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

## TL;DR

This paper explores the relationship between ideals in Hecke algebras and irreducible representations of reductive p-adic groups, providing explicit constructions via Bernstein center theory.

## Contribution

It explicitly constructs irreducible representations from maximal ideals in Hecke algebras using Bernstein center theory, enhancing understanding of their structure.

## Key findings

- Explicit construction of irreducible representations from Hecke algebra ideals
- Connection established between Bernstein center and representation theory
- Raises new questions for further research

## 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.