A supercharacter theory for involutive algebra groups

Abstract

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