Harish-Chandra Modules with the Unique Embedding Property

Abstract

Let $G$ be a connected semisimple real matrix group. In view of Casselman’s subrepresentation theorem, every irreducible admissible representation of $G$ may be realized as a submodule of some principal series representation. We give a classification of representations with a unique embedding into principal series, in the case of regular infinitesimal character. Our basic philosophy is to link the theory of asymptotic behavior of matrix coefficients with the theory of coherent continuation of characters. This is accomplished by using the "Jacquet functor" and the Kazhdan-Lusztig conjectures.