Irreducible characters of groups associated with finite nilpotent algebras with involution

An algebra group is a group of the form P=1+J where J is a finite-dimensional nilpotent associative algebra. A theorem of Z. Halasi asserts that, in the case where J is defined over a finite field F, every irreducible character of P is induced from a linear character of an algebra subgroup of P. If (J,σ) is a nilpotent algebra with involution, then σ naturally defines a group automorphism of P=1+J, and we may consider the fixed point subgroup CP(σ). Assuming that F has odd characteristic p, we show that every irreducible character of CP(σ) is induced from a linear character of a subgroup of the form CQ(σ) where Q is a σ-invariant algebra subgroup of P. As a particular case, the result holds for the Sylow p-subgroups of the finite classical groups of Lie type.