Cherednik and Hecke Algebras of Varieties with a Finite Group Action
Let G be a finite group of linear transformations of a finite dimensional complex vector space V. To this data one can attach a family of algebras Ht,c(V, G), parametrized by complex numbers t and conjugation invariant functions c on the set of complex reflections in G, which are called rational Che...
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| Format: | Article |
| Language: | English |
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National Research University, Higher School of Economics (HSE)
2019
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| Online Access: | https://hdl.handle.net/1721.1/122972 |
| _version_ | 1826210012549087232 |
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| author | Etingof, Pavel I |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I |
| author_sort | Etingof, Pavel I |
| collection | MIT |
| description | Let G be a finite group of linear transformations of a finite dimensional complex vector space V. To this data one can attach a family of algebras Ht,c(V, G), parametrized by complex numbers t and conjugation invariant functions c on the set of complex reflections in G, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety X with a finite group G of automorphisms of X. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero–Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case π2(X) ⊗ Q = 0) a flat deformation of the orbifold fundamental group of the orbifold X/G, which we call the Hecke algebra of X/G. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples. Keyword: Cherednik algebra; reflection hypersurface; Hecke algebra; variety with a finite group action |
| first_indexed | 2024-09-23T14:39:46Z |
| format | Article |
| id | mit-1721.1/122972 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T14:39:46Z |
| publishDate | 2019 |
| publisher | National Research University, Higher School of Economics (HSE) |
| record_format | dspace |
| spelling | mit-1721.1/1229722022-10-01T21:57:42Z Cherednik and Hecke Algebras of Varieties with a Finite Group Action Etingof, Pavel I Massachusetts Institute of Technology. Department of Mathematics Let G be a finite group of linear transformations of a finite dimensional complex vector space V. To this data one can attach a family of algebras Ht,c(V, G), parametrized by complex numbers t and conjugation invariant functions c on the set of complex reflections in G, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety X with a finite group G of automorphisms of X. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero–Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case π2(X) ⊗ Q = 0) a flat deformation of the orbifold fundamental group of the orbifold X/G, which we call the Hecke algebra of X/G. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples. Keyword: Cherednik algebra; reflection hypersurface; Hecke algebra; variety with a finite group action National Science Foundation (U.S.) (Grant DMS-9988796) National Science Foundation (U.S.) (Grant DMS-1502244) U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (RM1-2545-MO-03) 2019-11-20T13:48:43Z 2019-11-20T13:48:43Z 2017-10 2019-11-12T16:57:37Z Article http://purl.org/eprint/type/JournalArticle 1609-4514 https://hdl.handle.net/1721.1/122972 Etingof, Pavel. "Cherednik and Hecke Algebras of Varieties with a Finite Group Action." Moscow Mathematical Journal 17, 4 (October 2017): 635-666 © Independent University of Moscow. en http://dx.doi.org/10.17323/1609-4514-2017-17-4-635-666 Moscow Mathematical Journal Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf National Research University, Higher School of Economics (HSE) arXiv |
| spellingShingle | Etingof, Pavel I Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title | Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title_full | Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title_fullStr | Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title_full_unstemmed | Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title_short | Cherednik and Hecke Algebras of Varieties with a Finite Group Action |
| title_sort | cherednik and hecke algebras of varieties with a finite group action |
| url | https://hdl.handle.net/1721.1/122972 |
| work_keys_str_mv | AT etingofpaveli cherednikandheckealgebrasofvarietieswithafinitegroupaction |