Shintani descent for standard supercharacters of algebra groups

Abstract

Let [Formula: see text] be a finite-dimensional nilpotent algebra over a finite field [Formula: see text] with q elements, and let [Formula: see text]. On the other hand, let [Formula: see text] denote the algebraic closure of [Formula: see text], and let [Formula: see text]. Then [Formula: see text] is an algebraic group over [Formula: see text] equipped with an [Formula: see text]-rational structure given by the usual Frobenius map [Formula: see text], and [Formula: see text] can be regarded as the fixed point subgroup [Formula: see text]. For every [Formula: see text], the nth power [Formula: see text] is also a Frobenius map, and [Formula: see text] identifies with [Formula: see text]. The Frobenius map restricts to a group automorphism [Formula: see text], and hence it acts on the set of irreducible characters of [Formula: see text]. Shintani descent provides a method to compare F-invariant irreducible characters of [Formula: see text] and irreducible characters of [Formula: see text]. In this paper, we show that it also provides a uniform way of studying supercharacters of [Formula: see text] for [Formula: see text]. These groups form an inductive system with respect to the inclusion maps [Formula: see text] whenever [Formula: see text], and this fact allows us to study all supercharacter theories simultaneously, to establish connections between them, and to relate them to the algebraic group G. Indeed, we show that Shintani descent permits the definition of a certain “superdual algebra” which encodes information about the supercharacters of [Formula: see text] for [Formula: see text].