On the action of the center in the $\ell$-adic categorical Langlands program

Abstract

We discuss the action of the center in the categorical local Langlands program in the form of [FS24]. We provide a spectral description of the convolution action of a central torus in this situation. We show that the veracity of the $\ell$-adic categorical Langlands conjecture passes to the quotient by a central torus $D$ satisfying a minor cohomological condition in characteristic 0 and we show that under restriction to Langlands-Shahidi type parameters the conjecture passes to such a quotient in all characterstics and even integrally. In that situation we also show that the splitting of the semi-orthogonal decomposition descends. Finally we use these results to extend [Zou25] to $\mathrm{PGL}_n$.