Non Spherical Principal Series Representations Of A Semisimple Lie Group

Non-Spherical Principal Series Representations of a Semisimple Lie Group

Non-spherical principal series representations of a real semisimple Lie group are studied. These are representations, induced by a one-dimensional representation of a minimal parabolic subgroup, which have a one-dimensional subspace left stable by a maximal compact subgroup of the original group G. Necessary and sufficient conditions for such a representation to be irreducible, or to be cyclic, are found, in terms of parameters determined by certain rank one subgroups of G. A sufficient condition for such a representation to be unitary is found, and the condition is shown to be necessary in the rank one case.

Non-Spherical Principal Series Representations of a Semisimple Lie Group

Singular Unitary Representations and Discrete Series for Indefinite Stiefel Manifolds $U(p,q;{\mathbb F})/U(p-m,q;{\mathbb F})$

This memoir examines the basic problem of finding vanishing theorems for Harish-Chandra modules. The results of these difficult problems contribute in a meaningful way to the singular unitary representation theory of reductive groups.