Dimensions of modular irreducible representations of semisimple Lie algebras
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- by Roman Bezrukavnikov and Ivan Losev
- J. Amer. Math. Soc. 36 (2023), 1235-1304
- DOI: https://doi.org/10.1090/jams/1017
- Published electronically: March 16, 2023
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Abstract:
In this paper we classify and give Kazhdan-Lusztig type character formulas for equivariantly irreducible representations of Lie algebras of reductive algebraic groups over a field of large positive characteristic. The equivariance is with respect to a group whose connected component is a torus. Character computation is done in two steps. First, we treat the case of distinguished $p$-characters: those that are not contained in a proper Levi. Here we essentially show that the category of equivariant modules we consider is a cell quotient of an affine parabolic category $\mathcal {O}$. For this, we prove an equivalence between two categorifications of a parabolically induced module over the affine Hecke algebra conjectured by the first named author. For the general nilpotent $p$-character, we get character formulas by explicitly computing the duality operator on a suitable equivariant K-group.References
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Bibliographic Information
- Roman Bezrukavnikov
- Affiliation: Department of Mathematics, MIT, Cambridge, Massachusetts 02139
- MR Author ID: 347192
- Email: bezrukav@math.mit.edu
- Ivan Losev
- Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06511
- MR Author ID: 775766
- Email: ivan.loseu@yale.edu
- Received by editor(s): November 5, 2020
- Received by editor(s) in revised form: March 22, 2022, and August 30, 2022
- Published electronically: March 16, 2023
- Additional Notes: The first author was partially supported by the NSF under grant DMS-1601953. The second author was partially supported by the NSF under grant DMS-1501558.
- © Copyright 2023 American Mathematical Society
- Journal: J. Amer. Math. Soc. 36 (2023), 1235-1304
- MSC (2020): Primary 17B20, 17B35, 17B50
- DOI: https://doi.org/10.1090/jams/1017
- MathSciNet review: 4618958