Gaudin from Cherednik

Abstract

Let $W$ be a complex reflection group and $X$ the De Concini-Procesi wonderful model of its reflection arrangement. Inside Etingof's sheaf of Cherednik algebras on $X$ we define a subsheaf generated by the Dunkl operators that extend the standard ones on the hyperplane complement. We conjecture that at $t=0$ the image of the logarithmic vector fields in a natural module over this subsheaf is the sheaf of Gaudin algebras on $X$ defined by Aguirre, extending a construction of Bonnafé and Rouquier from the hyperplane complement to $X$. We prove the conjecture for $W=S_{3}$.