Character Values at Regular Elements

There has been a good deal of work in the last decade or so on computing upper bounds of irreducible character values for finite simple groups [G1, G2, GM, LS2, LP, MS-P, R]. Some of this work concerns permutation groups, some groups of Lie type. Some is aimed at worst-case behavior, some at the character values of typical or at least fairly well-behaved elements. Various applications have been given, to mixing behavior of random walks, to diameters of Cayley graphs, to the conjectures of Ore and Thompson, and to solving word equations in finite simple groups. The novelty of this paper is that it gives a uniform, elementary method that applies to sufficiently good elements in any finite group and which gives respectable results both for alternating groups and groups of Lie type. Let G be a finite group. We say that a ∈ G is regular if the centralizer Z(a) of a in G is abelian. We give an upper bound for the absolute value of irreducible characters evaluated at regular elements.