Irreducibility of Discrete Series Representations for Semisimple Symmetric Spaces

Abstract

This chapter discusses irreducibility of discrete series representations for semi-simple symmetric spaces. It explains the term translation principle that refers to the idea of studying infinite dimensional representations of reductive Lie algebras by investigating their tensor products with the rich, complicated, and well-understood family of finite-dimensional representations of a connected reductive Lie group. The chapter also describes a criterion for a translation functor to take irreducibles to irreducible. A translation functor is a sum of composites of elementary translational functors. To make good use of the translation functors, one needs a way to compute them effectively. This is provided by the conceptually more subtle idea of coherent families.