Abstract
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.
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Geck, M. Basic sets of Brauer characters of finite groups of Lie type, III. Manuscripta Math 85, 195–216 (1994). https://doi.org/10.1007/BF02568193
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DOI: https://doi.org/10.1007/BF02568193