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

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 ,ρ) be a type as constructed by Kim and Yu. We show that there exists a twisted Levi subgroup G⁰ ⊂ G and a type (K⁰, ρ⁰) for G⁰ such that the corresponding Hecke algebras H(G(F), (K, ρ)) and H(G⁰(F), (K⁰, ρ⁰)) 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.