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 ĝ of level −k−2h, where h is the dual Coxeter number of the underlying semisimple Lie algebra g and k ∈ C, and the category of finitely generated representations of the W–algebra associated to ĝ of level k. When k = −h this yields an equivalence between a category of representations of ĝ of central charge −h and the category Coh(OpLG(D )) of coherent sheaves on the space OpLG(D ) of G–opers on the punctured disc D, where G is the Langlands dual group to the algebraic group of adjoint type with Lie algebra 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 g and the categories of representations of the corresponding finitely generated W–algebras, and Kostant’s results on the classification of Whittaker modules over g. The purpose of this short note is to introduce a categorical equivalence, similar to the Skryabin equivalence between categories of generalized Gelfand-Graev representations of complex semisimple Lie algebras and categories of representations of finitely generated W–algebras (see the Appendix to [18]), in the setting of semi–infinite Hecke algebras defined in [21]. In particular, our result yields an equivalence between some categories of representations of affine Lie algebras and categories of representations of W–algebras associated to them. Skryabin’s equivalence is in fact an example of a quite general and simple categorical equivalence which can be established in the following situation (see e.g. the introduction in [22] for a review of equivalences of this king which occur in various settings). Let A be an associative algebra over a field k, A0 ⊂ A a subalgebra with a character χ : A0 → k. Denote by kχ the corresponding rank one representation of A0. Let Qχ = A⊗A0 kχ be the induced representation of A. Let Hk(A,A0, χ) = EndA(Qχ) opp be the algebra of A–endomorphisms of Qχ with the opposite multiplication. The algebra Hk(A,A0, χ) is called the Hecke algebra of the triple (A,A0, χ). In [20] a homological generalization of Hecke algebras of this type was introduced. It is this homological generalization which is called in [20] the Hecke algebra of the triple (A,A0, χ). In this paper we use a slightly different terminology. The appearance of the term “Hecke” is justified by the fact that if A is the group algebra of a Chevalley group over a finite field, A0 is the group algebra of a Borel subgroup in it, and χ is the trivial complex representation of the Borel subgroup then Hk(A,A0, χ) is the Iwahori–Hecke algebra (see [14]). 2000 Mathematics Subject Classification Primary 17B67; Secondary 81R10, 16G99, 22E57