A sharp upper bound for the size of Lusztig series

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 $G(q)$ and $q$ is large enough, we determine explicitly the maximum of the sizes of the Lusztig series of $G$.