Equivariant localization of -modules on the flag variety of the symplectic group of degree 4

We will show on the flag variety of the symplectic group of degree 4 over a field of positive characteristic p 5 that the direct image under the Frobenius morphism of any invertible sheaf defined by a p-regular weight is tilting. In particular, the derived localization theorem holds on the flag variety for the modules of finite type over the endomorphism ring of the direct image under the Frobenius morphism of the structure sheaf, which is locally a central reduction of the ring of crystalline differential operators.