Invariants and coinvariants of finite pseudoreflection groups, Jacobian determinants and Steenrod operations

Let rho : G hooked right arrow GL(n,F) be a representation of a finite group G over a finite field F and f(1),..., f(n) epsilon F[V](G) such that the ring of invariants is a polynomial algebra F[f(1),. . . ,f(n)]. It is known that in this case the algebra of coinvariants F[V](G) is a Poincare duality algebra, and if, moreover, the order of G is invertible in F, that a fundamental class is represented by the Jacobian determinant det[partial derivativef(i)/partial derivativez(j)], and is therefore a det(-1)-relative invariant. In this note we deduce what happens in the modular case. As a bonus we obtain a new criterion for an unstable algebra over the Steenrod algebra to be a ring of invariants.