The purpose of this talk is to define and study certain new algebras (depending on two non-negative intergers $n$ and $l$ and a bunch of continuous parameters) which we call cyclotomic double affine Hecke algebras. These are q-deformations of certain "partly spherical" subalgebras of cyclotomic rational Cherednik algebras. In this talk I am going to mention:

1) An algebraic definition of cyclotomic DAHA

2) A (partly conjectural) geometric definition in terms of equivariant K-theory

3) A relation of cyclotomic DAHA to quantization of certain multiplicative quiver varieties (which in this case also coincide with multiplicative bow varieties of Cherkis-Nakajima-Takayama)

4) An application to the study of the ring of q-quasi-invariants (Etingof-Rains conjecture).

This is a joint work with P.Etingof and M.Finkelberg.