Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question

1 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.

share|improve this answer
    
Welcome to MO, Ryan! –  Qiaochu Yuan Jun 5 '10 at 6:16
    
May I ask how you think of D-module on affine Grassmannian? –  Jiuzu 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? –  Jiuzu Hong Jun 6 '10 at 10:49
    
In Braverman's notes math.harvard.edu/~gaitsgde/grad_2009/Dmod_brav.pdf, 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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.