Geometrization of the Satake transform for mod p Hecke algebras
We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake trans...
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Cambridge University Press
2025-01-01
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| Series: | Forum of Mathematics, Sigma |
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| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424001300/type/journal_article |
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| author | Robert Cass Yujie Xu |
| author_facet | Robert Cass Yujie Xu |
| author_sort | Robert Cass |
| collection | DOAJ |
| description | We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a mod p Local Langlands Correspondence. |
| format | Article |
| id | doaj-art-5ebd17e2d3244378a8d78963e80b3365 |
| institution | Kabale University |
| issn | 2050-5094 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Cambridge University Press |
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| series | Forum of Mathematics, Sigma |
| spelling | doaj-art-5ebd17e2d3244378a8d78963e80b33652025-02-05T07:57:33ZengCambridge University PressForum of Mathematics, Sigma2050-50942025-01-011310.1017/fms.2024.130Geometrization of the Satake transform for mod p Hecke algebrasRobert Cass0https://orcid.org/0000-0003-2242-5758Yujie Xu1https://orcid.org/0000-0003-3023-3609University of Michigan, 530 Church St, Ann Arbor, MI 48109, U.S.A.Columbia University, 2990 Broadway, New York, NY 10027, U.S.A.; E-mail:We geometrize the mod p Satake isomorphism of Herzig and Henniart–Vignéras using Witt vector affine flag varieties for reductive groups in mixed characteristic. We deduce this as a special case of a formula, stated in terms of the geometry of generalized Mirković–Vilonen cycles, for the Satake transform of an arbitrary parahoric mod p Hecke algebra with respect to an arbitrary Levi subgroup. Moreover, we prove an explicit formula for the convolution product in an arbitrary parahoric mod p Hecke algebra. Our methods involve the constant term functors inspired from the geometric Langlands program, and we also treat the case of reductive groups in equal characteristic. We expect this to be a first step toward a geometrization of a mod p Local Langlands Correspondence.https://www.cambridge.org/core/product/identifier/S2050509424001300/type/journal_article14D2420C0814M15 |
| spellingShingle | Robert Cass Yujie Xu Geometrization of the Satake transform for mod p Hecke algebras Forum of Mathematics, Sigma 14D24 20C08 14M15 |
| title | Geometrization of the Satake transform for mod p Hecke algebras |
| title_full | Geometrization of the Satake transform for mod p Hecke algebras |
| title_fullStr | Geometrization of the Satake transform for mod p Hecke algebras |
| title_full_unstemmed | Geometrization of the Satake transform for mod p Hecke algebras |
| title_short | Geometrization of the Satake transform for mod p Hecke algebras |
| title_sort | geometrization of the satake transform for mod p hecke algebras |
| topic | 14D24 20C08 14M15 |
| url | https://www.cambridge.org/core/product/identifier/S2050509424001300/type/journal_article |
| work_keys_str_mv | AT robertcass geometrizationofthesataketransformformodpheckealgebras AT yujiexu geometrizationofthesataketransformformodpheckealgebras |