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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:
22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
22E15 General properties and structure of real Lie groups
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