Type A algebraic coherence conjecture of Pappas and Rapoport

The Pappas-Rapoport coherence conjecture, proved by Zhu, states that the dimensions of spaces of sections of certain line bundles coincide. The two sides of the equality correspond to the line bundles on spherical Schubert varieties in the affine Grassmannians and to the line bundles on unions of Schubert varieties in affine flag varieties. Algebraically the claim can be reformulated as an equality between dimensions of certain Demazure modules and certain sums of Demazure modules. The goal of this paper is to formulate an algebraic construction providing an explicit link between the above mentioned Demazure modules. Our construction works only in type A, but it is applicable to a much wider class of representations than whose popping up in the geometric coherence conjecture. In the general case one side of conjectural equality involves the affine Kostant-Kumar modules.