A sharp upper bound for the size of Lusztig series

Abstract

Abstract The paper is concerned with the character theory of finite groups of Lie type. The irreducible characters of a group G of Lie type are partitioned in Lusztig series. We provide a simple formula for an upper bound of the maximal size of a Lusztig series for classical groups with connected center; this is expressed for each group G in terms of its Lie rank and defining characteristic. When G is specified as G ⁢ ( q ) {G(q)} and q is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of G.