The center of Hecke algebras of types

Abstract

We describe the center of the Hecke algebra of a type attached to a Bernstein block under some hypothesis. When $\mathbf{G}$ 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 $\mathbf{G}$, 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 $\mathcal{H}(\mathbf{G}(F),K)$ when $K$ belongs to a nice family of compact open subgroups of $\mathbf{G}(F)$ (which includes all the Moy-Prasad filtrations of an Iwahori subgroup) via the theory of types.

arXiv comment27 pages