Multiplicities under basechange: finite field case

TL;DR

This paper investigates how multiplicities of group representation restrictions change under basechange, especially for finite fields, and applies geometric methods to compute these multiplicities for specific cases.

Contribution

It establishes a general relation for multiplicities under basechange and computes explicit examples for cuspidal representations transforming into principal series.

Findings

Derived a general proposition relating multiplicities under basechange.

Calculated multiplicities for certain cuspidal representations after basechange.

Used geometric methods to determine these multiplicities.

Abstract

A general proposition is proved relating multiplicities (of restriction of a representation of a group to a subgroup) under basechange, and used to calculate some multiplicities for cuspidal representations which become principal series representations under basechange for which multiplicities can be calculated by geometric methods.