Geometric Langlands correspondence near opers (Q2870996)

Let \(G\) be a complex, connected semi-simple Lie group and \(G^L\) its dual Langlands group. The geometric Langlands correspondence for a complex reductive group \(G\) and a smooth projective complex curve \(\Sigma\) proposed by A. Beilinson and V. Drinfeld as a meta-conjecture has the form of an equivalence of two categories (see [\textit{A. Beilinson and V. Drinfeld} ''Quantization of Hitchin's integrable system and Hecke eigensheaves'', preprint, available at \texttt{http://www.math.uchicago.edu/~mitya/langlands/hitchin/BD-hitchin.pdf}] and also see [\textit{E. Frenkel}, in: Frontiers in number theory, physics, and geometry II. On conformal field theories, discrete groups and renormalization. Papers from the meeting, Les Houches, France, March 9--21, 2003. Berlin: Springer. 389--533 (2007; Zbl 1196.11091)] for introduction to the Langlands program needed for better understanding the present paper). One category is the modified version of the derived category of quasicoherent sheaves on the moduli stack \(Loc_{G^L}\) of flat \(G^L\)-bundles on \(\Sigma\). Another category is the the modified version of the derived category of \(D\)-modules on the moduli stack \(Bun_G\) of \(G\)-bundles on \(\Sigma\).NEWLINENEWLINEA. Beilinson and V. Drinfeld established an equivalence between special pieces in the two categories. On one side, this is the category of those coherent sheaves on \(Loc_{G^L}\) that are supported scheme-theoretically at the locus \(Op_{G^L}\) of opers. On the other side this special piece is the category of \(D\)-modules on \(Bun_G\) admitting finite global presentations NEWLINE\[NEWLINE (D\otimes K^{-1/2})^{\oplus p}\rightarrow (D\otimes K^{-1/2})^{\oplus q} \rightarrow M, NEWLINE\]NEWLINE where \(K\) is the canonical line bundle of \(Bun_G\).NEWLINENEWLINEThis equivalence follows essentially from the following theorem of the same authors: there is a canonical isomorphism of algebras \(\Gamma (Bun^{\gamma}_G, D^s)\equiv {\mathbb C} [Op_{G^L}]\), where \(D^s=K^{1/2}\otimes D\otimes K^{-1/2}\) and \(Bun^{\gamma}_G\) is a component of \(Bun_G\) labelled by \(\gamma\in \pi_1G\).NEWLINENEWLINEThe authors of the present paper generalize this theorem. Namely, they prove that the derived endomorphism algebra \(\mathrm{Ext}_{D^s(Bun_G)}(D^s,D^s)\) is abstractly \(A_{\infty}\) isomorphic to the algebra \(\mathrm{Ext}_{Loc_{G^L}}^{\bullet}(O_{Op_{G^L}}, O_{Op_{G^L}})\). This formally implies an equivalence between the derived category of coherent sheaves on \(Loc_{G^L}\) supported at the formal neighbourhood of the locus of opers and the localization at \(D\) of the derived category of \(D\)-modules on \(Bun_G\), and an appropriate equivalence of abelian categories.NEWLINENEWLINEThe paper begins with the explanation of the \(GL(1)\) case of the geometric Langlands equivalence due to \textit{G. Laumon} [``Transformation de Fourier généralisée'', Preprint, \url{arXiv:alg-geom/9603004}] and \textit{M. Rothstein} [Acta Appl. Math. 42, No. 3, 297--308 (1996; Zbl 0848.58032)], and of the abelian Beilinson-Drinfeld construction. Then the authors recall the construction of \textit{N. Hitchin} [Duke Math. J. 54, 91--114 (1987; Zbl 0627.14024)], and its quantization due to Beilinson and Drinfeld. After that they prove the generalizations of the Beilinson-Drinfeld theorem and deduce several corollaries.