A supercharacter theory for involutive algebra groups

If JJ is a finite-dimensional nilpotent algebra over a finite field kk, the algebra group P=1+JP=1+J admits a (standard) supercharacter theory as defined in [16]. If JJ is endowed with an involution σ, then σ   naturally defines a group automorphism of P=1+JP=1+J, and we may consider the fixed point subgroup CP(σ)={x∈P:σ(x)=x−1}CP(σ)={x∈P:σ(x)=x−1}. Assuming that kk has odd characteristic p, we use the standard supercharacter theory for P   to construct a supercharacter theory for CP(σ)CP(σ). 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 André and Neto in [7] and [8] for the special case of the symplectic and orthogonal groups.