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 Homp(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.