A Reduction Over Finite Fields of the Tame Local Langlands Correspondence for SLN

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