The coherence conjecture of Pappas-Rapoport asserts the dimension equality between the global sections of a line bundle over a spherical Schubert variety in the affine Grassmannian and the global sections of another line bundle of the same central charge over certain union of Schubert varieties in a partial affine flag variety. This conjecture was proved by Xinwen Zhu. The main geometric theorem in Zhu's proof is that the global Schubert variety of parahoric Bruhat-Tits group scheme is flat over the base and has reduced fibers. In this talk, I will explain two applications of Zhu's geometric theorem on global Schubert variety. The first one is the determination of the smooth loci of spherical Schubert varieties in the twisted affine Grassmannian, which is a joint work with Marc Besson. The second one is that the dimension equality in the coherence conjecture can be updated to an isomorphism of representations of certain reductive group. This is an ongoing joint work with Huanhuan Yu.