Geometric Langlands correspondence near opers

Let G be a complex, connected semi-simple Lie group, L G its Langlands dual group, BunG the moduli stack of G-bundles on a smooth projective curve Sigma over C, LocLG the moduli stack of flat L G-bundles on Sigma. Beilinson and Drinfeld have constructed an equivalence between the category of coherent sheaves on Loc(LG) supported scheme-theoretically at the locus of opers and the category of D-modles on BunG admitting a certain global presentation. We generalize it to an equivalence between the derived category of coherent sheaves on Loc(LG) supported at the formal neighborhood of the locus of opers and the localization at D of the derived category of D-modules on BunG (and an appropriate equivalence of abelian categories).