A Galois side analogue of a theorem of Bernstein

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.