Let $\mathcal{ A}(q)$ be a finite-dimensional nilpotent algebra over a finite field $\mathbb{ F}_{ q}$ with $q$ elements, and let $G(q) = 1+\mathcal{ A}(q)$. On the other hand, let $\Bbbk$ denote the algebraic closure of $\mathbb{ F}_{ q}$, and let $\mathcal{ A} = \mathcal{ A}(q) \otimes_{ \mathbb{ F}_{ q}} \Bbbk$. Then $G = 1+\mathcal{ A}$ is an algebraic group over $\Bbbk$ equipped with an $\mathbb{ F}_{ q}$-rational structure given by the usual Frobenius map $F:G\to G$, and $G(q)$ can be regarded as the fixed point subgroup $G^{ F}$. For every $n \in \mathbb{ N}$, the $n$th power $F^{ n}:G\to G$ is also a Frobenius map, and $G^{ F^{ n}}$ identifies with $G(q^{ n}) = 1 + \mathcal{ A}(q^{ n})$. The Frobenius map restricts to a group automorphism $F:G(q^{ n})\to G(q^{ n})$, and hence it acts on the set of irreducible characters of $G(q^{ n})$. Shintani descent provides a method to compare $F$-invariant irreducible characters of $G(q^{ n})$ and irreducible characters of $G(q)$. In this paper, we show that it also provides a uniform way of studying supercharacters of $G(q^{ n})$ for $n \in \mathbb{ N}$. These groups form an inductive system with respect to the inclusion maps $G(q^{ m}) \to G(q^{ n})$ whenever $m \mid n$, 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 $G(q^{ n})$ for $n \in \mathbb{ N}$.
Categories: math.RT