[Paper Review] Radial components, prehomogeneous vector spaces, and rational Cherednik algebras

Thierry Levasseur|ArXiv.org|Jan 8, 2008

Advanced Algebra and Geometry27 references3 citations

TL;DR

This paper establishes a correspondence between the radial component map on G-invariant differential operators on a prehomogeneous vector space (G:V) with one-dimensional quotient V//G and the spherical subalgebra of a rational Cherednik algebra. It shows the image of the radial component map is isomorphic to this spherical subalgebra, whose multiplicity function is determined by the roots of the Bernstein-Sato polynomial of the invariant f. In multiplicity-free cases, it proves a Howe duality between G-representations in C[V] and lowest weight modules over the Lie algebra generated by f and its dual differential operator.

ABSTRACT

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.

Motivation & Objective

• To understand the structure of the algebra of G-invariant differential operators D(V)^G when the categorical quotient V//G is one-dimensional.
• To describe the image of the radial component map rad: D(V)^G → D(h/W) in terms of rational Cherednik algebras.
• To establish a Howe duality between representations of G in C[V] and lowest weight modules over the Lie algebra generated by f and its dual differential operator Δ in multiplicity-free cases.
• To extend results of Rubenthaler on parabolic prehomogeneous vector spaces to general prehomogeneous vector spaces with one-dimensional quotients.
• To apply the framework to Capelli-type representations, yielding new results on holonomic and equivariant D-modules.

Proposed method

• Define the radial component map rad: D(V)^G → D(h/W) as the composition of the G-invariant action and the isomorphism ψ: C[V]^G → C[h]^W induced by a Cartan subspace h ⊂ V.
• Show that the image of rad is isomorphic to the spherical subalgebra of a rational Cherednik algebra associated to the root system defined by the roots of the Bernstein-Sato polynomial of f.
• Use the theory of prehomogeneous vector spaces (PHVs) and polar representations to reduce the problem to a one-dimensional quotient, enabling the use of Weyl algebra structures.
• In multiplicity-free cases, construct a duality between G-representations in C[V] and lowest weight modules over the Lie algebra generated by f and Δ ∈ S(V), generalizing Howe duality.
• Apply the framework to Capelli-type representations, proving that the category of G × C-equivariant D-modules on V is equivalent to the category of θ-stable modules over a rational Cherednik algebra.
• Leverage results from Muro and Nang on special cases to support the conjecture that this equivalence holds universally for all Capelli-type PHVs with dim(V//G) = 1.

Experimental results

Research questions

• RQ1 What is the image of the radial component map rad: D(V)^G → D(h/W) when V//G is one-dimensional?
• RQ2 How is the spherical subalgebra of a rational Cherednik algebra related to the radial component map in prehomogeneous vector spaces with one-dimensional quotients?
• RQ3 Can a Howe duality be established between representations of G in C[V] and lowest weight modules over the Lie algebra generated by f and its dual differential operator Δ in multiplicity-free prehomogeneous vector spaces?
• RQ4 What are the implications of the Capelli-type condition on the structure of G-equivariant D-modules on V?
• RQ5 Is there a categorical equivalence between G × C-equivariant D-modules on V and θ-stable modules over a rational Cherednik algebra in Capelli-type prehomogeneous vector spaces with dim(V//G) = 1?

Key findings

• The image of the radial component map rad: D(V)^G → D(h/W) is isomorphic to the spherical subalgebra of a rational Cherednik algebra whose multiplicity function is determined by the roots of the Bernstein-Sato polynomial of f.
• In multiplicity-free prehomogeneous vector spaces with dim(V//G) = 1, the kernel of rad is described explicitly, and a Howe duality is established between G-representations in C[V] and lowest weight modules over the Lie algebra generated by f and Δ.
• For Capelli-type representations with dim(V//G) = 1, the category of G × C-equivariant D-modules on V is equivalent to the category of θ-stable modules over a rational Cherednik algebra.
• The conjecture that this equivalence holds universally for all Capelli-type PHVs with dim(V//G) = 1 is supported by known results of Muro and Nang in special cases.
• The framework provides a combinatorial classification of regular holonomic D-modules on V whose characteristic variety is contained in the nilpotent cone, when G is simply connected.
• The paper extends previous results of Rubenthaler from parabolic prehomogeneous vector spaces to general prehomogeneous vector spaces with one-dimensional quotients.