AbstractThe 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.
We shall consider finite groups of order of g which satisfy the following condition:(*) There exists a prime p dividing g such that if P ≠ 1 is an element of p-Sylow group ofthen the centralizer(P) of P incoincides with the centralizer() of in.This assumption is satisfied for a number of important classes of groups. It also plays a role in discussing finite collineation groups in a given number of dimensions.Of course (*) implies that is abelian. It is possible to obtain rather detailed information about the irreducible characters of groups in this class (§ 4).
A sharp upper bound for the size of Lusztig series
