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

Source abstract

Abstract We construct a surjection from the set of irreducible representations of $\mathrm{SL}_{N}(k)$, where $k$ is a finite field, to the set of inertia equivalence classes of tame Langlands parameters for $\mathrm{SL}_{N}(F)$, 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 $\mathrm{SL}_{N}(F)$, confirming a related conjecture of Vogan. This work extends several results previously established for $\mathrm{GL}_{N}$.