On the decomposition numbers of \(G_ 2(q)\) (Q1117316)

Let \(G=G_ 2(q)\) where q is a power of p. The r-blocks of G and the Brauer trees for r-blocks with cyclic defect groups where r is a prime not equal to 2, 3 or p were computed by \textit{J. Shamash} [J. Algebra 123, 378-396 (1989); Commun. Algebra (to appear)]. In this paper the author investigates the decomposition matrices for r-blocks with non-cyclic defect groups, i.e. r-blocks of maximal defect where r divides q-1 or \(q+1\). The decomposition numbers are determined completely when r divides q-1 (Theorem A). When r divides \(q+1\) they are determined up to some ambiguities (Theorem B). The method involves constructing projective characters of G by inducing from \(r'\)-subgroups or by tensoring defect 0 characters with ordinary characters, and then computing scalar products of the projective characters with characters in the given block. An interesting result is that the component of the Gelfand-Graev representation (which is projective for the prime r) which lies in a given r-block is always indecomposable. (Remark. The r-blocks of G for \(r=3\) are studied by the author and \textit{J. Shamash} [in 3-blocks and 3- modular characters of \(G_ 2(q)\), Preprint, RWTC Aachen].)