A refinement of the coherence conjecture of Pappas and Rapoport

The coherence conjecture of Pappas and Rapoport, proved by Zhu, asserts the equality of dimensions for the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and those of another line bundle over a certain union of Schubert varieties in a partial affine flag variety. In this paper, we enhance this equality of dimensions to an isomorphism of representations, which leads to interesting consequences in the setting of affine Demazure modules. Zhu's proof of coherence conjcture and our comparison theorem are established by introducing a parahoric Bruhat-Tits group scheme $\mathcal{G}$ over the affine line that is ramified at $0$. We further strengthen this comparison by equipping any line bundle on the global affine Grassmannian of $\mathcal{G}$ with a unique equivariant structure under the global jet group scheme of $\mathcal{G}$.