Equivariant Euler characteristics of discriminants of reflection groups

AbstractLet G be a finite, complex reflection group acting on a complex vector space V, and δ its disciminant polynomial. The fibres of δ admit commuting actions of G and a cyclic group. The virtual G × Cm character given by the Euler characteristics of a fibre is a refinement of the zeta function of the geometric monodromy, calculated in [8]. We show that this virtual character is unchanged by replacing δ by a slightly more general class of polynomials. We compute it explicitly, by studying the poset of normalizers of centralizers of regular elements in G, and the subspace arrangement given by the proper eigenspaces of elements of G. As a consequence we also compute orbifold Euler characteristics and find some new ‘case-free’ information about the discriminant