Representations of Chevalley Groups Arising from Admissible Lattices

The modules for a Chevalley group arising from admissible lattices in an irreducible module for the associated complex semisimple Lie algebra are studied. It is proved that the transpose of such a module is still in this collection and generically the cohomology modules of line bundles on the flag varieties are in this collection also. In the rank 1 case, all modules in this collection are indecomposable and we hope this is true in general. Introduction The purpose of this paper is to investigate the modules for a Chevalley group obtained by reduction from admissible lattices in an irreducible module for the associated complex Lie algebra g. These modules have been studied by W.J. Wong in [9]. It is well known that different admissible lattices in a module can give rise to different modules for the Chevalley group, though they have the same character. It is still not clear what kind of modules can be obtained in this way. The Weyl modules and the induced modules are obtained from the minimal and maximal admissible lattices respectively. The structure of these modules is still a mystery. In Section 1 we show that whenever M is a module arising from an admissible lattice, so is its transposeM tr ([2] 2.1). This is based on the contravariant form defined by W.J. Wong in [9]. Section 2 is based on the work of Andersen on the induction theory for Chevalley groups over the integers [1]. It is proved that wheneverH `(w) k (w·λ) is the only nonvanishing cohomology group of the line bundle, it can be obtained from an admissible lattice in V (λ). Besides these cohomology modules, there are still many modules arising from admissible lattices. It turns out that every quotient of a Weyl module is a submodule of such a module. In order to study these modules in general, we deal with sl2 in Section 3. In this case, we prove that all modules arising from admissible lattices in an irreducible module for sl2 are indecomposable for SL2. Communications with James E. Humphreys on this matter were very helpful. The calculation of the p-adic valuations of binomial coefficients in 3.4 is motivated from a conversation with Neal Koblitz. 1980 Mathematics Subject Classification (1985 Revision) Primary 20G05