On parameters of Hecke algebras for $p$-adic groups

TL;DR

This paper establishes explicit descriptions of Hecke algebra parameters for depth-zero types in p-adic groups, generalizing previous work and confirming a version of Lusztig's conjecture under certain splitting and divisibility conditions.

Contribution

It shows that affine Hecke algebras for depth-zero types are isomorphic to those for unipotent types in split groups over unramified extensions, enabling explicit parameter calculations.

Findings

Affine Hecke algebras for depth-zero types are isomorphic to unipotent types in split groups.

Parameters of Hecke algebras for arbitrary Bernstein blocks match those of unipotent blocks under specified conditions.

Confirmed a version of Lusztig's conjecture relating parameters of Hecke algebras in this context.

Abstract

Let $F$ be a non-archimedean local field with residue characteristic $p$ and $G$ be a connected reductive group defined over $F$. In earlier joint works with Jeffrey D. Adler, Jessica Fintzen, and Manish Mishra, we proved that the Hecke algebras attached to types constructed by Kim and Yu are isomorphic to the Hecke algebras attached to depth-zero types. Note that if $G$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the absolute Weyl group of $G$, such Hecke algebras cover the Hecke algebras attached to arbitrary Bernstein blocks. We also proved that for a depth-zero type $(K, \rho)$, the corresponding Hecke algebra $\mathcal{H}(G(F), (K, \rho))$ has an explicit description as a semi-direct product of an affine Hecke algebra $\mathcal{H}(W(\rho_M)_{\mathrm{aff}}, q)$ with a twisted group algebra $\mathbb{C}[\Omega(\rho_{M}), \mu]$, generalizing prior work of Morris. In this paper, we show that the affine Hecke algebra $\mathcal{H}(W(\rho_M)_{\mathrm{aff}}, q)$ appearing in the description of the Hecke algebra $\mathcal{H}(G(F), (K, \rho))$ attached to a depth-zero type $(K, \rho)$ is isomorphic to the one attached to a unipotent type for a connected reductive group splitting over an unramified extension of $F$. This makes it possible to calculate the parameters of the affine Hecke algebras for depth-zero types and types constructed by Kim and Yu explicitly. In particular, we prove a version of Lusztig's conjecture that the parameters of the Hecke algebra attached to an arbitrary Bernstein block agree with those of a unipotent Bernstein block under the assumption that $G$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the absolute Weyl group of $G$.