Generalized Jack Polynomials and the Representation Theory of Rational Cherednik Algebras

We study the rational Cherednik algebra Hc of type G(r, 1, n) by means of the Dunkl-Opdam operators introduced in [DuOp] and the generalized Jack polynomials introduced in [Gri]. Our main result is a characterization of the set of parameters c for which Hce+Hc 6= Hc, where e+ is the symmetrizing idempotent for G(r, 1, n). Such parameters are called aspherical. The proof is parallel to that in [Dun] for the case r = 1; in this case the result was first proved by Gordon-Stafford [GoSt] and Bezrukavnikov-Etingof [BeEt]. The parameter c is a tuple c = (c0, d0, . . . , dr−1) of complex numbers with d0+d1+ · · ·+dr−1 = 0 (see (1.4) for its relationship to Hc). For any l ∈ Z we define dl = dl′ if l = l ′ mod r and 0 ≤ l′ ≤ r−1. Our result is