Poisson Traces for Symmetric Powers of Symplectic Varieties
We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T*C[superscript n-1]/S[subscript n] associated with the type...
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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Oxford University Press
2015
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| Online Access: | http://hdl.handle.net/1721.1/92850 https://orcid.org/0000-0002-0710-1416 |
| _version_ | 1811073933967884288 |
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| author | Etingof, Pavel I. Schedler, Travis |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. Schedler, Travis |
| author_sort | Etingof, Pavel I. |
| collection | MIT |
| description | We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T*C[superscript n-1]/S[subscript n] associated with the type A Weyl group S[subscript n] and its reflection representation C[superscript n-1]. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional irreducible representations and prime ideals of quantizations of these varieties. Finally, motivated by these results, we pose conjectures on symplectic resolutions, and give related examples of the natural D-module. In an appendix, the second author computes the Poisson traces and associated D-module for the quotients T*C[superscript n]/D[subscript n] associated with type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems. |
| first_indexed | 2024-09-23T09:40:28Z |
| format | Article |
| id | mit-1721.1/92850 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T09:40:28Z |
| publishDate | 2015 |
| publisher | Oxford University Press |
| record_format | dspace |
| spelling | mit-1721.1/928502022-09-26T13:00:54Z Poisson Traces for Symmetric Powers of Symplectic Varieties Etingof, Pavel I. Schedler, Travis Massachusetts Institute of Technology. Department of Mathematics Etingof, Pavel I. We compute the space of Poisson traces on symmetric powers of affine symplectic varieties. In the case of symplectic vector spaces, we also consider the quotient by the diagonal translation action, which includes the quotient singularities T*C[superscript n-1]/S[subscript n] associated with the type A Weyl group S[subscript n] and its reflection representation C[superscript n-1]. We also compute the full structure of the natural D-module, previously defined by the authors, whose solution space over algebraic distributions identifies with the space of Poisson traces. As a consequence, we deduce bounds on the numbers of finite-dimensional irreducible representations and prime ideals of quantizations of these varieties. Finally, motivated by these results, we pose conjectures on symplectic resolutions, and give related examples of the natural D-module. In an appendix, the second author computes the Poisson traces and associated D-module for the quotients T*C[superscript n]/D[subscript n] associated with type D Weyl groups. In a second appendix, the same author provides a direct proof of one of the main theorems. National Science Foundation (U.S.) (Grant DMS-1000113) 2015-01-14T14:58:28Z 2015-01-14T14:58:28Z 2013-03 2011-09 Article http://purl.org/eprint/type/JournalArticle 1073-7928 1687-0247 http://hdl.handle.net/1721.1/92850 Etingof, P., and T. Schedler. “Poisson Traces for Symmetric Powers of Symplectic Varieties.” International Mathematics Research Notices (March 21, 2013). https://orcid.org/0000-0002-0710-1416 en_US http://dx.doi.org/10.1093/imrn/rnt031 International Mathematics Research Notices Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Oxford University Press arXiv |
| spellingShingle | Etingof, Pavel I. Schedler, Travis Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title | Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title_full | Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title_fullStr | Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title_full_unstemmed | Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title_short | Poisson Traces for Symmetric Powers of Symplectic Varieties |
| title_sort | poisson traces for symmetric powers of symplectic varieties |
| url | http://hdl.handle.net/1721.1/92850 https://orcid.org/0000-0002-0710-1416 |
| work_keys_str_mv | AT etingofpaveli poissontracesforsymmetricpowersofsymplecticvarieties AT schedlertravis poissontracesforsymmetricpowersofsymplecticvarieties |