A generalized semi-infinite Hecke equivalence and the local geometric Langlands correspondence
aa r X i v : . [ m a t h . R T ] F e b A GENERALIZED SEMI–INFINITE HECKE EQUIVALENCE AND THELOCAL GEOMETRIC LANGLANDS CORRESPONDENCE
A. SEVOSTYANOV
Abstract.
We introduce a class of equivalences, which we call generalized semi–infinite Heckeequivalences, between certain categories of representations of graded associative algebras whichappear in the setting of semi–infinite cohomology for associative algebras and categories of rep-resentations of related algebras of Hecke type which we call semi–infinite Hecke algebras. As anapplication we obtain an equivalence between a category of representations of a non–twisted affineLie algebra b g of level − k − h ∨ , where h ∨ is the dual Coxeter number of the underlying semisimpleLie algebra g and k ∈ C , and the category of finitely generated representations of the W–algebraassociated to b g of level k . When k = − h ∨ this yields an equivalence between a category of rep-resentations of b g of central charge − h ∨ and the category Coh(Op L G ( D × )) of coherent sheaveson the space Op L G ( D × ) of L G –opers on the punctured disc D × , where L G is the Langlandsdual group to the algebraic group of adjoint type with Lie algebra g . This can be regarded asa version of the local geometric Langlands correspondence. The above mentioned equivalencesgeneralize to the case of affine Lie algebras the Skryabin equivalence between the categories ofgeneralized Gelfand-Graev representations of g and the categories of representations of the corre-sponding finitely generated W–algebras, and Kostant’s results on the classification of Whittakermodules over g . The purpose of this short note is to introduce a categorical equivalence, similar to the Skryabinequivalence between categories of generalized Gelfand-Graev representations of complex semisimpleLie algebras and categories of representations of finitely generated W–algebras (see the Appendixto [18]), in the setting of semi–infinite Hecke algebras defined in [21]. In particular, our result yieldsan equivalence between some categories of representations of affine Lie algebras and categories ofrepresentations of W–algebras associated to them.Skryabin’s equivalence is in fact an example of a quite general and simple categorical equivalencewhich can be established in the following situation (see e.g. the introduction in [22] for a review ofequivalences of this king which occur in various settings). Let A be an associative algebra over afield k , A ⊂ A a subalgebra with a character χ : A → k . Denote by k χ the corresponding rankone representation of A . Let Q χ = A ⊗ A k χ be the induced representation of A .Let Hk( A, A , χ ) = End A ( Q χ ) opp be the algebra of A –endomorphisms of Q χ with the oppositemultiplication. The algebra Hk( A, A , χ ) is called the Hecke algebra of the triple ( A, A , χ ). In [20]a homological generalization of Hecke algebras of this type was introduced. It is this homologicalgeneralization which is called in [20] the Hecke algebra of the triple ( A, A , χ ). In this paper we usea slightly different terminology.The appearance of the term “Hecke” is justified by the fact that if A is the group algebra of aChevalley group over a finite field, A is the group algebra of a Borel subgroup in it, and χ is thetrivial complex representation of the Borel subgroup then Hk( A, A , χ ) is the Iwahori–Hecke algebra(see [14]). Primary 17B67; Secondary 81R10, 16G99, 22E57
Key words and phrases.
Hecke type algebra, Whittaker vector, W–algebra, Langlands correspondence .
The use of Hecke algebras is due to the observation that for any representation V of A the algebraHk( A, A , χ ) naturally acts in the space of Whittaker vectorsWh χ ( V ) = Hom A ( Q χ , V ) = Hom A ( k χ , V )by compositions of homomorphisms, and for any Hk( A, A , χ )–module W Q χ ⊗ Hk(
A,A ,χ ) W is a left A –module. Let Hk( A, A , χ ) − mod be the category of left Hk( A, A , χ )–modules and A − mod χA the category of left A –modules of the form Q χ ⊗ Hk(
A,A ,χ ) W , where W ∈ Hk(
A, A , χ ) − mod, withmorphisms induces by morphisms of left Hk( A, A , χ )–modules. Then the Skryabin equivalence isan example of categorical equivalences established in the following simple theorem. Theorem 1. A − mod χA is a full subcategory in the category of left A –modules, and the functors Hom A ( Q χ , · ) and Q χ ⊗ Hk(
A,A ,χ ) · yield mutually inverse equivalences of the categories, (1) A − mod χA ≃ Hk(
A, A , χ ) − mod . Proof.
