Multiplicities under basechange: Finite field case

Abstract

A general proposition is proved relating multiplicities (for the restriction of a representation of a group to a subgroup) under basechange for groups over finite fields, and used to calculate certain multiplicities for cuspidal representations of general linear groups which become principal series representations under basechange for which the multiplicities can be calculated by geometric methods. For E/F quadratic extension of finite fields, we use this method to calculate which 1 dimensional representations of GLn(E) appear in a cuspidal representation of GL2n(F). We also calculate which 1 dimensional representations of GLn(F)×GLn(F) appear in a cuspidal representation of GL2n(F).