Bruhat-Tits buildings, representations of $p$-adic groups and Langlands correspondence

TL;DR

This paper explores the relationship between Bruhat-Tits buildings, representations of p-adic groups, and the local Langlands correspondence, extending recent results to more general representations and clarifying the structure of L-packets.

Contribution

It generalizes recent results on supercuspidal representations to arbitrary representations, linking enhanced L-parameters with Springer correspondence and describing their role in the local Langlands correspondence.

Findings

Enhanced L-parameters with semisimple cuspidal support correspond to Springer correspondence.

Every L-packet contains at least one representation with non-singular supercuspidal support.

Results hold for groups splitting over tamely ramified extensions with certain conditions on p.

Abstract

The Bruhat-Tits theory is a key ingredient in the construction of irreducible smooth representations of $p$-adic reductive groups. We describe generalizations to arbitrary such representations of several results recently obtained in the case of supercuspidal representations, in particular regarding the local Langlands correspondence and the internal structure of the $L$-packets. We prove that the enhanced $L$-parameters with semisimple cuspidal support are those which are obtained via the (ordinary) Springer correspondence. Let ${\mathbf G}$ be a connected reductive group over a non-archimedean field $F$ of residual characteristic $p$. In the case where ${\mathbf G}$ splits over a tamely ramified extension of $F$ and $p$ does not divide the order of the Weyl group of ${\mathbf G}$, we show that the enhanced $L$-parameters with semisimple cuspidal support correspond to the irreducible smooth representations of ${\mathbf G}(F)$ with non-singular supercuspidal support via the local Langlands correspondence constructed by Kaletha, under the assumption that the latter satisfies certain expected properties. As a consequence, we obtain that every compound $L$-packet of ${\mathbf G}(F)$ contains at least one representation with non-singular supercuspidal support.