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

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.