Ž . JOURNAL OF ALGEBRA 195, 634649 1997 ARTICLE NO. JA977048 On the Quadratic Type of Some Simple Self-Dual Modules over Fields of Characteristic Two Rod Gow Department of Mathematics, Uni  ersity College, Belfield, Dublin 4, Ireland and Wolfgang Willems Fachbereich Mathematik, Uni  ersitat Mainz, Saarstrasse 21, 55099, Mainz, Germany ¨ Communicated by Walter Feit Received December 3, 1996 INTRODUCTION 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   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  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  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  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 G and contains B  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 G 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 634 0021-869397 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.