Character Sums Associated to Finite Coxeter Groups

The main result of this paper is a character sum identity for Coxeter arrangements over finite fields which is an analogue of Macdonald’s conjecture [19] proved by Opdam [21]. 0. Introduction Throughout this paper F will denote a finite field of characteristic p different from 2. We first slightly reformulate Macdonald’s conjecture in a form which has a direct analogue over F. Let G be a finite subgroup of GLn(R) generated by reflections and let q be a positive definite quadratic form which is invariant under G. Let AG be the associated arrangement consisting of the reflection hyperplanes. Let l1, . . . , lN be equations for the N different reflection hyperplanes. Set ∆(x) = ( ∏N i=1 li) 2 and let d1, . . . , dn be the degrees of G. Macdonald’s conjecture [19] is the following equality,