Cyclotomic double affine Hecke algebras (with an appendix by Hiraku Nakajima and Daisuke Yamakawa)

TL;DR

This paper introduces cyclotomic double affine Hecke algebras (DAHA), providing multiple algebraic descriptions, a new q-deformation, and connections to quiver varieties and Coulomb branches, with applications to q-deformed quasiinvariants.

Contribution

It offers four equivalent descriptions of cyclotomic rational Cherednik algebras, defines a new q-deformation called cyclotomic DAHA, and links these algebras to geometric and physical structures.

Findings

Multiple algebraic descriptions of cyclotomic rational Cherednik algebra.

Introduction of a new q-deformation called cyclotomic DAHA.

Connections established between cyclotomic DAHA and quiver varieties, Coulomb branches, and q-deformed quasiinvariants.

Abstract

We show that the partially spherical cyclotomic rational Cherednik algebra (obtained from the full rational Cherednik algebra by averaging out the cyclotomic part of the underlying reflection group) has four other descriptions: (1) as a subalgebra of the degenerate DAHA of type A given by generators; (2) as an algebra given by generators and relations; (3) as an algebra of differential-reflection operators preserving some spaces of functions; (4) as equivariant Borel-Moore homology of a certain variety. Also, we define a new $q$-deformation of this algebra, which we call cyclotomic DAHA. Namely, we give a $q$-deformation of each of the above four descriptions of the partially spherical rational Cherednik algebra, replacing differential operators with difference operators, degenerate DAHA with DAHA, and homology with K-theory, and show that they give the same algebra. In addition, we show that spherical cyclotomic DAHA are quantizations of certain multiplicative quiver and bow varieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new flatness results for various kinds of spaces of $q$-deformed quasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added in version 2), the authors explain the relations between multiplicative bow varieties and (various versions of) multiplicative quiver varieties for a cyclic quiver.