Non Commutative Harmonic Analysis and Lie Groups: Proceedings of the International Conference Held in Marseille Luminy, 21–26 June, 1982 PDF
Preview Non Commutative Harmonic Analysis and Lie Groups: Proceedings of the International Conference Held in Marseille Luminy, 21–26 June, 1982
Lecture Notes ni Mathematics Edited yb .A Dold dna .B nnamkcE 1020 noN evitatummoC Harmonic sisylanA dna eiL Groups Proceedings of the International Conference Held ni Marseille ,ynimuL 21-26 June, 1982 Edited yb .J Carmona dna .M Vergne galreV-regnirpS Berlin Heidelberg New York Tokyo 1983 srotidE seuqaJ anomraC d' Universit6 Aix Marseille ,II de Departement seuqitam~htaM de Luminy 70 Route .L ,pmahcaL 13288 Cedex Marseille ,2 ecnarF Mich6le engreV CNRS de Universit6 Paris VII, REU seuqitam6htaM ,2 75221 Jussieu, Place Paris 05, Cedex ecnarF AMS Subject Classifications :)0891( 71 B35, 71 B45, 22D10, E30, 22 E46, 22 43A25 ISBN 3-540-12717-8 Berlin Heidelberg Springer-Verlag New York oykoT ISBN 0-387-12717-8 Springer-Verlag New Heidelberg York Berlin oykoT Library of Congress Cataloging in Publication Data. Main entry under title: Non commutative harmonic analysis and Lie groups. (Lecture notes in mathematics; 1,020) English and French. Proceedings of the Fifth Colloque "Analyse harmonique non commutative et grou- pes de Lie," held at Marseille-Luminy, June 21-26, 1982. .1 Harmonic analysis-Congresses. 2. Lie groups-Congresses. .I Carmona, Jacques, 1934-. .1I Vergne, Mich~le. III. Colloque =Analyse harmonique non commutative et groupes de Lie" (Sth: 1982: Marseille France) .VI Series: Lecture notes in mathematics (Springer-Verlag); 1,020. QA3.L28 no. 1,020 510s [512'.55] 11171-38 [QA403] ISBN 0-387-12717-8 (U.S.) This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1983 Printed in Germany Printing and binding: E}eltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210 ECAFERP eL em~iuqnic Colloque "Analyse euqinomraH noN evitatummoC et sepuorG ed Lie" s'est tenu a Marseille-Luminydu 12 ua 62 Juin 2891 snad le cadre ud uaevuon Centre International ed sertnocneR .seuqitam~htaM seL participants noteront euq la listedes articles publics ci-dessous en dnopserroc sap tnemet~Ipmoc xua secnerefnoc se~tnes~rp durant le Colloque. C'est le cas, ne particulier, pour sed travaux dont la publication d~taill~e ~tait euv~rp rap ailleurs, uo ed r~sultats s~d plusieurs coauteurs. ertuO les participants a cette rencontre, snonet suon ~ -remer cier I'U.E.R. ed Marseille-Luminy et le Centre International ed -nocneR tres seuqitam~htaM qui tno rendu possible la tenue ed ec Colloque, ainsi euq le secretariat ud Laboratoire ed seuqitam~htaM qui a assur~ la -aperp ration ud present .emulov seuqcaJ ANOMRAC Mich~le ENGREV NON COMMUTATIVE HARMONIC ANALYSIS AND LIE GROUPS BALDONI SILVA M.W. LZ index and unitary representations ................ 1 CARMONA Jaques Sur la classification des modules admissibles i rr(~duct ibles .................................................... 11 GUILLEMONAT Alain Representations sph~riques singuli~res ............... 35 HERB Rebecca The Plancherel Theorem for semisimple Lie groups without compact Cartan Subgroups ..................... 73 JOSEPH Anthony Completion functors in the Q- category ............ 80 KNAPP Anthony W. Minimal K- type formula .................................. 107 KNAPP A.W. ,8 The role of basic cases in classification : SPEH B. Theorems about unitary representations applicable to SU(N,2) .......................................... 