A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence

We introduce a class of equivalences, which we call generalized semi-infinite Hecke equivalences, between certain categories of representations of graded associative algebras which appear in the setting of semi-infinite cohomology for associative algebras and categories of representations of related algebras of Hecke type which we call semi-infinite Hecke algebras. As an application we obtain an equivalence between a category of representations of a non-twisted affine Lie algebra $\widehat{\mathfrak g}$ of level $-2h^\vee-k$, where $h^\vee$ is the dual Coxeter number of the underlying semisimple Lie algebra $\mathfrak g$ and $k\in \mathbb{C}$, and the category of finitely generated representations of the W-algebra associated to $\widehat{\mathfrak g}$ of level $k$. When $k=-h^\vee$ this yields an equivalence between a category of representations of $\widehat{\mathfrak g}$ of central charge $-h^\vee$ and the category ${\rm Coh}({\rm Op}_{^LG}(D^\times))$ of coherent sheaves on the space ${\rm Op}_{^LG}(D^\times)$ of $^LG$-opers on the punctured disc $D^\times$, where $^LG$ is the Langlands dual group to the algebraic group of adjoint type with Lie algebra $\mathfrak g$. This can be regarded as a version of the local geometric Langlands correspondence. The above mentioned equivalences generalize to the case of affine Lie algebras the Skryabin equivalence between the categories of generalized Gelfand-Graev representations of $\mathfrak g$ and the categories of representations of the corresponding finitely generated W-algebras, and Kostant's results on the classification of Whittaker modules over $\mathfrak g$.