Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Abstract

Abstract. Let (G : V ) be a finite dimensional representation of a connected reductive complex Lie group (G : V ). Denote by G the derived subgroup ofG and assume that the categorical quotient V / /G is one dimensional, i.e. C[V ] G = C[f ] for a non constant polynomial f . In this situation there exists a homomorphism rad : D(V ) G → A 1 (C), the radial component map, where A 1 (C) is the first Weyl algebra. We show that the image of rad 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 (G : V ) is also multiplicity free we describe the kernel of rad and prove a Howe duality result between representations of G occuring in C[V ] and lowest weight modules over the Lie algebra generated by f and the "dual" differential operator ∆ ∈ S(V ); this extends results of H. Rubenthaler obtained when (G : V ) is a parabolic prehomogeneous vector space. If (G : V ) 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 M. Muro or P. Nang in special cases of the representation (G : V ).

D-modules on representations of Capelli type

“…By the way, using these results, T. Levasseur [27,Theorem 4.15] gives a duality (of Howe type) correspondence between (multplicity-free) representations (with a one-dimensional quotient) of G and lowest weight modules over the Lie algebra generated by f and ∆ (which is infinite dimensional when the degree of f is ≥ 3 We should note that when (G, V ) is irreducible, then Ω r = 0, the two sided ideal J = Σ r−1 j=0 AΩ j = Σ r−1 j=0 Ω j A, and (26)…”

“…Definition 2 (see Levasseur [27] ) A mutiplicity-free-space (G, V ) is said to have a one-dimensional quotient if there exists a non constant polynomial…”

“…First, we are interesting in the description of the algebras of G-invariant differential operators on a multiplicity-free space following the work by Benson -Ratcliff [1], Howe -Umeda [14], Knopp [25] and Levasseur [27]. Actually, the isomorphism m is G-invariant, hence the algebra of G-invariant differential operators decomposes as a direct sum of one-dimensional irreducible G-modules CE γ :…”

“…the algebra of polynomial coefficients G ′ -invariant differential operators. This algebra is well understood (see [14], [27]), in particular it contains θ the Euler vector field on V . Note that R. Howe and T. Umeda [14] have proved that when (G, V ) is of Capelli type, the algebraĀ is a polynomial algebra on a canonically defined set of generators.…”

“…⊂Ā denotes the two sided ideal annihilator of G ′ -invariant polynomials on V , we consider A the quotient algebraĀ/J , going modulo a suitable idealJ ofĀ described in section 4:J is the preimage in A of the ideal in A/J defined by specific relations (28), (29), (30), (31) [25] and Levasseur [27], we will deduce that the quotient algebra A is generated by the following three operators and relations (see Corollary 11): θ the Euler vector field on V , f the multiplication by the polymonial f (x) of degree n, and the differential operator ∆ := f ∂ ∂x as above satisfying the Bernstein-Sato equations:…”