Irreducible characters of groups associated with finite nilpotent algebras with involution

Abstract

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 C P ( σ ) . Assuming that F has odd characteristic p , we show that every irreducible character of C P ( σ ) is induced from a linear character of a subgroup of the form C Q ( 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.