Finite Dimensional Modules for Rational Cherednik Algebras

We construct and study some finite dimensional modules for rational Cherednik algebras for the groups G(r, p, n) by using intertwining operators and a commutative family of operators introduced by Dunkl and Opdam. The coinvariant ring and an analog of the ring constructed by Gordon in the course of proving Haiman’s conjectures on diagonal coinvariants are special cases. We study a certain hyperplane arrangement that determines a weight basis for the irreducible modules in an explicit fashion. Using the same methods we also construct a basis of eigenfunctions for the coinvariant ring for G(r, p, n) whose leading terms are, in the case of G(1, 1, n), the descent monomials studied by Garsia and Stanton.