Elliptic Double Affine Hecke Algebras
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 t...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2020 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2020
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/211009 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Elliptic Double Affine Hecke Algebras. Eric M. Rains. SIGMA 16 (2020), 111, 133 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862536164754325504 |
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| author | Rains, Eric M. |
| author_facet | Rains, Eric M. |
| citation_txt | Elliptic Double Affine Hecke Algebras. Eric M. Rains. SIGMA 16 (2020), 111, 133 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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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| first_indexed | 2026-03-12T16:05:00Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-211009 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-12T16:05:00Z |
| publishDate | 2020 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Rains, Eric M. 2025-12-22T09:26:51Z 2020 Elliptic Double Affine Hecke Algebras. Eric M. Rains. SIGMA 16 (2020), 111, 133 pages 1815-0659 2020 Mathematics Subject Classification: 33D80; 39A70; 14A22 arXiv:1709.02989 https://nasplib.isofts.kiev.ua/handle/123456789/211009 https://doi.org/10.3842/SIGMA.2020.111 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. 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 authors 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. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Elliptic Double Affine Hecke Algebras Article published earlier |
| spellingShingle | Elliptic Double Affine Hecke Algebras Rains, Eric M. |
| title | Elliptic Double Affine Hecke Algebras |
| title_full | Elliptic Double Affine Hecke Algebras |
| title_fullStr | Elliptic Double Affine Hecke Algebras |
| title_full_unstemmed | Elliptic Double Affine Hecke Algebras |
| title_short | Elliptic Double Affine Hecke Algebras |
| title_sort | elliptic double affine hecke algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211009 |
| work_keys_str_mv | AT rainsericm ellipticdoubleaffineheckealgebras |