Shintani descent for standard supercharacters of algebra groups

TL;DR

The paper studies Shintani descent in the context of standard supercharacters for finite algebra groups $G(q)=1+\\mathcal{A}(q)$ and extends it to the inductive family $G(q^{n})$, forming a uniform framework across $n$. It defines a Shintani descent map $\operatorname{Sh}_{n}$ that identifies $F$-invariant supercharacters of $G(q^{n})$ with supercharacters of $G(q)$ via the trace map $\operatorname{Tr}_{n}$ and the norm map $\operatorname{Nm}_{n}$, and proves that $\operatorname{Sh}_{n}$ restricts to a bijection $\operatorname{SCh}(G(q^{n}))^{F} \to \operatorname{SCh}(G(q))$. The authors then organize the family of finite algebra groups into an inductive system and introduce the superdual algebra $\operatorname{scf}(G)$, defined as the direct limit of the spaces of superclass functions, capturing all supercharacters $\operatorname{SCh}(G(q^{n}))$ for $n\in\mathbb{N}$. They further construct the Serre dual $\widehat{\mathcal{A}}$ and prove a canonical bijection between the global supercharacters $\operatorname{SCh}(G)$ and isomorphism classes of super $\mathbb{C}[G]$-modules, integrating the representation-theoretic data across all $n$. The results connect Shintani descent, orbit methods, and duality in a unified framework for algebra groups and their supercharacter theories.

Abstract

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 $n \in \mathbb{N}$.

Theorems & Definitions (16)

• Lemma 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• Proposition 3.1
• proof
• Lemma 3.2