Hecke algebras for $p$-adic reductive groups and Local Langlands Correspondence for Bernstein blocks

Abstract: We study the endomorphism algebras attached to Bernstein components of reductive $p$-adic groups and construct a local Langlands correspondence with the appropriate set of enhanced $L$-parameters, using certain "desiderata" properties for the LLC for supercuspidal representations of proper Levi subgroups. We give several applications of our LLC to various reductive groups with Bernstein blocks cuspidally supported on general linear groups. In particular, for Levi subgroups of maximal parabolic of the split exceptional group $G_2$, we compute the explicit weight functions for the corresponding Hecke algebras, and show that they satisfy a conjecture of Lusztig's. Some results from $\S 4$ are used by the same authors to construct a full local Langlands correspondence in \cite{AX-LLC}. Moreover, we also prove a reduction to depth zero case result for the Bernstein components attached to regular supercuspidal representations of Levi subgroups.