119 LIPSMAN Ronald L. On the existence of a Generalized Weil Representation ..................................................... 161 WILLIAMS Floyd L. Solution of a conjecture of Langlands .................. 179 2 L index and unitary representations M. W. Baldonl Silva §I. Introduction. In this paper we investigate the representations that contribute to the index of the Dirac operator. Let G be a connected real semislmple Lie group with finite center, and let K be a maximal compact subgroup of the same rank as .G We assume that G C GC, where C G is a linear complexlficatlon of G. Let ~0 and ~ be the Lie algebra of G and K respectively. Passing if needed to a suitable double covering of G, we can assume that the isotropy representation K + So(~o/~) lift to Spin(~0/<). Since dlm(G/K) is even, the spin representation (s,S) of Spin(~0/~) breaks up into two half &, + ~ + spin representations s s, , and correspondingly S = S @ S-. Let n be an irreducible finite dimensional representation of K and let n V be the corresponding space. If we let ~ denote respectively the homogeneous vector bundle on G/K defined by the - K modules E- + = V ~ S- + then we may identify the 2 L cross- section of such a bundle as (L2(G)QE±) K and we have a corresponding Dirae operator + Z : (L2(G)OE+) K ÷ (L2(G)QE-) K defined as in [P]. Since ~+ is G invarlant, it drops down to an elliptic differential operator ~r on F\G/K with coefficients in the bundle ~± = ~+ ~-/r, r being a discrete torsion-free subgroup of G of finite covolume. The 2 L index of ~r is finite ([M], [A]). More precisely let A G denote the set of all equivalence classes of irreducible unitary representations ~ of 2 G and write mF(~) for the multiplicity of ~ in Ld(F\G). Then + mr(n) mid( E,)~(H(KmOH index ~F = Z +) - dim HomK(H(~),E-) ) H(~) being the representation space of ~. The aim of this paper is, fixed a K-module ,n to describe all the ~'s for which m(n,n) = dim HomK(H(~),E )+ dim HomK(H(~),E-) # 0 and to compute it, when the real rank of G is one. If ~ is an irreducible unitary representation with regular integral infinitesimal character, then [VI] gives a necessary condition for m(~,n) not to be zero. This condition says that is obtained in terms of parabolic induction via the Vogan-Zuckerman functor and allows one to proceed and compute m(~,n) as shownin theorem 2.1 below. If the real rank of G is one, this necessary condition is also sufficient in view of the classification of [B-B] of the irreducible unitary representations for real rank one groups. Thus one obtains a complete classification. If ~ has singular integral infinitesimal character we have to use in some cases a more direct approach. The new results in this paper are joint work with D. Barhasch. .2§ Notation and main results. fI H is a real Lie group we will write ~0 for the corresponding Lie algebra and J = (~0)~ for the complexificat ion. Let now G be a Lie group satisfying the same assumption as in the introduction, and let e be a Cartan involution of ~0 with Cartan decomposition ~0 = ~0 @ "oS Fix a Cartan subalgebra ~0 of yO contained in ~ and let K and T be the analytic 0+ subgroups corresponding to and . Let ~ = A(~,~) be the roots of ~ relative to ~, and let A~ )p = A(/,~) be the corresponding compact roots. fI V ~ ~ si a ~-invariant subspace, write A(V) for the roots of ~ in ,V and p(V) = 1/2 E ~(V ) .