A description of the integral depth-$r$ Bernstein center

TL;DR

This paper describes the structure of the depth-$r$ Bernstein center for reductive groups over non-archimedean fields, linking it to parahoric Hecke algebras and introducing a new invariant for irreducible representations.

Contribution

It provides a new description of the depth-$r$ Bernstein center as a limit of parahoric Hecke algebras and constructs the depth-$r$ Deligne-Lusztig parameter for irreducible representations.

Findings

The depth-$r$ Bernstein center is described as a limit of standard parahoric Hecke algebras.

Maps from stable functions on Moy-Prasad filtration quotients to the Bernstein center are constructed.

The depth-$r$ Deligne-Lusztig parameter equals the semi-simple part of minimal $K$-types.

Abstract

In this paper we give a description of the depth-$r$ Bernstein center for non-negative integers $r$ of a reductive simply connected group $G$ over a non-archimedean local field as a limit of depth-$r$ standard parahoric Hecke algebras. Using the description, we construct maps from the algebra of stable functions on the $r$-th Moy-Prasad filtration quotient of hyperspecial parahorics to the depth-$r$ Bernstein center and use them to attach to each depth-$r$ irreducible representation $\pi$ an invariant $\theta(\pi)$, called the depth-$r$ Deligne-Lusztig parameter of $\pi$. We show that $\theta(\pi)$ is equal to the semi-simple part of minimal $K$-types of $\pi$.