A supercharacter theory for involutive algebra groups

If $\mathcal{J}$ is a finite-dimensional nilpotent algebra over a finite field $\mathbb{k}$, the algebra group $P=1+\mathcal{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $\mathcal{J}$ is endowed with an involution $\widehat{\varsigma }$, then $\widehat{\varsigma }$ naturally defines a group automorphism of $P=1+\mathcal{J}$, and we may consider the fixed point subgroup $C_{P}(\widehat{\varsigma}) = {x\in P : \widehat{\varsigma}(x) = x^{-1}}$\mathrm{.}Assumingthat$\Bbbk$ has odd characteristic $p$, we use the standard supercharacter theory for $P$ to construct a supercharacter theory for $C_P(\widehat{\varsigma })$. In particular, we obtain a supercharacter theory for the Sylow $p$-subgroups of the finite classical groups of Lie type, and thus extend in a uniform way the construction given by Andr'e and Neto for the special case of the symplectic and orthogonal groups.