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

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.