Kra\'skiewicz-Pragacz modules and Ringel duality

Abstract

Kra\'skiewicz and Pragacz introduced representations of the upper-triangular Lie algebras whose characters are Schubert polynomials. In a previous work the author studied the structure of Kra\'skiewicz-Pragacz modules using the theory of highest weight categories. From the results there, in particular we obtain a certain highest weight category whose standard modules are KP modules. In this paper we show that this highest weight category is self Ringel-dual: this leads to an interesting symmetry relation on Ext groups between KP modules. We also show that the tensor product operation on b-modules is compatible with Ringel duality functor.