~ Fix a system of positive compact roots A+(~. Given a e stable (cf. V[ )]2 parabolic subalgebra q = £ + u of ~ with £ ~ +, we fix a positive system A+(£ ~) ~_ ~+(~) for Let L be the subgroup corresponding to £0 = ~ £ ~0" Irreducible K types or ~ L K types will be parametrized by the highest weight with respect to A+(~) or 4+(£~). + Fix a positive system of roots A for A such that c = 2p + + -- E eA+(~ ) ~ is dominant for A and normalize s and s such ± that on T', the regular elements of ,T the characters of s satisfy: 4 ch (s+-s-)IT' = H e( a/2-e-a/2). )~(+A-+AG~ We want to compute m(~,n) = dim HomK(H(~ ) ,Vn~S +) - dim HomK(H(~) ,VnQS-) for ~ a irreducible unitary representation of G and for ~ a fixed K-module, H(~) and V(q) be&ng the corresponding representation spaces. Without loss of generality we may and do exclude the case that is a discrete series or a nondegenerate limit of discrete series. Let ch ~ be the distribution character of ,~ and let ch be the character of ,n n being a K-module. Let D = ~aCA+ (e~/2_e-~/2) . We recall the following facts from [A-S]. If X denote the infinitesimal character of ,~ then )i m(~,n) # 0 only if X = X ~+Pc )2 m(~,n) = (-l)qa where a is defined by = Z^ a ch y q = i/2(dim -dim~) ch(s+-s-) ch~ 'TI y~K 'TI Let @ be a system of positive roots that makes n + Pc dominant, and set ~(~) = n + Pc - P(@), where p(~) : 1/2 E ~@ .~ We denote by ~* the Vogan-Zuckerman functor as defined in [V2]. Theorem 2.1. Suppose G has real rank one and n + Pc is regular. Then the irreducible unitary representations for which m(~,n) # 0 are given by the following procedure. Let ~ = ~ + u be a e-stable parabolic subalgebra such that = ~(n) satisfies (~,~) = 0 for ~ ~ A(A) and (~,~) > 0 for sL ~A(u). Then ~ =~ ~% and m(~,n) = (-i)--(-i) '-A+~', where qL = i/2(dim .~ - dim ~0~ )' and s = dim(uO~). Remark. In [VI] it is proved in full generality, i.e., without the rank one assumption, the fact that if m(~,n) # 0 then there exists a e-stable parabolic subalgebra and a '% as in the theorem so that ~ = ~s ~, L This is the necessary condition to which we alluded in the introduction. Proof: By the classification of the irreducible unitary representations of real rank one groups of [B-B] all the repre- sentations in the conclusion of the theorem are unitary and conversely all the unitary irreducible representations with regular integral infinitesimal character are obtained in this way. Therefore it is enough to prove that any representation of the sL form ~ ~%, with 'X as in the theorem has m(~,n) = 0 unless '~ = ~ and in this case m(~,n) = (-l)qL(-l) I-A+O~I. Thus suppose that ~ = ~ + u is a e-stable parabolic subalgebra and '% e T A is such that (%',=) = 0 for a ~ A(~) and (%',~) ~ 0 for ~ e A(u). Let ~(~) be a positive system for A(A) that makes 2p(£~) = 7 A+(£f]~) ~ dominant, and set = $(£) U A(u) . Then ~ = ~s ~, e is an irreducible unitary iL representation by [B-B] and ~ ~, = 0 for i < s by [V2]. We want to compute m(~,n) for such a representation. L For any irreducible representation ~ of L we denote by e(~ L) the virtual character e(~ L) = E (-l)ich(~ L) i and refer to it as the Euler characteristic. The Euler characteristic is a well defined map from the ring of virtual characters of L to the ring of virtual characters of G. Furthermore it commutes with coherent continuation. Suppose ch ~, = E jc ch L(yj) where cj are integers and ~L(vj) is a discrete series or a principal series representation. Then e(~,) L iff cje(L(yj)). We note that yj + p(u) = w(%'+0(~)) with w E W(A), W(A) being the Weyl group of .£ Since (%'+p(~),~) > 0 for ~ ~ A(u), we obtain (yj+p(u),~)) 0 for ~ ~ A(u). It follows from [V ,2 theorem 8.2.15] that
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