Structure of Hecke algebras arising from types

TL;DR

This paper characterizes the structure of Hecke algebras associated with certain types in p-adic groups, showing they are semi-direct products of affine Hecke algebras and relate to depth-zero types, with implications for Bernstein blocks.

Contribution

It provides an explicit structural description of Hecke algebras for a broad class of types, including all depth-zero types, and establishes isomorphisms to simpler Hecke algebras, extending previous results.

Findings

Hecke algebras are semi-direct products of affine Hecke algebras and twisted group algebras.

Hecke algebras are isomorphic to those of certain reductive subgroups with depth-zero representations.

The results apply to all depth-zero types and are compatible with other type constructions.

Abstract

Let $G$ denote a connected reductive group over a nonarchimedean local field $F$ of residue characteristic $p$, and let $\mathcal{C}$ denote an algebraically closed field of characteristic $\ell \neq p$. If $\rho$ is an irreducible, smooth $\mathcal{C}$-representation of a compact, open subgroup $K$ of $G(F)$, then the pair $(K,\rho)$ gives rise to a Hecke algebra $\mathcal{H}(G(F),(K, \rho))$. For a large class of pairs $(K,\rho)$, we show that $\mathcal{H}(G(F),(K, \rho))$ is a semi-direct product of an affine Hecke algebra with explicit parameters with a twisted group algebra, and that it is isomorphic to $\mathcal{H}(G^0(F),(K^0, \rho^0))$ for some reductive subgroup $G^0 \subset G$ with compact, open subgroup $K^0$ and depth-zero representation $\rho^0$ of $K^0$. The class of pairs that we consider includes all depth-zero types. In describing their Hecke algebras, we thus recover a result of Morris as a special case. In a second paper, we will show that our class also contains all the types constructed by Kim and Yu, and hence we obtain as a corollary that arbitrary Bernstein blocks are equivalent to depth-zero Bernstein blocks under minor tameness assumptions. The pairs to which our results apply are described in an axiomatic way so that the results can be applied to other constructions of types by only verifying that the relevant axioms are satisfied. The Hecke algebra isomorphisms are given in an explicit manner and are support preserving.