Highest weight modules at the critical level and noncommutative Springer resolution
In the article by Bezrukavnikov and Mirkovic a certain non-commutative algebra A was defined starting from a semi-simple algebraic group, so that the derived category of A-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let gˇ...
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| Format: | Article |
| Language: | en_US |
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American Mathematical Society
2013
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| Online Access: | http://hdl.handle.net/1721.1/80850 https://orcid.org/0000-0001-5902-8989 |
| _version_ | 1826189330622709760 |
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| author | Bezrukavnikov, Roman Lin, Qian |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Bezrukavnikov, Roman Lin, Qian |
| author_sort | Bezrukavnikov, Roman |
| collection | MIT |
| description | In the article by Bezrukavnikov and Mirkovic a certain non-commutative algebra A was defined starting from a semi-simple algebraic group, so that the derived category of A-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution.
Let gˇ be the Langlands dual Lie algebra and let [˄ over g] be the corresponding affine Lie algebra, i.e. [˄ over g] is a central extension of gˇ ⊗ C((t)).
Using results of Frenkel and Gaitsgory we show that the category of [˄ over g] modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a central reduction of A. This implies that numerics of Iwahori integrable modules at the critical level is governed by the canonical basis in the K-group of a Springer fiber, which was conjecturally described by Lusztig and constructed by Bezrukavnikov and Mirkovic. |
| first_indexed | 2024-09-23T08:13:23Z |
| format | Article |
| id | mit-1721.1/80850 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T08:13:23Z |
| publishDate | 2013 |
| publisher | American Mathematical Society |
| record_format | dspace |
| spelling | mit-1721.1/808502022-09-30T08:22:27Z Highest weight modules at the critical level and noncommutative Springer resolution Bezrukavnikov, Roman Lin, Qian Massachusetts Institute of Technology. Department of Mathematics Bezrukavnikov, Roman Lin, Qian In the article by Bezrukavnikov and Mirkovic a certain non-commutative algebra A was defined starting from a semi-simple algebraic group, so that the derived category of A-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let gˇ be the Langlands dual Lie algebra and let [˄ over g] be the corresponding affine Lie algebra, i.e. [˄ over g] is a central extension of gˇ ⊗ C((t)). Using results of Frenkel and Gaitsgory we show that the category of [˄ over g] modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a central reduction of A. This implies that numerics of Iwahori integrable modules at the critical level is governed by the canonical basis in the K-group of a Springer fiber, which was conjecturally described by Lusztig and constructed by Bezrukavnikov and Mirkovic. National Science Foundation (U.S.) (Grant DMS-0854764) National Science Foundation (U.S.) (Grant DMS-1102434) 2013-09-23T13:23:52Z 2013-09-23T13:23:52Z 2012 2010-08 Article http://purl.org/eprint/type/JournalArticle 9780821853177 9780821885369 1098-3627 0271-4132 http://hdl.handle.net/1721.1/80850 Bezrukavnikov, Roman, and Qian Lin. Highest weight modules at the critical level and noncommutative Springer resolution. American Mathematical Society, 2012. https://orcid.org/0000-0001-5902-8989 en_US http://dx.doi.org/10.1090/conm/565/11188 Algebraic Groups and Quantum Groups Creative Commons Attribution-Noncommercial-Share Alike 3.0 http://creativecommons.org/licenses/by-nc-sa/3.0/ application/pdf American Mathematical Society arXiv |
| spellingShingle | Bezrukavnikov, Roman Lin, Qian Highest weight modules at the critical level and noncommutative Springer resolution |
| title | Highest weight modules at the critical level and noncommutative Springer resolution |
| title_full | Highest weight modules at the critical level and noncommutative Springer resolution |
| title_fullStr | Highest weight modules at the critical level and noncommutative Springer resolution |
| title_full_unstemmed | Highest weight modules at the critical level and noncommutative Springer resolution |
| title_short | Highest weight modules at the critical level and noncommutative Springer resolution |
| title_sort | highest weight modules at the critical level and noncommutative springer resolution |
| url | http://hdl.handle.net/1721.1/80850 https://orcid.org/0000-0001-5902-8989 |
| work_keys_str_mv | AT bezrukavnikovroman highestweightmodulesatthecriticallevelandnoncommutativespringerresolution AT linqian highestweightmodulesatthecriticallevelandnoncommutativespringerresolution |