Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Abstract

Let be a finite-dimensional representation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situation there exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that the image of is isomorphic to the spherical subalgebra of a rational Cherednik algebra whose multiplicity function is defined by the roots of the Bernstein-Sato polynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representations of G occurring in and lowest-weight modules over the Lie algebra generated by f and the “dual” differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space. If satisfies a Capelli-type condition, some applications are given to holonomic and equivariant D-modules on V. These applications are related to results proved by Muro [34, 37, 38] or Nang [40, 42, 44] in special cases of the representation .