Stratification of the discriminant in reflection groups

Abstract

LetV be a complex vector space and letG⊂GL(V) be a finite unitary reflection group. Letγ:V →V/G be the orbit map. We show that the image of the orbit stratification underγ is determined by a matrix of polynomials called the discriminant matrix, and that this stratification agrees with the stratification ofV/G obtained from logarithmic vector fields tangent to the discriminant hypersurface.