Elliptic Double Affine Hecke Algebras

TL;DR

This paper constructs elliptic analogues of affine Hecke algebras for Coxeter groups acting on abelian varieties and applies these to deform symmetric powers of rational surfaces and potentially their Hilbert schemes.

Contribution

It introduces a new elliptic construction of affine Hecke algebras for Coxeter groups and uses it to produce novel noncommutative deformations of geometric objects.

Findings

Constructed elliptic affine Hecke algebras for Coxeter groups.

Deformed symmetric powers of rational surfaces noncommutatively.

Proposed a conjectural deformation of Hilbert schemes.

Abstract

We give a construction of an affine Hecke algebra associated to any Coxeter group acting on an abelian variety by reflections; in the case of an affine Weyl group, the result is an elliptic analogue of the usual double affine Hecke algebra. As an application, we use a variant of the $\tilde{C}_n$ version of the construction to construct a flat noncommutative deformation of the $n$th symmetric power of any rational surface with a smooth anticanonical curve, and give a further construction which conjecturally is a corresponding deformation of the Hilbert scheme of points.