On the Quadratic Type of Some Simple Self-Dual Modules over Fields of Characteristic Two

Abstract

Let G be a finite group and let K be an algebraically closed field of Ž characteristic 2. Let V be a non-trivial simple self-dual KG-module we . say that V is self-dual if it is isomorphic to its dual V * . It is a theorem of w x Fong 4, Lemma 1 that in this case there is a non-degenerate G-invariant alternating bilinear form, F, say, defined on V = V. We say that V is a KG-module of quadratic type if F is the polarization of a non-degenerate w x G-invariant quadratic form defined on V. In a previous paper 6 , the present authors described some methods to decide if such a module V is of w x quadratic type. One of the main results of 6 is the following. Suppose that Ž . G is a group with a split B, N -pair of odd characteristic p. Let V be a simple self-dual KG-module that is not of quadratic type and let w be the Brauer character of V. Then there exists a complex irreducible character of G that occurs as a constituent of the induced character 1 and contains B w as a modular constituent with non-zero multiplicity. This result suggests that we should investigate the decomposition modulo 2 of the irreducible characters in 1 when G is a group of Lie type of odd characteristic and B see which real-valued irreducible Brauer characters occur as constituents. The decomposition modulo 2 of these characters is not known in complete