# Local counterpart of the NON-Hitchin Hecke eigen-sheaves ?

Insight of Beilinson and Drinfeld at early 90-ies - that Hitchin's D-modules are Hecke eigen-D-modules. However they are NOT all Hecke-eigensmodules and actually they are only the half-dimensional Lagrangian "sub-manifold" in the space of ALL Hecke-modules. They have clear LOCAL description - they come from the center of universal enveloping of loop algebra g((t)).

What about the rest Hecke-eigen-D-modules - what is their local counterpart ? I.e. how to describe them in terms of corresponding loop algebra (or may be one need adels ?)

PS

D-module = perverse sheaf by Riemann-Hilbert, so sometimes people write "eigen-sheaves", sometimes "eigen-modules"

PSPS

Let me try to provide some background for the question.

Langlands correspondence relates the represenations of Galois (="fundamental") group of a curve "X" to kind of "automorphic form" with the requirement that zeta-functions on both sides coincide.

If we are over complex-numbers we can think that representations of fundamental group coming from flat connections on curve "X". Among all connections there is half-dimensional Lagrangian subset which comes from "opers". In the simplest GLn case oper is just the differential operator on the curve X (modula some details). So you can repesentation of the fundamental group taking the monodromy of the solution of this differential operator.

So we see on "Galois side" certain distinguished half-dimensional subset. What is its Langlands dual ? Answer - Hitchin's D-modules. Let me try to give a flavour what is about.

"Automorphic side" is somewhat more involved. But let me point some simple facts which are related to my question. Consider the loop algebra g((t)), on the "automorphic side" "stands" certain irreps of g((t)) . Irreps are parametrized by the values of central elements of U(g((t))). (By "Schur lemma" central elements act as scalars in irreps).

It is quite simple to understand the map: irreps of g((t)) <-> Hitchin's D-modules

Fix curve X. Moduli stack BunX of vector bundles on X can be presented as $BunX= G_{out}$\$G((t))/G_{in}$. Where two subgroups correspond to G-valued function which are holomorphic inside small disk in X and outside small disk. (This just presenting vector bundle by gluing function).

Lie algebra can be identified with left(right) invariant vector fields on G((t)). So the center of U(g((t))) corresponds to certain differential operators on G((t)).

MAIN POINT since center of U(g((t))) is left AND right invariant differential operators they can be pushed down to the factor $BunX= G_{out}$\ $G((t))/G_{in}$. Claim (BeilinsonDrinfeld) the differential operators obtained in this way are Hitchin's quantum Hamiltoniams (i.e. their symbols will defined the classical Hitchin's integrable system).

So when we have irrep $V$ of g((t)) it defines scalar $\lambda_{H,V}$ for each element $H$ of the center of U(g((t))). Hitchin's D-modules are defined by $AllDIFF/ (allH-\lambda_{H,V})$, speaking more down-to-earth just we consider the joint eigen functions of all Hitchin's hamiltonians with some eigenvalues $\lambda_{H}$, these eigenvalues parametrize all possible Hitchin's D-modules.

One may look at http://arxiv.org/abs/0711.2236 page 30 section 8.2

• The beginning of your PS needs some reinforcing adjectives to be a true statement. – S. Carnahan Feb 6 '12 at 7:52
• @Scott What reinforcements ? – Alexander Chervov Feb 6 '12 at 9:04
• I think your description of Riemann-Hilbert is a great mnemonic, but its generality could be misleading. The Kashiwara-Mebkhout theorem states that for $X$ a complex manifold, $RHom_D(-,\mathcal{O}_X)[\dim X]$ is an equivalence between the category of regular holonomic $D$-modules on $X$ and the category of complex perverse sheaves on $X$. I just wanted to point out that one may encounter $D$-modules that are not regular or holonomic, one may encounter perverse sheaves with alternative coefficients, and one may find oneself working with geometric objects other than complex manifolds. – S. Carnahan Feb 6 '12 at 10:53
• Isn't the same assertion true for smooth algebraic varieties over the field of complex numbers? With algebraic D-modules and perverse sheaves with respect to stratifications by algebraic subvarieties? – Leonid Positselski Feb 6 '12 at 11:35
• There is a subtlety in the notion of perverse sheaf in the algebraic category: If you demand that the restriction to a stratum is étale locally constant rather than analytically locally constant, then the theorem fails. The regular holonomic equation $df/dz= \alpha zf$ on $\mathbb{C}^\times$ has analytic solution $f = z^\alpha$, which is not a section of an étale locally constant sheaf for irrational $\alpha$. – S. Carnahan Feb 7 '12 at 4:11