Shintani descent for algebraic groups and almost characters of unipotent groups

TL;DR

This paper extends Shintani descent to all algebraic groups over finite fields, defining almost characters for neutrally unipotent groups and linking them to Frobenius traces, confirming positivity of related modular categories.

Contribution

It generalizes Shintani descent to disconnected algebraic groups and establishes the well-defined nature of almost characters for neutrally unipotent groups.

Findings

Almost characters coincide with Frobenius trace functions.

Modular categories from character sheaves are positive integral.

Extension of Shintani descent to all algebraic groups.

Abstract

In this paper, we extend the notion of Shintani descent to general (possibly disconnected) algebraic groups defined over a finite field $\mathbb{F}_q$. For this, it is essential to treat all the pure inner $\mathbb{F}_q$-rational forms of the algebraic group at the same time. We prove that the notion of almost characters (introduced by T. Shoji using Shintani descent) is well defined for any neutrally unipotent algebraic group, i.e. an algebraic group whose neutral connected component is a unipotent group. We also prove that these almost characters coincide with the "trace of Frobenius" functions associated with Frobenius-stable character sheaves on neutrally unipotent groups. In the course of the proof, we also prove that the modular categories that arise from Boyarchenko-Drinfeld's theory of character sheaves on neutrally unipotent are in fact positive integral confirming a conjecture due to Drinfeld.