zbMATH — the first resource for mathematics

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.

22E46 Semisimple Lie groups and their representations
22E30 Analysis on real and complex Lie groups
22E15 General properties and structure of real Lie groups
Full Text: Numdam EuDML
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.