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On the unitary dual of some classical Lie groups. (English) Zbl 0692.22007
For complex classical Lie groups Barbasch gave a complete classification of the unitary dual. The parametrization is given in terms of representations containing a fundamental K-type. These representations are either unipotent, complementary series or edges of complementary series [cf. D. Barbasch: Invent. Math. 96, 103-176 (1989; see the preceding review Zbl 0692.22006)].
For real groups there is no complete understanding of the unitary dual up to now. However, the definition of special unipotent representations with integral infinitesimal character $$\gamma$$, given for complex Lie groups by D. Barbasch and D. Vogan [Ann. Math., II. Ser. 121, 41-110 (1985; Zbl 0582.22007)], applies for real groups as well. Vogan conjectured that a unitary representation of a real reductive Lie group can be obtained by cohomological parabolic induction from a special unipotent representation of a subgroup. If the integral infinitesimal character $$\gamma$$ is regular then the special unipotent representations involved are one dimensional and the conjecture reads as follows: Given an irreducible unitary Harish-Chandra module X with regular integral infinitesimal character. Then there are a $$\theta$$-stable parabolic subalgebra $${\mathfrak q}$$ and a unitary one dimensional character $$\lambda$$ of the Levi L of $${\mathfrak q}$$ such that $$X=A_{{\mathfrak q}}(\lambda)$$, i.e. X occurs in the complete list of irreducible unitary representations with non-vanishing relative Lie algebra cohomology given by D. Vogan and G. Zuckerman [Compos. Math. 53, 51-90 (1984; see the following review Zbl 0692.22008)]. In the paper under review this conjecture is proved when G is $$SL_ n({\mathbb{R}})$$, $$Sp_ n({\mathbb{R}})$$ or SU(p,q). The proof in the case $$SL_ n({\mathbb{R}})$$ is different from the one previously given by B. Speh [in: Non-commutative harmonic analysis and Lie groups, Lect. Notes Math. 880, 483-505 (1981; Zbl 0552.22005)] and, moreover, needed for the general case. The proof is by induction on the dimension of G and uses Vogan’s embedding result.
Reviewer: J.Schwermer

##### MSC:
 2.2e+47 Semisimple Lie groups and their representations 2.2e+31 Analysis on real and complex Lie groups 2.2e+16 General properties and structure of real Lie groups
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