Pramod N. Achar, Shotaro Makisumi, S. Riche, G. Williamson
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Williamson","doi":"10.1090/JAMS/905","DOIUrl":null,"url":null,"abstract":"<p>We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\">\n <mml:semantics>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\ell</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> in terms of <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l\">\n <mml:semantics>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n <mml:annotation encoding=\"application/x-tex\">\\ell</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>-Kazhdan–Lusztig polynomials, for <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l greater-than h\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n <mml:mo>></mml:mo>\n <mml:mi>h</mml:mi>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\ell > h</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula> the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if <inline-formula content-type=\"math/mathml\">\n<mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"script l greater-than-or-equal-to 2 h minus 2\">\n <mml:semantics>\n <mml:mrow>\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\n <mml:mo>≥<!-- ≥ --></mml:mo>\n <mml:mn>2</mml:mn>\n <mml:mi>h</mml:mi>\n <mml:mo>−<!-- − --></mml:mo>\n <mml:mn>2</mml:mn>\n </mml:mrow>\n <mml:annotation encoding=\"application/x-tex\">\\ell \\ge 2h-2</mml:annotation>\n </mml:semantics>\n</mml:math>\n</inline-formula>. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/JAMS/905","citationCount":"77","resultStr":"{\"title\":\"Koszul duality for Kac–Moody groups and characters of tilting modules\",\"authors\":\"Pramod N. Achar, Shotaro Makisumi, S. Riche, G. Williamson\",\"doi\":\"10.1090/JAMS/905\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l\\\">\\n <mml:semantics>\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> in terms of <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l\\\">\\n <mml:semantics>\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>-Kazhdan–Lusztig polynomials, for <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l greater-than h\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n <mml:mo>></mml:mo>\\n <mml:mi>h</mml:mi>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell > h</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula> the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if <inline-formula content-type=\\\"math/mathml\\\">\\n<mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"script l greater-than-or-equal-to 2 h minus 2\\\">\\n <mml:semantics>\\n <mml:mrow>\\n <mml:mi>ℓ<!-- ℓ --></mml:mi>\\n <mml:mo>≥<!-- ≥ --></mml:mo>\\n <mml:mn>2</mml:mn>\\n <mml:mi>h</mml:mi>\\n <mml:mo>−<!-- − --></mml:mo>\\n <mml:mn>2</mml:mn>\\n </mml:mrow>\\n <mml:annotation encoding=\\\"application/x-tex\\\">\\\\ell \\\\ge 2h-2</mml:annotation>\\n </mml:semantics>\\n</mml:math>\\n</inline-formula>. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.</p>\",\"PeriodicalId\":54764,\"journal\":{\"name\":\"Journal of the American Mathematical Society\",\"volume\":\"1 1\",\"pages\":\"\"},\"PeriodicalIF\":3.5000,\"publicationDate\":\"2017-06-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://sci-hub-pdf.com/10.1090/JAMS/905\",\"citationCount\":\"77\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/JAMS/905\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/JAMS/905","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 77
摘要
我们建立了一个关于特征为r \ well的连通约化群的不可分解倾斜模的特征公式，它是用r \ well -Kazhdan-Lusztig多项式表示的，对于r \ well - h \ well - > h为Coxeter数。利用Andersen的结果，我们可以推导出当r≥2h−2 \ well \ge 2h-2时简单模的特征公式。我们的结果是将Bezrukavnikov和Yun建立的一元Koszul对偶等价推广到模系数的结果。
Koszul duality for Kac–Moody groups and characters of tilting modules
We establish a character formula for indecomposable tilting modules for connected reductive groups in characteristic ℓ\ell in terms of ℓ\ell-Kazhdan–Lusztig polynomials, for ℓ>h\ell > h the Coxeter number. Using results of Andersen, one may deduce a character formula for simple modules if ℓ≥2h−2\ell \ge 2h-2. Our results are a consequence of an extension to modular coefficients of a monoidal Koszul duality equivalence established by Bezrukavnikov and Yun.
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