A REMARK ON RATIONAL CHEREDNIK ALGEBRAS AND DIFFERENTIAL OPERATORS ON THE CYCLIC QUIVER

Abstract

We show that the spherical subalgebra $U_{k,c}$ of the rational Cherednik algebra associated to $S_n C_{\ell}$, the wreath product of the symmetric group and the cyclic group of order $\ell$, is isomorphic to a quotient of the ring of invariant differential operators on a space of representations of the cyclic quiver of size $\ell$. 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.