On the number of anisotropic simple submodules in modules with a form

Let G be a finite group, and V a finite-dimensional semisimple G-module over a finite field. Assume that V is endowed with a nonsingular bilinear form which is symmetric or symplectic, and which is invariant under the action of G. In this setting, we compute the number of anisotropic simple submodules of V.Received: 25 May 2004