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On the unitary dual of the classical Lie groups. II: Representations of \(SO(n,m)\) inside the dominant Weyl chamber. (English) Zbl 0782.22010
There is the conjecture due to Vogan 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 is regular then the special unipotent representations involved are one dimensional, and the conjecture says: Suppose \(X\) is an irreducible unitary Harish-Chandra module of \(G\) whose infinitesimal character minus half the sum of the positive roots is dominant. Then there are a \(\theta\)-stable parabolic subalgebra \(\mathfrak q\) of the complexified Lie algebra \(\mathfrak g\) and a unitary one dimensional character \(\lambda\) of the Levi subgroup \(L\) of \(\mathfrak q\) such that \(X\) is isomorphic to the Zuckerman module \(R^{\mathfrak g}_{\mathfrak q}(\mathbb{C}_ \lambda)\).
In part I [ibid. 68, 251-303 (1988; Zbl 0692.22007)] this conjecture was proved when \(G\) is \(SL_ n(\mathbb{R})\), \(Sp_ n(\mathbb{R})\) or \(SU(p,q)\). The paper under review deals with the case \(SO(n,m)\). The proof is by reduction to a subgroup of \(G\) of smaller dimension. The main tool used is the Dirac operator inequality as e.g., stated in Part I.

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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References:
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