On Lusztig’s Character Formula for Chevalley Groups of Type Al


Ibraev S.S. Kainbaeva L. Yensebayeva G.M. Ibrayeva A.A. Parmenova M.Z. Yeshmurat G.K.
December 2024Multidisciplinary Digital Publishing Institute (MDPI)

Mathematics
2024#12Issue 23

For a Chevalley group (Formula presented.) over an algebraically closed field (Formula presented.) of characteristic (Formula presented.) with the irreducible root system (Formula presented.) Lusztig’s character formula expresses the formal character of a simple (Formula presented.) -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 (Formula presented.) of the field (Formula presented.) Lusztig’s character formula holds. The known lower bound of the characteristic (Formula presented.) is much larger than the Coxeter number (Formula presented.) of the root system (Formula presented.) Observations show that for simple modules with restricted highest weights of small Chevalley groups such as those of types (Formula presented.), and (Formula presented.), Lusztig’s character formula holds for all (Formula presented.). For large Chevalley groups, no other examples are known. In this paper, for (Formula presented.) of type (Formula presented.) we give some series of simple modules for which Lusztig’s character formula holds for all (Formula presented.). Using this result, we compute the cohomology of (Formula presented.) 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.

Chevalley group , Jantzen filtration , Jantzen sum formula , Kazhdan–Lusztig polynomial , Lusztig’s character formula , simple module

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Department of Physics and Mathematics, Institute of Natural Science, Korkyt Ata Kyzylorda Univesity, Aiteke bie St. 29A, Kyzylorda, 120014, Kazakhstan

Department of Physics and Mathematics

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