Differential operators and Cherednik algebras

Abstract

We establish a link between two geometric approaches to the representation theory of rational Cherednik algebras of type A: one based on a noncommutative Proj construction [GS1]; the other involving quantum hamiltonian reduction of an algebra of differential operators [GG]. In this paper, we combine these two points of view by showing that the process of hamiltonian reduction intertwines a naturally defined geometric twist functor on D-modules with the shift functor for the Cherednik algebra. That enables us to give a direct and relatively short proof of the key result [GS1, Theorem 1.4] without recourse to Haiman's deep results on the n! theorem [Ha1]. We also show that the characteristic cycles defined independently in these two approaches are equal, thereby confirming a conjecture from [GG].

Generalized Jack polynomials and the representation theory of rational Cherednik algebras

“…Writing e for the symmetrizing idempotent (corresponding to the trivial character), Proposition 5.5 of Berest-Chalykh [1] states that for certain choices of ξ , we have an isomorphism eH c e ∼ = e ξ H c e ξ for a parameter c depending on c and ξ , so combining this with our results allows one to establish Morita equivalences between rational Cherednik algebras at different parameters. We hope that by combining these equivalences with the techniques of the papers [7,8] and [5], one may obtain a tighter relationship between the Cherednik algebra of type G(r, 1, n) and certain quiver varieties, including the Hilbert schemes of points on resolutions of type A surface singularities.…”

Harish-Chandra bimodules over rational Cherednik algebras

“…σ∈Sn sign(σ)σ. The bimodule Q c+1 c is HC, this follows from [GGS,Theorem 1.7]. The functor F : eH c e(n)-mod → eH c+1 e(n)-mod given by F (M ) = Q c+1 c ⊗ eHce M is then an equivalence of categories, [BE,Corollary 4.3].…”

The Bernstein–Sato $b$-function of the Space of Cyclic Pairs

“…Let W = S n be the Weyl group corresponding to gl n . Consistent with the notation in [5], we define X reg to consist of pairs (M, v) where M ∈ g rs . Lastly define h reg ⊂ h to be those points which avoid the root hyperplanes.…”

“…Since radial reduction maps the standard Laplacian operator ∆ to the Calogero-Moser operator on h [5], we can think of X and f as describing a broader, yet simpler, precursor situation to the one studied in [11]. More generally, the function f and the space X are relevant to the study of mirabolic D-modules and rational Cherednik algebras [2] - [4].…”