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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?

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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.

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Welcome to MO, Ryan! – Qiaochu Yuan Jun 5 '10 at 6:16
May I ask how you think of D-module on affine Grassmannian? – Hong Jun 5 '10 at 19:18
The affine grassmannian is a "strict ind-scheme": there is a sequence of closed subschemes $S_1 \subset S_2 \subset \dots \subset \operatorname{Gr}_G$ of which the grassmannian is the union. In fact, we can take them all to be projective schemes of finite type. For the purposes of geometric Satake, we declare a D-module (or perverse sheaf) to be supported on one of the finite-dimensional pieces. Kashiwara's theorem is used to ensure that this is independent of the choice of the $S_i$'s and that this makes sense when they are not necessarily smooth. – Ryan Reich Jun 5 '10 at 19:42
I'm actually quite confused by this kind of definition. On some singular variety, by Kashiwara's lemma, we do have some realization of D-module on it. But is it really the definition? We have so many realizations, which one you will take? Then on affine Grassmannian. D-module on affine Grassmannian, look more like some complicated construction, not really the definition. Is it possible to have some more intrinsic point of view? – Hong Jun 6 '10 at 10:49
In Braverman's notes, section 7.3, this issue is discussed. You should especially read the exercises. As for the grassmannian, I am afraid that as far as I know, one must always define everything in this inductive way. One can, of course, make explicit the choice of the schemes Si; for example, take them to be unions of the closures of the orbits of G([[t]]), which is especially well suited for the geometric Satake problem. – Ryan Reich Jun 6 '10 at 14:50

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