Quantization of Hitchin’s fibration and Langlands program. (English) Zbl 0864.14007

Beilinson, A. A.; Drinfeld, V. G.

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].

Reviewer: A.Rudakov (Waltham)

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)

Keywords:

Hecke eigenvalue; \(D\)-modules; dominant weight; moduli stack; Langlands dual group; microweight