Geometric Satake correspondence allows us to construct Langlands dual group in a canonical way. In Mirkovic-Vilonen's paper, they prove that category of spherical perverse sheaves is an commutative tensor category, and hypercohomology functor gives a fiber functor. By Tannakian formalism, we can construct an algebraic group, which is exactly the Langlands dual group.
We can also realize the category to be the category of spherical D-module on affine Grassmannian. Now my quesion is, is there any nice construction of fiber functor on this category, without using Riemann-Hilbert correspondence?
Edit: On a smooth finite dimensional variety, given a D-module, one can associate a deRham complex, and then take hypercohomology. The problem is that on smooth variety, we have sheaf of differential forms, which is canonical. However, on affine Grassmannian, D-module is actually not concret, so we can't associate a deRham complex canonically, I mean it depends on the realization of D-module.
Can someone answer this question?