given in i. We use this alternative construction
in the present paper. Let p e X and let Gp : {xEG : (ad*x)p = p}. As
detailed in (4, pp. 197-199), we may construct a complex hermitian
line bundle with connection (L,V) as follows: Let X be any unitary
character of Gp such that d X = 2nip (if one exists). Form the complex
hermitian line bundle G x ~ over G using the usual hermitian structure
on r Let G act on G x ~ on the right by means of
P
(2.1) (x,k)y = (xY,X(y)-ik)
and on G by right translation. Then G (cid:141) ~/Gp becomes a complex
hermitian line bundle L over G/Gp. G/Gp is identified with X in the
usual way by means of the map ~ : x ~ (ad~x)p of G onto X. Let
also denote the canonical projection of G x ~ onto L.
Let ep be the left invariant 1-form on G whose value at 1 is p.
Let ~(cid:141) = C- {0} and let L x = L- {0-section}. Then (4, p. 199) the
1-form (~p, ~ ) on G (cid:141) cx is ~*~ for a unique connection form
on L x , and the connection V associated to e by means of
(2.2) V~s = 2~i<s~,~>s(q),
where q e X, s is a section of L x near q, and ~ e (TX)q , leaves the
3
hermitian structure of L invariant.
Let f be a ~-valued function on G. Then there is a section sf
of L over X such that ~(x,f(x)) = sf(~(x)) if and only if
(2.3) f(x,y) = X(y)-if(x) for x (cid:12)9 G, y (cid:12)9 Gp,
and in that case sf determines f uniquely. Moreover sf never vanishes
if and only if f never vanishes, and in that case ~*sf*~
= r 2-~ ), where r = (x,f(x)). Setting ~(f) = sf*~, we get
+ i df
(2.4) ~*s(f) = ep 2-~-F"
i df
(Note that neither ep nor ~-~ -~- come from 1-forms on X via ~ : G § X
although their sum does.)
Which 1-forms 6 on X are of the form ~(f) for a smooth never-
vanishing function f on G? Clearly we must have d8 = m, where m is
the canonical symplectic 2-form on X due to KirillOV (4, p. 182).
Moreover we must have, by (2.3) and (2.4),
-2~if~(~*8-~p)
(2.5) X(Y) : e for y (cid:12)9 G .
P
^
where y is any piecewise smooth arc in G from 1 to y. The following
proposition says that this is sufficient.
Proposition 2.6: Let 8 be a smooth 1-form on X such that d8 = m.
Then ~'8 - e is closed on G, so that integrations of it are independent
P
of path. Set
2~i/x(~.8 - ~p)
(2.7) f(x) = e I for x ~ Gp,
where the integration is over any arc from 1 to x. Then
(a) (flGp) -I is a character A of Gp,
(b) dA = 2~ip, and
(c) f(xy) : A(y)-if(x) for x e G, y e Gp.
Proof: To show that ~*B - a is closed, we show that da = ~*m. Let
P P
~, ~ be two left invariant vector fields on G. Then according to
(4, p. 96) dep(~,n) = <ep,n,~>, since <~p,~> and <ep,O> are con-
stant functions. But this is just the definition of ~*m. Since G is
simply connected, integrations of ~'8 - ap are independent of path.
Our next step is to show
-2~i/xY(~*B-~p)
(2.8) A(y) = e x for all x (cid:12)9 G, y (cid:12)9 G .
P
This will imply that
2~i/x(~*8-~p) 2~i/xY(~*8 - ~p)
f(xy) = e 1 e x = A(y)-if(x)
for x E G, y (cid:12)9 Gp, which gives (c) and, specializing x to Gp, (a)
as well. Now IXy~p is obviously independent of x since ~p is left
X
invariant. So let ~ be any arc in G from 1 to y ~ Gp and set y = ~y.
y is closed. Set r = I 8 for x (cid:12)9 G and let ~ (cid:12)9 g. ~ defines
(ad*x)-iy
a vector field, also called ~, on X by means of
(~)(q) : d~ ((ad*exp t~)-lq)lt=0
for ~ (cid:12)9 C(X) and q (cid:12)9 X. Then
(2.9) (~)(x) = I(ad,x)_ly 8(~)8 for x ~ G,
where 8(~) denotes the Lie derivative with respect to .~ Now 8(E)8
= di(~)B + i(~)dS, where i(~) is the left interior product with respect
to ~. The first term is exact. The second term is just i(~)m, which
is also exact by (4, Proposition 5.3.1). Since the integral in (2.9)
is over a closed arc, it follows that ~r = 0 for all ~ (cid:12)9 g. Since G is
connected, r is constant, which proves (2.8).
As to (b), let y (cid:12)9 (Gp)0, the identity component of Gp, and for
this y let ~ be as above, with ~ in (Gp) .0 Then y = ~o~ is a constant
arc, which implies that
2~iI~p
(2.10) A(y) = e for y ~ (Gp) .0
Differenting (2.10) with respect to ~ ~ 9p, the Lie algebra of Gp, and
evaluating at y = i gives (b).
We close this section by recalling two facts relating sections sf
and their corresponding 1-forms ~(f). Firstly, let fl and f2 be two
never-vanishing functions satisfying (2.3). Then (4, Proposition
1.9.1) implies
e 2~ir where de = s(f2 ) -~-~.
(2.11) <sf2,Sfl>
In particular, Nsfll is constant if and only if ~(f) is real.
Secondly, let F be a polarization of X. Then sf is covariant constant
with respect to F if and only if <e(f),F> = .0
.3 The example.
Let G be the simply connected covering group of SL(2,~). g
a b
consists of all matrices of the form c -a with a,b,c eR, which
we will denote by (a,b,c). We have the commutation rule
(3.1) (a,b,c),(a',b',c') = (bc'-cb', 2(ab'-ba'),2(ca'-ac')).
g possesses an ad G invariant symmetric bilinear form B defined by
1
(3.2) B((a,b,c),(a',b',c')) = aa' +~(bc + cb').
Using B to identify g with g~, we replace ad ~ by ad in the orbit
method.
Let us look at the orbit X = {~ E g : B(~,~) = 12}, where I ~ .0
We calculate the Kirillov form ~ on X.
