Basic sets of Brauer characters of finite groups of Lie type. III
This is the third in a series of papers investigating basic sets of Brauer characters of finite groups of Lie type [for part I see the author and \textit{G. Hiss}, J. Reine Angew. Math. 418, 173-188 (1991; Zbl 0771.20008) and for part II cf. J. Lond. Math. Soc., II. Ser. 47, 255-268 (1994; Zbl 0797.20013)]. In the first two papers the author asserted that ordinary basic sets always exist provided that \(\ell\) is good for \(G\), where \(G\) is a connected reductive group over \(\mathbb{F}_ q\); and that \(\ell\) does not divide the order of the group of components of the centre of \(G\). In this paper he studies the case where \(G\) is of classical type with connected centre and \(\ell = 2\), \(q\) is odd. Let \(G(\mathbb{F}_ q)\) be a finite classical group corresponding to \(G\). The author shows that there exists a set of irreducible characters of \(G(\mathbb{F}_ q)\) such that the corresponding matrix of scalar products with the characters of Kawanaka's generalized Gelfand-Graev representations is square unitriangular by using in an essential way Lusztig's theory of character sheaves. As an application he proves that there exists an ordinary basic set of 2- modular Brauer characters and that the decomposition matrix of the principal 2-block of \(G(\mathbb{F}_ q)\) has lower unitriangular shape.
decomposition matrix of principal 2-block
scientific article; zbMATH DE number 714699
scientific article