Fiber functor of category of D-module on affine Grassmannian.

Question

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?

Answer 1

The very short answer is that you just replace hypercohomology with global de Rham cohomology of a D-module.

The short answer is to read Theorem 3.5 in Mirkovic-Vilonen (I am referring to the arXiv paper here) and use their definition of the functor $F$ via the weight functors $F_\nu$.

The longer answer is that you can phrase this construction in a slightly more geometric way. Suppose you choose in $G$ a Borel subgroup $B$ and denote by $T$ the quotient of $B$ by its unipotent part. Then the maps $B \to G$ and $B \to T$ induce maps on the affine grassmannians, giving a diagram $\operatorname{Gr}\_G \xleftarrow{b} \operatorname{Gr}\_B \xrightarrow{t} \operatorname{Gr}\_T$. Recall the structure of $\operatorname{Gr}_T \cong X_*(T)$ (at least, topologically and as a group), and let's call (as in the paper) $2\rho$ the sum of the positive roots of $G$ with respect to $B$. Then that Theorem 3.5 can be understood as saying that $t_* b^! \mathcal{F}[2\rho(\lambda)]$ is a vector space (inside the derived category of vector spaces) whenever $\mathcal{F}$ is a spherical perverse sheaf (or D-module) on $\operatorname{Gr}_G$.

That is, we can define $F(\mathcal{F})$ to be the vector space $t_* b^! \mathcal{F}[2\rho(\lambda)]$ on the component $\{\lambda\}$ of $\operatorname{Gr}_T$; it is a vector space graded by $X_*(T)$ and this gives a faithful exact tensor functor from spherical D-modules to $\mathbf{Rep}({}^L T)$ (thus, a little more specific than just a fiber functor).

You might want to read the notes (written by me) from February 16th and 23rd at this seminar page.