Derived equivalences for trigonometric double affine Hecke algebras

TL;DR

This paper constructs translation functors for trigonometric double affine Hecke algebras that induce derived category equivalences when parameters differ by integers, extending known results from rational Cherednik algebras.

Contribution

It introduces a new method to establish derived equivalences for trigonometric DAHAs based on parameter shifts, generalizing previous rational cases.

Findings

Constructed translation functors between module categories for different parameters.

Proved these functors induce derived category equivalences.

Extended Losev's rational Cherednik algebra results to the trigonometric setting.

Abstract

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 $c'$ which differ by integers, we construct the translation functor from $\mathbf{H}_{c}\operatorname{-Mod}$ to $\mathbf{H}_{c'}\operatorname{-Mod}$ 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.