On Ideals Defining Irreducible Representations of Reductive $p$--adic Groups

Abstract

Let $G$ be a reductive $p$--adic group. Assume that $L\subset G$ is an open--compact subgroup, and ${\mathcal{H}}_L$ is the Hecke algebra of $L$--biinivariant complex functions on $G$. It is a well--known and standard result on how to prove existence of a complex smooth irreducible $G$--module out of a maximal left ideal $I\subset {\mathcal{H}}_L$. Using theory on Bernstein center we make this construction explicit. This leads us to some very interesting questions.

accepted for publication in Contemporary Mathematics "Representation Theory XVI"