Poincaré Duality and Steinberg’s Theorem on Rings of Coinvariants

Let k be a field, possibly of finite characteristic, V a finite dimensional vector space over k of dimension r, and W ⊂ Autk(V ) a finite subgroup. There is a natural action of W on the k-dual V # of V , as well as on the symmetric algebra S = S(V ). The algebra S is isomorphic to a polynomial algebra over k on r generators, and we are interested in the question of when the invariant algebra R = S is also isomorphic to such a polynomial algebra. It is well–known (see for instance Serre [8]) that R is polynomial if W is generated by reflections and the characteristic of k is zero or prime to the order of W . In a slightly different direction, Steinberg [10] has shown that R is polynomial if k is the field of complex numbers and the quotient algebra P∗ = S ⊗R k satisfies Poincaré duality (1.3). Steinberg’s result was extended by Kane [4, 5] to other fields of characteristic zero, and by T.-C. Lin [6] to the case in which k is a finite field of characteristic prime to the order of W . In this note we use elementary methods to prove Steinberg’s result for fields of characteristic 0 or of characteristic prime to the order of W . This gives a new proof even in the characteristic zero case.