A Galois side analogue of a theorem of Bernstein

TL;DR

This paper establishes a finiteness result for Galois representations analogous to Bernstein's theorem on supercuspidal representations, deepening the understanding of the local Langlands correspondence.

Contribution

It proves a Galois-side analogue of Bernstein's theorem, showing finiteness of certain Galois representations up to unramified twists.

Findings

Finiteness of Galois representations analogous to Bernstein's theorem

Extension of Bernstein's finiteness results to the Galois side

Enhanced understanding of the local Langlands correspondence

Abstract

Let $G$ be a connected reductive group defined over a non archimedean local field $k$. A theorem of Bernstein states that for any compact open subgroup $K$ of $G(k)$, there are, up to unramified twists, only finitely many $K$-spherical supercuspidal representations of $G(k)$. We prove an analogous result on the Galois side of the Langlands correspondence.