A supercharacter theory for involutive algebra groups

If $mathscr{J}$ is a finite-dimensional nilpotent algebra over a finite field $Bbbk$, the algebra group $P = 1+mathscr{J}$ admits a (standard) supercharacter theory as defined by Diaconis and Isaacs. If $mathscr{J}$ is endowed with an involution $widehat{varsigma}$, then $widehat{varsigma}$ naturally defines a group automorphism of $P = 1+mathscr{J}$, and we may consider the fixed point subgroup $C_{P}(widehat{varsigma}) = {xin P : widehat{varsigma}(x) = x^{-1}}$. Assuming that $Bbbk$ 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 Andre and Neto for the special case of the symplectic and orthogonal groups.