Invariant differential operators and the generalized symmetric group

Abstract

In this paper we study the decomposition of the direct image of π₊(O_X) the polynomial ring O_X as a D-module, under the map π: spec O_X →spec O_X^G(r,n), where O_X^G(r,n) is the ring of invariant polynomial under the action of the wreath product G(r, p):= Z/rZ ~S_n. We first describe the generators of the simple components of π₊(O_X) and give their multiplicities. Using an equivalence of categories and the higher Specht polynomials, we describe a D-module decomposition of the polynomial ring localized at the discriminant of π. Furthermore, we study the action invariants, differential operators, on the higher Specht polynomials