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

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.