Rings of generalized and stable invariants of pseudoreflections and pseudoreflection groups

Let rho: G hooked right arrow GL(n, F) be a representation of a finite group G over the field F and F[V] the space of polynomial functions on V = F-n. We associate to G an ideal L(infinity)(G) subset of F[V] called the ideal of stable invariants of G (or more precisely rho). If L subset of GL(n, F) is a set of pseudoreflections we associate to L an ideal I(L) subset of F[V] called the ideal of generalized invariants of L. When G is a pseudoreflection group we investigate I(L) for various choices of L subset of G and the relation between L(infinity)(G) and I(L). To a representation rho: G hooked right arrow GL(n, F) of a finite group, respectively to a set L subset of GL(n, F) of pseudoreflections, we also associate a ring <(gr)over bar>(J infinity(G)), respectively <(gr)over bar>(I(L)). We show that <(gr)over bar>(I(L)), is always a polynomial algebra over F, and whenever rho(G) is generated by semisimple pseudoreflections L that <(gr)over bar>(J infinity) congruent to <(gr)over bar>(I(L)). (C) 1996 Academic Press, Inc.