an:00203004
Zbl 0782.22010
Salamanca-Riba, Susana A.
On the unitary dual of the classical Lie groups. II: Representations of \(SO(n,m)\) inside the dominant Weyl chamber
EN
Compos. Math. 86, No. 2, 127-146 (1993).
00013709
1993
j
22E46 22E30 22E15
unitary representation; real reductive Lie group; cohomological parabolic induction; special unipotent representations; irreducible unitary Harish- Chandra module; positive roots; complexified Lie algebra; Zuckerman module; Dirac operator inequality
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.
J.Schwermer (Eichst??tt)
Zbl 0692.22007