Let ${𝒜}(q)$ be a finite-dimensional nilpotent algebra over a finite field ${𝔽}_q$ with q elements, and let $G(q)=1+{𝒜}(q)$. On the other hand, let $𝕜$ denote the algebraic closure of ${𝔽}_q$, and let ${𝒜}={𝒜}(q){\otimes }_{{𝔽}_q}𝕜$. Then $G=1+{𝒜}$ is an algebraic group over $𝕜$ equipped with an ${𝔽}_q$-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 nth power $F^n:G\to G$ is also a Frobenius map, and $G^{F^n}$ identifies with $G(q^n)=1+{𝒜}(q^n)$. 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 $m|n$, 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 $n\in \mathbb{N}$.

让${𝒜}（q）$是有限域上的有限维幂零代数${𝔽}_q$与q元素，并让$G（q）=1+{𝒜}（q）$。另一方面，让$𝕜$表示代数闭包${𝔽}_q$， 然后让${𝒜}={𝒜}（q）{\otimes }_{{𝔽}_q}𝕜$。然后$G=1+{𝒜}$是一个代数群$𝕜$配备有${𝔽}_q$-由通常的弗罗贝尼乌斯图给出的合理结构$F：G\to G$， 和$G（q）$可以看作不动点子群$G^F$。对于每一个$n\epsilon \mathbb{N}$，n次方$F^n：G\to G$也是 Frobenius 映射，并且$G^{F^n}$认同于$G（q^n）=1+{𝒜}（q^n）$。 Frobenius 映射限制群自同构$F：G（q^n）\to G（q^n）$，因此它作用于不可约特征集$G（q^n）$。 Shintani 血统提供了一种比较F不变不可约特征的方法$G（q^n）$和不可约特征$G（q）$。在本文中，我们表明它还提供了一种研究超级字符的统一方法$G（q^n）$为了$n\epsilon \mathbb{N}$。这些组形成了关于包含图的归纳系统$G（q^{米}）\to G（q^n）$每当$米|n$，这一事实使我们能够同时研究所有超级字符理论，建立它们之间的联系，并将它们与代数群G联系起来。事实上，我们表明 Shintani 血统允许定义某种“超对偶代数”，该代数编码有关超级字符的信息$G（q^n）$为了$n\epsilon \mathbb{N}$。