  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.
