182

В Kostdnt

aeF^a^ . Thus if Fis an irreducible Harish-Chandra module, say with respect to a representation cr, then V" is another such module with respect to a sentation a" where if юеУ = У and ueU one puts o''[u)v = o{a~^[u))v. Now let V', i = l,2, ...,d, a^eF^^^, have the property that V is equivalent to K"^ for only one value of i and put

( 6 . 8 . 1 ) V=V'@-''@V^

where V'^V"\ Since aeG([ one has Uy=Uy in the notation §3.1; and hence V admits an infinitesimal character.

If we forget about continuity one has the following result. It was proved first by Casselman and Zuckerman for the case where G = Sl(n, IR).

Theorem 6.8.1. Let G be any connected quasi-split semi-simple Lie group and let U be an enveloping algebra of the Lie algebra of G. Let V be any irreducible Harish- Chandra module for U and let Uy^U be the annihilator of V in U. Let V be defined by (6.8.1) and let V' be the full dual to V so that V' is a U-module. Let Wh V' be the space of Wlnttaker vectors in V'. Then Wh V' фО if and only if Uy is a minimal primitive ideal. Furthermore in such a case one has

dimWhF' = |^|

where W is the then (little) Weyl group W{q, a).

Proof If Uy = Uy is not a minimal primitive ideal then Wh V' = 0 by Theorem 3.9. Assume Uy is a minimal primitive ideal. By the subquotient theorem we can assume that there is a quotient component Я of a principal series representation such that H^ = V. But then {H\= V. Furthermore as in the proof of Theorem 6.7.2 we can assume Н = Щ ^ for some (a, v)eM^^^ x a' where 71^ ^ is a principal series representation of G^^^ and the notation is as in Proposition 6.6.3. One then has an isomorphism (X^ vLA^a, v)m-1 = ^ of (7-modules. But now as noted in the proof of Theorem 5.5, Z^^ is a finitely generated U{n) module and hence

dim Wh (X;, „) = X dim Wh ((X,, J,/(X,.,),_ ,)')

I

by (4.5.11) and (4.5.14). But dim Wh((X^ ДДХ^ Д_ i)') = 0 for i^m by Theorem 3.9 and Proposition 6.6.3. Thus dimWh(Z; J = dimWhF'. But dimWh(X; ,) = 11^1 by Theorem 5.5. Q.E.D.

6 . 9 . Now ^/ is a character of N and hence it defines by induction a sentation of G. One natural question is, what is the "decomposition" of this representation. Towards this end we contribute the next theorem. For simplicity, instead of just considering rj we take into account the action of F^^^. For any aeF^^^ let r|'^ : N-^ (C Ы the unitary character defined by putting r]''{z) = rj{a''^(z)) for aeF^^^ and zeN. Now let a^eF^^^, i = 1, ..., /c, be representatives of the cosets o^F in F^3^ where F = F^^^n(G cent G^) and let

rj^ - . N - ^CL'