Radial components, prehomogeneous vector spaces, and rational Cherednik algebras

Let V be a finite dimensional representation of the connected complex reductive group H. Denote by G the derived subgroup of H and assume that the categorical quotient of V by G is one dimensional. In this situation there exists a homomorphism, denoted by rad, from the algebra A of G-invariant differential operators on V to the first Weyl algebra. We show that the image of rad is isomorphic to the spherical subalgebra of a Cherednik algebra, whose parameters are determined by the b-function of the relative invariant associated to the prehomogeneous vector space (H : V). If (H : V) is furthemore assumed to be multiplicity free we obtain a Howe duality between a set of representations of G and modules over a subalgebra of the associative Lie algebra A. Some applications to holonomic modules and H-equivariant D-modules on V are also given.