Let
W, W ′ ∈ Hk(
A, A , χ ) − mod. Then by the Frobenius reciprocity and by the definitionof the algebra Hk( A, A , χ )Hom A ( Q χ , Q χ ⊗ Hk(
A,A ,χ ) W ) = Hom A ( k χ , Q χ ⊗ Hk(
A,A ,χ ) W ) == Hom A ( k χ , Q χ ) ⊗ Hk(
A,A ,χ ) W == Hom A ( Q χ , Q χ ) ⊗ Hk(
A,A ,χ ) W = Hk( A, A , χ ) opp ⊗ Hk(
A,A ,χ ) W = W. This implies the second claim of this theorem.By the formula above we also haveHom A ( Q χ ⊗ Hk(
A,A ,χ ) W ′ , Q χ ⊗ Hk(
A,A ,χ ) W ) == Hom Hk(
A,A ,χ ) ( W ′ , Hom A ( Q χ , Q χ ⊗ Hk(
A,A ,χ ) W )) == Hom Hk(
A,A ,χ ) ( W ′ , W ) , and hence A − mod χA is a full subcategory in the category of left A –modules. (cid:3) We call the equivalence established in Theorem 1 a generalized Hecke equivalence.Skryabin considered a generalized Hecke equivalence in the following situation. Let g be a complexsemisimple Lie algebra, e ∈ g a non–zero nilpotent element in g . By the Jacobson–Morozov theoremthere is an sl –triple ( e, h, f ) associated to e , i.e. elements f, h ∈ g such that [ h, e ] = 2 e , [ h, f ] = − f ,[ e, f ] = h . Fix such an sl –triple.Let χ be the element of g ∗ which corresponds to e under the isomorphism g ≃ g ∗ induced by theKilling form. Under the action of ad h we have a decomposition(2) g = ⊕ i ∈ Z g ( i ) , where g ( i ) = { x ∈ g | [ h, x ] = ix } . The skew–symmetric bilinear form ω on g ( −
1) defined by ω ( x, y ) = χ ([ x, y ]) is non–degenerate. Fixan isotropic Lagrangian subspace l of g ( −
1) with respect to ω .Let(3) m = l ⊕ M i ≤− g ( i ) . Note that m is a nilpotent Lie subalgebra of g and χ ∈ g ∗ restricts to a character χ : m → C . Denoteby C χ the corresponding one–dimensional U ( m )–module.The associative algebra W e ( g ) = End U ( g ) ( U ( g ) ⊗ U ( m ) C χ ) opp = Hk( U ( g ) , U ( m ) , χ ) is called theW–algebra associated to the nilpotent element e . The algebra W e ( g ) was introduced in [15] in casewhen e is principal nilpotent and in [16] when the grading (2) is even. In paper [6] the algebras EMI–INFINITE HECKE EQUIVALENCE 3 W e ( sl n ) are defined using cohomological BRST reduction, and the simple equivalent algebraic defi-nition for arbitrary nilpotent element e given above first appeared in [18]. The equivalence of thesetwo definitions follows, for instance, from a general property of homological Hecke–type algebras (see[20, 21]). An explicit computation establishing this equivalence can also be found in the Appendixin [8].Theorem 1 immediately yields a categorical equivalence U ( g ) − mod χU ( m ) ≃ Hk( U ( g ) , U ( m ) , χ ) − mod = W e ( g ) − mod . But in fact Skryabin proved a much stronger statement: the category U ( g ) − mod χU ( m ) can be describedas the category of g –modules on which x − χ ( x ) acts locally nilpotently for any x ∈ m , and anymodule from this category is m –injective.In the particular case when m = n ⊂ g is a maximal nilpotent subalgebra, and χ : n → C is anon–singular character of n which does not vanish on all simple root vectors in n this result wasalready obtained by Kostant in [15] for the purpose of the study of Whittaker and principal seriesrepresentations of g . This situation is particularly close to the definition of W–algebras associatedto affine Lie algebras we are interested in.Kostant showed that the algebra End U ( g ) ( U ( g ) ⊗ U ( n ) C χ ) opp is canonically isomorphic to thecenter Z ( U ( g )) of the universal enveloping algebra U ( g ),(4) End U ( g ) ( U ( g ) ⊗ U ( n ) C χ ) opp ≃ Z ( U ( g )) . Using this result he proved that modules from the category U ( g ) − mod χU ( n ) are in one–to–onecorrespondence with representations of Z ( U ( g )) which is a polynomial algebra in rank g generators.One of our objectives it to obtain a counterpart of this statement for affine Lie algebras. It willimply a version of the local geometric