Hecke algebras for tame supercuspidal types

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 $F$. In~2001, Yu constructed types which are called {\it 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 {\it regular supercuspidal type} is isomorphic to the group algebra of a certain abelian group.