Finiteness for Hecke algebras of $p$-adic groups

TL;DR

This paper proves the finiteness of Hecke algebras and their centers for p-adic groups, confirming long-standing conjectures and establishing a key link via the Fargues-Scholze morphism, with implications for the local Langlands correspondence.

Contribution

It establishes the finite generation of Hecke algebras and their centers over their centers, and proves second adjointness for smooth representations, using the Fargues-Scholze morphism.

Findings

Hecke algebras are finitely generated over their centers.

Centers of Hecke algebras are finitely generated R-algebras.

Second adjointness holds for smooth representations with invertible p.

Abstract

Let $G$ be a reductive group over a non-archimedean local field $F$ of residue characteristic $p$. We prove that the Hecke algebras of $G(F)$ with coefficients in a ${\mathbb Z}_{\ell}$-algebra $R$ for $\ell$ not equal to $p$ are finitely generated modules over their centers, and that these centers are finitely generated $R$-algebras. Following Bernstein's original strategy, we then deduce that "second adjointness" holds for smooth representations of $G(F)$ with coefficients in any ring $R$ in which $p$ is invertible. These results had been conjectured for a long time. The crucial new tool that unlocks the problem is the Fargues-Scholze morphism between a certain "excursion algebra" defined on the Langlands parameters side and the Bernstein center of $G(F)$. Using this bridge, our main results are representation theoretic counterparts of the finiteness of certain morphisms between coarse moduli spaces of local Langlands parameters that we also prove here, which may be of independent interest