# Globalizing Feigin--Frenkel duality

Let $$\mathfrak{g}$$ be a semisimple Lie algebra, $$\mathfrak{g}^L$$ be its Langlands dual. Feigin--Frenkel duality says $$W^k(\mathfrak{g})=W^{k_L}(\mathfrak{g}^L)$$ if $$r'(k+h^{'})(k_L+h'_L)=1$$, where $$r'$$ is the maximum number of edges between two vertices in the Dynkin diagram of $$\mathfrak{g}$$, $$h'$$ (resp. $$h'_L$$) is the dual Coxeter number of $$\mathfrak{g}$$ (resp. $$\mathfrak{g}_L$$).

A degeneration of this result when $$k_L$$ tends to infinity was used as local input by Beilinson--Drinfeld to establish some results on geometric Langlands (which was globalized using the theory of chiral algebras). Regarding the non-degenerate result, they say "We do not know if this “doubly quantized” picture can be globalized." Has the non-degenerate result been globalized in some way since then?

According to Gaitsgory [Gai2], a Ran space version of this statement can be used to prove the existence of the quantum geometric Langlands correspondence $$\mathbb{L}_\kappa$$ discussed in the Introduction.