Hecke algebras for tame supercuspidal types

TL;DR

This paper proves Yu's conjecture that Hecke algebras for tame supercuspidal types are isomorphic to those for certain Levi subgroups, and shows that Hecke algebras for regular supercuspidal types are group algebras.

Contribution

It confirms Yu's conjecture on the isomorphism of Hecke algebras for tame supercuspidal types and characterizes Hecke algebras for regular supercuspidal types.

Findings

Hecke algebras for tame supercuspidal types are isomorphic to those of twisted Levi subgroups.

Hecke algebra for a regular supercuspidal type is isomorphic to a group algebra.

The results verify conjectures in the representation theory of p-adic groups.

Abstract

Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal types and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a regular supercuspidal type is isomorphic to the group algebra of a certain abelian group.