Reduction to depth zero for tame p-adic groups via Hecke algebra isomorphisms

TL;DR

This paper demonstrates an explicit isomorphism between Hecke algebras associated with certain types in tame p-adic groups and their Levi subgroups, reducing complex representation problems to depth-zero cases.

Contribution

It constructs explicit Hecke algebra isomorphisms for tame p-adic groups, enabling reduction of representation theory problems to depth-zero cases under certain conditions.

Findings

Hecke algebras for types in G and G^0 are isomorphic.

Bernstein blocks are equivalent to depth-zero blocks when p does not divide the Weyl group order.

The isomorphism includes a description as semi-direct products of affine Hecke algebras and twisted group algebras.

Abstract

Let $F$ be a nonarchimedean local field of residual characteristic $p$. Let $G$ denote a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Let $(K ,\rho)$ be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup $G^0 \subset G$ and a type $(K^0, \rho^0)$ for $G^0$ such that the corresponding Hecke algebras $\mathcal{H}(G(F), (K, \rho))$ and $\mathcal{H}(G^0(F), (K^0, \rho^0))$ are isomorphic. If $p$ does not divide the order of the absolute Weyl group of $G$, then every Bernstein block is equivalent to modules over such a Hecke algebra. Hence, under this assumption on $p$, our result implies that every Bernstein block is equivalent to a depth-zero Bernstein block. This allows one to reduce many problems about (the category of) smooth, complex representations of $p$-adic groups to analogous problems about (the category of) depth-zero representations. Our isomorphism of Hecke algebras is very explicit and also includes an explicit description of the Hecke algebras as semi-direct products of an affine Hecke with a twisted group algebra. Moreover, we work with arbitrary algebraically closed fields of characteristic different from $p$ as our coefficient field. This paper relies on a prior axiomatic result about the structure of Hecke algebras by the same authors and a key ingredient consists of extending the quadratic character of Fintzen--Kaletha--Spice to the support of the Hecke algebra, which might be of independent interest.