Hecke algebra with respect to the pro-$p$-radical of a maximal compact open subgroup for $GL(n,F)$ and its inner forms

TL;DR

This paper describes the structure of the Hecke algebra associated with the pro-$p$-radical of a maximal compact subgroup in inner forms of general linear groups over non-archimedean fields, including a presentation and categorical equivalence.

Contribution

It provides a presentation by generators and relations for the Hecke algebra $\\mathscr{H}(G,K^1)$ and establishes an equivalence between certain smooth representations and modules over this algebra.

Findings

Presentation of the Hecke algebra by generators and relations.

Categorical equivalence for level-0 smooth representations.

Description of the algebra's structure in terms of double cosets.

Abstract

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, that is the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$ that are bi-invariant for $K^1$ and whose supports are a finite union of $K^1$-double cosets. We produce a presentation by generators and relations of this algebra. Finally we prove that the level-$0$ subcategory of the category of smooth representations of $G$ over a unitary commutative ring $R$ such that $p\in R^{\times}$ is equivalent to the category of modules over $\mathscr{H}(G,K^1)\otimes_\mathbb{Z} R$.