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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:
 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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##### References:
 [1] A. Borel and N. Wallach : Continuous cohomology, discrete subgroups and representations of reductive subgroups , in Annals of Mathematics Studies Vol. 94, Princeton University Press, 1980. · Zbl 0443.22010 [2] S. Salamanca-Riba : On the unitary dual of some classical Lie groups , Compositio Math. 68 (1988), 251-303. · Zbl 0692.22007 · numdam:CM_1988__68_3_251_0 · eudml:89938 [3] B. Speh and D. Vogan : Reducibility of generalized principal series representations , Acta Math. 145 (1980), 227-229. · Zbl 0457.22011 · doi:10.1007/BF02414191 [4] D. Vogan : Representations of Real Reductive Lie Groups , Birkhäuser, Boston-Basel- Stuttgart, 1981. · Zbl 0469.22012 [5] D. Vogan : Unitarizability of certain series of representations , Annals Math. 120 (1984),141-187. · Zbl 0561.22010 · doi:10.2307/2007074 [6] G. Zuckerman : On Construction of Representations by Derived Functors . Handwritten notes, 1977.
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