On the unicity of types in special linear groups

TL;DR

This paper characterizes typical representations of maximal compact subgroups in special linear groups over non-archimedean fields, linking them to simple types and providing explicit descriptions of conjugacy classes and an inertial Langlands correspondence.

Contribution

It establishes a precise connection between typical representations and Bushnell--Kutzko simple types in $ ext{SL}_N(F)$, and explicitly describes the inertial Langlands correspondence for tame representations.

Findings

Classifies conjugacy classes of typical representations

Provides explicit description of inertial Langlands correspondence

Shows that typical representations are induced from simple types

Abstract

Let $F$ be a non-archimedean local field. We show that any representation of a maximal compact subgroup of $\mathbf{SL}_N(F)$ which is typical for an essentially tame supercuspidal representation must be induced from a Bushnell--Kutzko maximal simple type. From this, we explicitly count and describe the conjugacy classes of such typical representations, and give an explicit description of an inertial Langlands correspondence for essentially tame irreducible $N$-dimensional projective representations of the Weil group of $F$.