The center of Hecke algebras of types

TL;DR

This paper characterizes the center of Hecke algebras associated with types in Bernstein blocks for certain reductive groups over non-archimedean local fields, extending understanding of their algebraic structure.

Contribution

It provides a description of the Bernstein center of Hecke algebras for types satisfying specific hypotheses, generalizing previous results for a broad class of groups.

Findings

Describes the center of Hecke algebras under certain hypotheses.

Applies results to specific compact open subgroups including Moy-Prasad filtrations.

Extends the understanding of the Bernstein center in the context of types.

Abstract

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When $\bf G$ is a connected reductive group over non-archimedean local field $F$ that splits over a tamely ramified extension of $F$ and the residue characteristic of $F$ does not divide the order of the absolute Weyl group of $\bf G$, the works of Kim-Yu and Fintzen associate a type to each Bernstein block and our hypothesis is satisfied for such types. We use our results to give a description of the Bernstein center of the Hecke algebra $\mathcal{H}({\bf G } (F),K)$ when $K$ belongs to a nice family of compact open subgroups of ${\bf G}(F)$ (which includes all the Moy-Prasad filtrations of an Iwahori subgroup) via the theory of types.