Sato–Tate equidistribution for families of automorphic representations through the stable trace formula

Abstract

Shin and Templier studied families of automorphic representations with local restrictions: roughly, Archimedean components contained in a fixed $L$-packet of discrete series and non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components as either the weight of the $L$-packet or level went to infinity. We extend their weight-aspect results to families where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet.

We do this by using a so-called “hyperendoscopy” version of the stable trace formula of Ferarri. The main technical difficulties are first, defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and second, bounding the values of transfers of unramified functions. We also present an extension to noncuspidal groups of Arthur’s simple trace formula since it does not seem to appear elsewhere in the literature.