Modulo $\ell$-representations of $p$-adic groups $\mathrm{SL_n}(F)$

TL;DR

This paper constructs maximal simple cuspidal types for Levi subgroups of SL_n over non-archimedean fields in characteristic l, establishing the uniqueness of supercuspidal support in certain cases, advancing modular representation theory of p-adic groups.

Contribution

It introduces a method to construct maximal simple cuspidal types for Levi subgroups of SL_n and proves the uniqueness of supercuspidal support in specific residual characteristic settings.

Findings

Construction of maximal simple cuspidal types for Levi subgroups of SL_n

Proof of uniqueness of supercuspidal support in characteristic p cases

Advancement in modular representation theory of p-adic groups

Abstract

Let $k$ be an algebraically closed field of characteristic $l\neq p$. We construct maximal simple cuspidal $k$-types of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$, when $F$ is a non-archimedean locally compact field of residual characteristic $p$. We show that the supercuspidal support of irreducible smooth $k$-representations of Levi subgroups $\mathrm{M}'$ of $\mathrm{SL}_n(F)$ is unique up to $\mathrm{M}'$-conjugation, when $F$ is either a finite field of characteristic $p$ or a non-archimedean locally compact field of residual characteristic $p$.