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Quantization of Hitchin’s fibration and Langlands program. (English) Zbl 0864.14007
Boutet de Monvel, Anne (ed.) et al., Algebraic and geometric methods in mathematical physics. Proceedings of the 1st Ukrainian-French-Romanian summer school, Kaciveli, Ukraine, September 1-14, 1993. Dordrecht: Kluwer Academic Publishers. Math. Phys. Stud. 19, 3-7 (1996).
Let $$X$$ be a smooth projective algebraic curve over $$\mathbb{C}$$ of genus $$g>1$$, $$G$$ a semisimple algebraic group and $$\text{Bun}_G$$ the moduli stack of $$G$$-bundles on $$X$$.
The article is a short note that describes the construction of $$D$$-modules $${\mathcal E}_\eta$$ on $$\text{Bun}_G$$ and provides the necessary definitions to state a kind of a Hecke eigenvalue property for $${\mathcal E}_\eta$$: $h_*(f^!{\mathcal E}_\eta \otimes {\mathcal M}_\lambda) ={\mathcal E}_\eta [n+1] \otimes V^\lambda_R.$ In the above $$\lambda\in P_+$$ is a dominant coweight of $$G$$, $$n=(\lambda, 2\rho)$$, the second tensor product means external tensor product, $$\eta$$ is used to denote a $$\widehat G$$-bundle $$R$$ with a connection and a filtration $$F$$ for the Langlands dual group $$\widehat G$$ that are to satisfy an additional condition and are given a special name “$$\widehat G$$-opers”. $$V^\lambda_R$$ is a local system on $$X$$ corresponding to the irreducible representation $$V^\lambda$$ of $$\widehat G$$ with the highest weight $$\lambda$$ and $$\widehat G$$-bundle $$R$$. The maps $$h$$ and $$f$$ define the Hecke correspondence which is twisted by a $$D$$-module $${\mathcal M}_\lambda$$. To give all the definitions would mean to repeat most of the note. The authors claim they have proven this when $$\lambda$$ is a microweight of $$\widehat G$$ and proclaim the general case as a conjecture.
Denote by $$B(G,X)$$ the ring of polynomial functions on $$V$$. As a preliminary step to construct $${\mathcal E}_\eta$$ the authors define an algebra $$A(G,X)$$ and a morphism $$\varphi: A(G,X)\to H^0(\text{Bun}_G, {\mathcal D})$$ that provides a quantization of the Hitchin morphism $$B(G,X)\to$${Functions on $$T^*\text{Bun}_G\}$$, where $${\mathcal D}$$ stays for the sheaf of differential operators on $$\text{Bun}_G$$ twisted by a half-canonical bundle. The algebra $$A(G,X)$$ appears to be the coordinate ring of the manifold of $$\widehat G$$-opers.
For the entire collection see [Zbl 0833.00031].

MSC:
 14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 32S40 Monodromy; relations with differential equations and $$D$$-modules (complex-analytic aspects)