Jacobians of Reflection Groups

Abstract

Steinberg showed that when a finite reflection group acts on a real or complex vector space of finite dimension, the Jacobian determinant of a set of basic invariants factors into linear forms which define the reflecting hyperplanes. This result generalizes verbatim to fields whose characteristic is prime to the order of the group. Our main theorem gives a generalization of Steinberg's result for arbitrary fields using a ramification formula of Benson and Crawley-Boevey. As an intermediate result, we show that every finite group which fixes a hyperplane pointwise has a polynomial ring of invariants.