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

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.