A Reduction over finite fields of the tame local Langlands correspondence for SLn

TL;DR

This paper proves a conjecture by Vogan for SLn, creating a connection between irreducible representations over finite fields and tame Langlands parameters for p-adic fields, extending known results from GLn.

Contribution

It constructs a surjection linking irreducible representations of SLn over finite fields to tame Langlands parameters, and parametrizes the fibers, extending prior GLn results.

Findings

Established a surjection from SLn representations to Langlands parameters.

Parametrized the fibers of the surjection.

Extended results from GLn to SLn.

Abstract

We establish a conjecture formulated by Vogan for SLn. Specifically, we construct a surjection from the set of irreducible representations of SLn(k), where k is a finite field, to the inertia equivalence classes of tame Langlands parameters for SLn(F), where F is a p-adic field with residue field k. Additionally, we provide a parametrization of the fibers of this surjection and examine its compatibility with the Local Langlands Correspondence for SLn. This work extends several results previously established for GLn to the context of SLn.