Derived equivalences for trigonometric double affine Hecke algebras

The trigonometric double affine Hecke algebra ${\mathbf{H}}_c$ for an irreducible root system depends on a family of complex parameters $c$ Given two families of parameters $c$ and $c^{\mathrm{\prime }}$ which differ by integers, we construct the translation functor from ${\mathbf{H}}_c\mathrm{-Mod}$ to ${\mathbf{H}}_{c^{\mathrm{\prime }}}\mathrm{-Mod}$ and prove that it induces equivalence of derived categories. This is a trigonometric counterpart of a theorem of Losev on the derived equivalences for rational Cherednik algebras.