Let G be a group of Lie type
over the field of q elements. Let χ be a nonlinear irreducible character of G and x
a noncentral element of G. Examination of character tables suggests that
|χ(x)∕χ(1)|≤ C∕q, where C is a universal constant independent of χ, x, and G.
This order of magnitude is attained when, for example, χ is the doubly
transitive permutation character of GL(n,q) and x centralizes a hyperplane of
PG(n − 1,q); |χ(x)∕χ(1)| then approaches 1∕q as n →∞. In this paper, we
establish a bound of the above type when x is a semisimple element which has
prime order modulo Z(G). However, we must exclude certain groups G in
characteristic 2 and 3. The most serious exclusions are the groups of type Cn in
characteristic 2. Our proof, which is summarized below, does not use Deligne-Lusztig
theory.
