On elliptic Dunkl operators

TL;DR

This paper introduces elliptic Dunkl operators on abelian varieties with finite group actions, proving their commutativity and exploring their role in representations of elliptic Cherednik and double affine Hecke algebras.

Contribution

It generalizes elliptic Dunkl operators to abelian varieties, establishes their commutativity, and connects them to representations of elliptic Cherednik and double affine Hecke algebras.

Findings

Operators commute, enabling new algebraic structures.

Representations from category O of elliptic Cherednik algebras are constructed.

Finite dimensional monodromy representations of differential equations are obtained.

Abstract

We attach elliptic Dunkl operators to an abelian variety with a finite group action. This generalizes elliptic Dunkl operators for Weyl groups, defined by Buchstaber, Felder, and Veselov in 1994. We show that these operators commute, and use them to define representations from category O of elliptic Cherednik algebras. We also consider the monodromy representations of differential equations defined by elliptic Dunkl operators, and show that they yield finite dimensional rrepresentations of generalized double affine Hecke algebras.