Progrès récents sur les représentations supercuspidales

Abstract

Let $G$ be a reductive group over a nonarchimedean local field $F$. In the quest for a classification of irreducible smooth representations of $G$, it is critical to understand the case of supercuspidal representations -- those whose matrix coefficients are compactly supported modulo the center. Progress in understanding these representations has been continuous over the past fifty years. In "tame" cases where the residual characteristic of $F$ is big enough for $G$, J.-K. Yu described in 2001 a general construction of supercuspidal representations, building on a large body of work. But recent developments have made the general picture much more complete and much clearer. For instance, the work of J. Fintzen, T. Kaletha and L. Spice provides (in the tame case) a classification of supercuspidal representations, an explicit formula for "almost all" their characters, and an explicit construction of a local Langlands correspondence for supercuspidal $L$-packets. While the basic constructions involve Bruhat--Tits buildings and representations of finite groups, the resulting character formulas and the description of $L$-packets have striking parallels with the case of real groups