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

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 A1,A2, A3,B2, B3, and C3, 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 Al, 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.