ON REPRESENTATIONS OF SPESIAL LINEAR GROUPS WITH SINGULAR HIGHEST WEIGHTS

Abstract

In this paper, we study representations with singular highest weights of semi-simple algebraic groups of type  over an algebraically closed field  of positive characteristic  where  is the Coxeter number. We consider only the representations whose highest weights belong to the restricted region of restricted weights. It is known that, in the restricted region, the theory of representations of semi-simple simply connected algebraic groups and their Lie algebras are equivalent. Therefore, the results obtained in this paper for algebraic groups will also be valid for representations of their Lie algebras. In the case of regular highest weights, the structure of similar representations is known. Singular highest weights fall on the walls of affine alcoves, and in this case, the usual structure of representations with regular highest weights is collapsed. Currently such representations with singular highest weights are poorly understood. We give a description of the structure of some Weyl modules with highest singular weights. The obtained results also makes it possible to describe the characters of the corresponding simple modules and extensions of simple modules. As an example of application of the obtained results, we give a description of extensions of some simple modules with singular highest weights.