On Lusztig’s Character Formula for Chevalley Groups of Type A l

Sherali S. Ibraev, Larissa Kainbaeva (), Gulzat M. Yensebayeva, Anar A. Ibrayeva, Manat Z. Parmenova and Gulnur K. Yeshmurat
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Sherali S. Ibraev: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan
Larissa Kainbaeva: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan
Gulzat M. Yensebayeva: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan
Anar A. Ibrayeva: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan
Manat Z. Parmenova: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan
Gulnur K. Yeshmurat: Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda 120014, Kazakhstan

Mathematics, 2024, vol. 12, issue 23, 1-19

Abstract: For a Chevalley group G over an algebraically closed field K of characteristic p > 0 with the irreducible root system R , Lusztig’s character formula expresses the formal character of a simple G -module by the formal characters of the Weyl modules and the values of the Kazhdan–Lusztig polynomials at 1. It is known that, for a sufficiently large characteristic p of the field K , Lusztig’s character formula holds. The known lower bound of the characteristic p is much larger than the Coxeter number h of the root system R . Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types A 1 , A 2 , A 3 , B 2 , B 3 , and C 3 , Lusztig’s character formula holds for all p ≥ h . For large Chevalley groups, no other examples are known. In this paper, for G of type A l , we give some series of simple modules for which Lusztig’s character formula holds for all p ≥ h . Using this result, we compute the cohomology of G with coefficients in these simple modules. To prove the results, Jantzen’s filtration properties for Weyl modules and the properties of Kazhdan–Lusztig polynomials are used.

Keywords: Chevalley group; simple module; Kazhdan–Lusztig polynomial; Lusztig’s character formula; Jantzen filtration; Jantzen sum formula (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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