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We show that the partially spherical cyclotomic rational Cherednik algebra(obtained from the full rational Cherednik algebra by averaging out thecyclotomic part of the underlying reflection group) has four otherdescriptions: (1) as a subalgebra of the degenerate DAHA of type A given bygenerators; (2) as an algebra given by generators and relations; (3) as analgebra of differential-reflection operators preserving some spaces offunctions; (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 thepartially spherical rational Cherednik algebra, replacing differentialoperators with difference operators, degenerate DAHA with DAHA, and homologywith K-theory, and show that they give the same algebra. In addition, we showthat spherical cyclotomic DAHA are quantizations of certain multiplicativequiver and bow varieties, which may be interpreted as K-theoretic Coulombbranches of a framed quiver gauge theory. Finally, we apply cyclotomic DAHA toprove new flatness results for various kinds of spaces of $q$-deformedquasiinvariants. In the appendix by H. Nakajima and D. Yamakawa (added inversion 2), the authors explain the relations between multiplicative bowvarieties and (various versions of) multiplicative quiver varieties for acyclic quiver.

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