Basic sets of Brauer characters of finite groups of Lie type,III

LetG(F ^q ) be a finite classical group whereq is odd and the centre ofG is connected. We show that there exists a set of irreducible characters ofG(F ^q ) such that the corresponding matrix of scalar products with the characters of Kawanaka’s generalized Gelfand-Graev representations is square unitriangular. This uses in an essential way Lusztig’s theory of character sheaves. As an application we prove that there exists an ordinary basic set of 2-modular Brauer characters and that the decomposition matrix of the principal 2-block ofG(F ^q ) has a lower unitriangular shape.