The image of the KZ functor for Cherednik algebras of varieties with finite group actions

TL;DR

This paper demonstrates the essential surjectivity of the KZ functor for Cherednik algebras associated with varieties with finite group actions and explores the structure of category O in this context.

Contribution

It establishes the essential surjectivity of the KZ functor for certain Cherednik algebra modules and investigates category O for Cherednik algebras on Riemann surfaces and elliptic curves.

Findings

KZ functor is essentially surjective for specific Cherednik algebra modules

Conditions identified for non-vanishing of category O on Riemann surfaces

Analysis of Cherednik algebras on products of elliptic curves

Abstract

We prove that the KZ functor from a certain category of modules for the Cherednik algebra to finite dimensional modules over the Hecke algebra is essentially surjective. Then we begin to use this result to study the analog of category O for Cherednik algebras on Riemann surfaces and on products of elliptic curves. In particular we give conditions on the parameters under which these categories are nonzero.