Rational Cherednik algebras of $G(\ell ,p,n)$ from the Coulomb perspective

Abstract

We prove a number of results on the structure and representation theory of the rational Cherednik algebra of the imprimitive reflection group $G(\ell ,p,n)$. In particular, we: (1) show a relationship to the Coulomb branch construction of Braverman, Finkelberg and Nakajima, and 3-dimensional quantum field theory; (2) show that the spherical Cherednik algebra carries the structure of a principal Galois order; (3) construct a graded lift of category $\mathcal{O}$ and the larger category of Dunkl-Opdam modules, whose simple modules have the properties of a dual canonical basis and (4) give the first classification of simple Dunkl-Opdam modules for the rational Cherednik algebra of the imprimitive reflection group $G(\ell ,p,n)$.

42 pages; v1: preliminary version, comments encouraged v2: edited for submission; only minor changes for clarity, etc