Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 16 (2020), 111, 133 pages      arXiv:1709.02989      https://doi.org/10.3842/SIGMA.2020.111
Contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory

Elliptic Double Affine Hecke Algebras

Eric M. Rains
Department of Mathematics, California Institute of Technology, USA

Received December 19, 2019, in final form October 16, 2020; Published online November 05, 2020

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.

Key words: elliptic curves; Hecke algebras; noncommutative deformations.

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