# Supercharacter theory of an algebra group

Suppose $F$ is a finite field of size $q$, a prime power with underlying prime $p$. Suppose $N$ is a nilpotent associative finite-dimensional algebra over $F$. Suppose, further, that $G$ is the algebra group corresponding to $N$, i.e., $G = 1 + N$..
The supercharacter theory of $G$ is a supercharacter theory defined as follows.
Note that $G$ is a subgroup of the unitization $N + F$, hence acts on $N + F$ by both left and right multiplication. The ideal $N$ is invariant under both actions, so we get a left and a right $G$-action on $N$ that commute with each other. Consider the double orbits of $N$ under these actions, i.e., sets of the form $GxG, x \in N$. The corresponding sets $1 + GxG$, which are subsets of $G$, are the superconjugacy classes.
Note that the conjugation action of $G$ on $N$ corresponds to its conjugation action on itself, and the orbits for this conjugation action are contained inside the double orbits, so the superconjugacy classes are unions of conjugacy classes.