JACK POLYNOMIALS ATTACHED TO REPRESENTATIONS OF G(r, p, n)

The rational Cherednik algebra H is a certain algebra of differential-reflection operators attached to a complex reflection group. There is a category O of modules for this algebra which is a highest weight category. For the infinite family G(r, p, n) of complex reflection groups, the algebra H contains a subalgebra isomorphic to a (generalized) degenerate affine Hecke algebra, and our strategy is to study the standard modules in category O by means of this subalgebra. We use the Okounkov-Vershik approach to the representations of G(r, p, n) to compute the spectra of the standard modules in O with respect to the polynomial subalgebra of the affine Hecke algebra. The eigenbasis consists of a generalization of the non-symmetric Jack polynomials. As an application, we show that when the parameters are chosen “coprime to the Coxeter number of G(r, p, n)”, category O has an especially simple structure, with exactly one non-semisimple block. In the final section we compute the norms of the generalized Jack polynomials with respect to the contravariant form. This formula determines the radical of the standard modules in the cases in which the Jack polynomials are all well defined.