Langlands correspondence.Firstly we are going to generalize the categorical equivalence established in Theorem 1 to thesetting in which the semi–infinite cohomology of associative algebras and the corresponding versionsof Hecke algebras are defined (see [1, 2, 3, 17, 21]). We start by recalling the initial setup.Let A be a Z –graded associative algebra over a field k , A = M n ∈ Z A n . For N ∈ N let I N be the left ideal in A generated by A n , n ≥ N . Define a topology on A in whicha basis of open neighborhoods of 0 is formed by the left ideals I N , N ∈ N . The multiplication in A is continuous in this topology, and hence one can define the restricted completion b A of A as theinverse limit b A = lim ← A/I N with the multiplication induced from A .The category of left (right) A –modules with morphisms being homomorphisms of A –modules isdenoted by A − mod (mod − A ). For both of these categories the set of morphisms between twoobjects is denoted by Hom A ( · , · ). For Z –graded A –modules M, M ′ , ∈ Ob A − mod (Ob mod − A ), M = L n ∈ Z M n , M ′ = L n ∈ Z M ′ n we shall also use the space of homomorphisms of all possibledegrees with respect to the gradings on M and M ′ introduced byhom A ( M, M ′ ) = M n ∈ Z Hom A ( M, M ′ h n i ) , where the module M ′ h n i is obtained from M ′ by grading shift as follows: M ′ h n i k = M ′ k + n , and Hom A ( M, M ′ ) stands for set of A –homomorphisms from M to M ′ preserving the gradings. A. SEVOSTYANOV
In this paper we shall deal with the full subcategory of A − mod (mod − A ) whose objects aresmooth modules M ∈ Ob A − mod (Ob mod − A ), i.e. for any v ∈ M there exists N ∈ N such that av = 0 for any n ≥ N and any a ∈ A n . This subcategory is denoted by ( A − mod) ((mod − A ) ).The action of A on any object of the category ( A − mod) induces an action of the completion b A .We also denote by Vect k the category of vector spaces over k .Following [1] we shall impose additional restrictions on the algebra A . Namely, in the rest of thispaper we suppose that A satisfies the following conditions: (i) A contains two graded subalgebras N and B .(ii) N is positively graded.(iii) N = k .(iv) dim N n < ∞ for any n ∈ N . In particular, N is naturally augmented. (v) B is negatively graded.(vi) The multiplication in A defines isomorphisms of graded vector spaces (5) B ⊗ N → A and N ⊗ B → A. We call the decompositions (5) the triangular decompositions for the algebra A . Note that thecompositions of the triangular decomposition maps and of their inverse maps yield linear mappings(6) N ⊗ B → B ⊗ N,B ⊗ N → N ⊗ B. (vii) Mappings (6) are continuous in the following sense: for every m, n ∈ Z there exist k + , k − ∈ Z such that N m ⊗ B n → M k − ≤ k ≤ k + B n − k ⊗ N m + k and B n ⊗ N m → M k − ≤ k ≤ k + N m − k ⊗ B n + k . Next, we recall the definition the semiregular bimodule for the algebra A . The notion of thesemiregular bimodule was introduced by Voronov (see [23]) in the Lie algebra case and generalizedin [1] to the case of graded associative algebras satisfying conditions (i)–(vii). In the semi–infiniteversion of the Hecke algebra theory this bimodule plays the role of the regular representation. In par-ticular, the semiregular bimodule naturally appears in the definition of the semi–infinite modificationof Hecke algebras.First consider the right graded N -module N ∗ = hom k ( N, k ), where the action of N on N ∗ isdefined by ( n · f )( n ′ ) = f ( nn ′ ) for any f ∈ N ∗ , n ∈ N. The right A –module S A = N ∗ ⊗ N A is called the right semiregular representation of A (see [23], Sect. 3.2; [1], Sect. 3.4).Clearly, S A = N ∗ ⊗ B as a right B -module. The space S A = N ∗ ⊗ B is non–positively graded,and hence S A ∈ (mod − A ) . EMI–INFINITE HECKE EQUIVALENCE 5
