A remark on rational cherednik algebras and differential operators on the cyclic quiver

Abstract

We show that the spherical subalgebra Uk,c of the rational Cherednik algebra associated to Sn wreath product sign C ℓ, the wreath product of the symmetric group and the cyclic group of order ℓ, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size ℓ. This confirms a version of [5Conjecture 11.22] in the case of cyclic groups. The proof is a straightforward application of work of Oblomkov [12] on the deformed Harish-Chandra homomorphism, and of Crawley-Boevey, [3] and [4], and Gan and Ginzburg [7] on preprojective algebras. © 2006 Glasgow Mathematical Journal Trust.