**Date & Time:** Wednesday, February 26, 2014, 16:00-17:00.

**Venue:** Ramanujan Hall

**Speaker:** M. S. Raghunathan, IIT Bombay

**Title:** Imbedding Harish-Chandra Modules in Principal Series

**Abstract:** Let *G* be a connected semisimple Lie group with a finite centre and *g* its Lie algebra. Let *K* be a maximal compact subgroup. Let *U* be the enveloping algebra of *g*. A *(g, K)*-module is a module *M* over *U* with an action of *K* on it compatible with the *U*-module structure. A *(g, K)*-module *M* is a Harish-Chandra module if it decomposes as a *K*-module into a direct sum of finite dimensional irreducible representations of *K* with each irreducible representation occuring with finite multiplicity. Let *P* be a minimal parabolic subgroup of *G* and *p* its Lie algebra. Let ρ be a finite dimensional continuous irreducible representation of *P* on a vector space *E*. The representation I(ρ) of *U* induced by ρ is the *U*-module Hom_{p}*(U,E)* - *U* is regarded as a *p*-*U*-bimodule with *p* acting on the left and *U* acting on the right. A theorem of Casselman and Miličić asserts that any irreducible Harish-Chandra module imbeds in I(ρ) for a suitable ρ. In this talk we give a simple algebraic proof of this theorem in the special case when *G* has rank 1.