Non-Spherical Principal Series Representations of a Semisimple Lie Group

Book excerpt: 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.