Rational points and cohomology of discriminant varieties

Abstract

Let G be a finite unitary reflection group acting in a complex vector space V= C n . The discriminant variety X G of G is defined as the space of regular orbits of G on V. Classical examples include the varieties of complex polynomials of degree n with distinct (resp. non-zero distinct) roots. The normaliser G of G in GL( V) acts on X G ; in this work we determine the action of G/G on the cohomology of X G . In the classical cases this amounts to computing the cohomology of X G with certain local coefficient systems. Our methods are to compute equivariant weight polynomials by means of explicit counting of the rational points of certain varieties over finite fields, and then to exploit the weight purity of the relevant varieties. We obtain some power series identities as a byproduct.