On R. Steinberg's theorem on algebras of coinvariants

Let G be a finite group and :G!GL(n,F) a representation of G over the field F. Steinberg’s Theorem states that if F = C, the coinvariant algebra C[V ]G is a Poincar´e duality algebra if and only if the invariant algebra C[V ]G is a polynomial algebra. The final result on the extension of this theorem to the nonmodular case (i.e. when |G| 2 F×) is due to W. G. Dwyer and C. W. Wilkerson [“Poincar´e duality and Steinberg’s theorem on rings of coinvariants”, preprint, hopf. math.purdue.edu/Dwyer-Wilkerson/GorensteinCoinvariants.pdf]. In the paper under review the author places their result in a more general context and gives new characteristic free applications to invariant theory. He proves that for a representation over the field F such that F[V ]G is a Poincar´e duality algebra of formal dimension d, F[V ]G is a polynomial algebra if and only if HomF[V ]G(F[V ],F[V ]) contains a nonzero element of degree −d. This theorem, together with some results on the twisted derivations s of Demazure, associated to refections s 2 GL(n,F), implies that if G is a reflection group such that F[V ]G is a Poincar´e duality algebra and the trivial representation 1G occurs only once as a subrepresentation in F[V ]G, then F[V ]G is a polynomial algebra.