A MODULAR ANALOG OF A THEOREM OF R. STEINBERG ON COINVARIANTS OF COMPLEX PSEUDOREFLECTION GROUPS

Let (rho):G(hookrightarrow) GL(n, F) be a representation of a finite group over the field F, V = F^n the corresponding G-module, and F[V] the algebra of polynomial functions on V. The action of G on V extends to F[V], and F[V]^G, respectively F[V]_G, denotes the ring of invariants, respectively coinvariants. The theorem of Steinberg referred to in the title says that when F = C, dim C (Tot(C[V]_G))=|G| if and only if G is a complex reflection group. Here Tot(F[V]_G) denotes the direct sum of all the homogeneous components of the graded algebra F[V]_G and |G| is the order of G. Chevalleyʹs theorem tells us that the ring of invariants of a complex pseudoreflection representation G(hookrightarrow) GL(n, C) is polynomial algebra, and the theorem of Shephard and Todd yields the converse. Combining these results gives: dim_F(Tot(C[V]_G) = |G| if and only if C[V]^G is a polynomial algebra. The purpose of this note is to show that the two conditions