Now we obtain another realization for the right semiregular representation. Consider anotherright A -module S ′ A = hom B ( A, B ), where B acts on A and B by right multiplication. The rightaction of A on the space S ′ A is given by( a · f )( a ′ ) = f ( aa ′ ) , f ∈ hom B ( A, B ) , a ∈ A. Lemma 1. ( [1] , Lemma 3.5.1)
Fix a decomposition (7) A = N ⊗ B provided by the multiplication in A . Let φ : S A → S ′ A be the map defined by φ ( f ⊗ a )( a ′ ) = f (( aa ′ ) N )( aa ′ ) B , where f ⊗ a ∈ S A , a ′ ∈ A and aa ′ = ( aa ′ ) N ( aa ′ ) B is decomposition (7) of the element aa ′ . Then φ is a morphism of right A –modules. We shall suppose that the algebra A satisfies the following additional condition: (viii) The morphism φ : S A → S ′ A constructed in the previous lemma is an isomorphism ofright A –modules. Finally we have two realizations of the right A –module S A :(8) S A = N ∗ ⊗ N A, and(9) S A = hom B ( A, B ) . Now we define the structure of a left module on S A commuting with the right semiregular actionof A . First observe that using realizations (8) and (9) of the right semiregular representation onecan define natural left actions of the algebras N and B on the space S A induced by the naturalleft action of N on N ∗ and the left regular representation of B , respectively. Clearly, these actionscommute with the right action of the algebra A on S A . Therefore we have natural inclusions ofalgebras N ֒ → hom A ( S A , S A ) , B ֒ → hom A ( S A , S A ) . Denote by A ♯ the subalgebra in hom A ( S A , S A ) generated by N and B . Proposition 1. ( [1] , Corollary 3.3.3, Lemma 3.5.3 and Corollary 3.5.3) A ♯ is a Z –gradedassociative algebra satisfying conditions (i)–(vii). Moreover, S A ∈ ( A ♯ − mod) and (10) S A = A ♯ ⊗ N N ∗ =(11) = hom B ( A ♯ , B ) as a left A ♯ –module. Using Proposition 1 the space S A is equipped with the structure of an A ♯ − A bimodule. Thisbimodule is called the semiregular bimodule associated to the algebra A . The left action of thealgebra A ♯ on the space S A is called the left semiregular action.Let M ∈ mod − A be a right A –module and M ′ ∈ A ♯ − mod a left A ♯ –module. Consider thesubspace M ⊗ N M ′ in the tensor product M ⊗ M ′ defined by M ⊗ N M ′ = { m ⊗ m ′ ∈ M ⊗ M ′ : mn ⊗ m ′ = m ⊗ nm ′ for every n ∈ N } . Following [21], we define the semiproduct M ⊗ NB M ′ of modules M ∈ mod − A and M ′ ∈ A ♯ − modas the image of the subspace M ⊗ N M ′ ⊂ M ⊗ M ′ under the canonical projection M ⊗ M ′ → M ⊗ B M ′ ,(12) M ⊗ NB M ′ = Im( M ⊗ N M ′ → M ⊗ B M ′ ) . A. SEVOSTYANOV
The semiproduct ⊗ NB is a mixture of the tensor product ⊗ B over B and of the functor ⊗ N of“ N –invariants”. The semiproduct of modules naturally extends to a functor ⊗ NB : (mod − A ) × ( A ♯ − mod) → Vect k . The semiproduct functor is a generalization of the functor of semivariants (see [23],Sect. 3.8).Let, as above, A be an associative Z –graded algebra over a field k . Suppose that the algebra A contains a graded subalgebra A , and both A and A satisfy conditions (i)–(viii). We denoteby N, B and N , B the graded subalgebras in A and A , respectively, providing the triangulardecompositions of these algebras (see condition (vi)).Let S − Ind AA be the functor of semi-infinite inductionS − Ind AA : A ♯ − mod → A ♯ − moddefined on objects by S − Ind AA ( V ) = S A ⊗ N B V, V ∈ A ♯ − mod , the structure of a left A ♯ -module on S A ⊗ N B V being induced by the left semiregular action of A ♯ on S A . In the Lie algebra case this functor was introduced in [24] and the definition given above firstappeared in [21].Note that the functor S − Ind AA sends objects of the category A ♯ − mod to objects of the category( A ♯ − mod) and Z –graded A ♯ –modules to Z –graded A ♯ –modules.Now assume that the algebra A ♯ is augmented, ε : A ♯ → k . We denote this one–dimensional A ♯ –module by k ε . Definition 1.
