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Elliptic Double Affine Hecke Algebras

Rains, Eric M. (2020) Elliptic Double Affine Hecke Algebras. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 16 . Art. No. 111. ISSN 1815-0659. doi:10.3842/SIGMA.2020.111.

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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 C_n version of the construction to construct a flat noncommutative deformation of the nth 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.

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Additional Information:© 2020 the Authors. Creative Commons Attribution-ShareAlike License Received December 19, 2019, in final form October 16, 2020; Published online November 05, 2020. This paper is a contribution to the Special Issue on Elliptic Integrable Systems, Special Functions and Quantum Field Theory. The full collection is available at The author would particularly like to thank P. Etingof both for asking the original seed question (with an important assist from A. Okounkov!) and hosting the author's sabbatical at MIT (which the author would also like to thank, naturally) where much of the basic approach was worked out, with great assistance from conversations with not only Etingof but also (regarding various geometrical issues) B. Poonen. Thanks also go to T. Graber and E. Mantovan for helpful and encouraging conversations regarding the constructions of Section 2 as well as various general algebraic geometric questions, and to O. Chalykh for asking some fruitful questions about residue conditions. The author's work presented here was supported in part by grants from the National Science Foundation, DMS-1001645 and DMS-1500806.
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Subject Keywords:elliptic curves; Hecke algebras; noncommutative deformations
Classification Code:2020 Mathematics Subject Classification: 33D80; 39A70; 14A22
Record Number:CaltechAUTHORS:20170922-134529235
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81753
Deposited By: Tony Diaz
Deposited On:22 Sep 2017 21:25
Last Modified:15 Nov 2021 19:45

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