Cyclotomic double affine Hecke algebras

We show that the partially spherical cyclotomic rational Cherednik algebra (ob- tained 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 ratio- nal 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 va- rieties, which may be interpreted as K-theoretic Coulomb branches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA to prove new fitness results for various kinds of spaces of q-de- formed quasiinvariants.