The algebra (13) Hk ∞ ( A, A , ε ) = Hom A ♯ (S − Ind AA ( k ε ) , S − Ind AA ( k ε )) opp is called the semi–infinite Hecke algebra of the triple ( A, A , ε ) , and the Z –graded algebra (14) Hk ∞ , • ( A, A , ε ) = hom A ♯ (S − Ind AA ( k ε ) , S − Ind AA ( k ε )) opp is called the graded semi–infinite Hecke algebra of the triple ( A, A , ε ) . Obviously, Hk ∞ , • ( A, A , ε ) ⊂ Hk ∞ ( A, A , ε ), and Hk ∞ ( A, A , ε ) is a completion of Hk ∞ , • ( A, A , ε ).In [21] a homological version of the algebra Hk ∞ , • ( A, A , ε ) is called a semi–infinite Hecke algebra.As shown in Proposition 3.1.1 in [21] in “good cases” the homological zero degree component of thehomological version of the algebra Hk ∞ , • ( A, A , ε ) coincides with the algebra defined by formula(14).Denote Q ∞ ε = S − Ind AA ( k ε ), so that(15) Hk ∞ ( A, A , ε ) = Hom A ♯ ( Q ∞ ε , Q ∞ ε ) opp . In complete analogy with the situation described in the beginning of the paper, for any represen-tation V ∈ A ♯ − mod the algebra Hk ∞ ( A, A , ε ) naturally acts in the spaceWh ∞ χ ( V ) = Hom A ♯ ( Q ∞ ε , V )by compositions of homomorphisms. We call this space the space of semi–infinite Whittaker vectors.One can define an obvious graded version of this space for Z –graded A –modules.Let (Hk ∞ ( A, A , ε ) − mod) fg be the full subcategory in Hk ∞ ( A, A , ε ) − mod the objects of whichare finitely generated modules from Hk ∞ ( A, A , ε ) − mod.For any Hk ∞ ( A, A , ε )–module W ∈ Hk ∞ ( A, A , ε ) − mod EMI–INFINITE HECKE EQUIVALENCE 7 Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W is an object of the category ( A ♯ − mod) . Let A ♯ − mod ε, ∞ A ♯ be the categoryof left A ♯ –modules of the form Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W , where W ∈ (Hk ∞ ( A, A , ε ) − mod) fg , withmorphisms induces by morphisms in the category (Hk ∞ ( A, A , ε ) − mod) fg . Then we have thefollowing analogue of Theorem 1. Theorem 2. A ♯ − mod ε, ∞ A ♯ is a full subcategory in the category ( A ♯ − mod) , and the functors Hom A ♯ ( Q ∞ ε , · ) and Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) · yield mutually inverse equivalences of the categories, (16) A ♯ − mod ε, ∞ A ♯ ≃ (Hk ∞ ( A, A , ε ) − mod) fg . Proof.
Let
W, W ′ ∈ (Hk ∞ ( A, A , ε ) − mod) fg . First note that, since Q ∞ ε is Z –graded with zeropositive degree components, we obviously have Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W ∈ ( A ♯ − mod) .Next, since W is finitely generated over Hk ∞ ( A, A , ε ) we haveHom A ♯ ( Q ∞ ε , Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W ) = Hom A ♯ ( Q ∞ ε , Q ∞ ε ) ⊗ Hk ∞ ( A,A ,ε ) W == Hk ∞ ( A, A , ε ) ⊗ Hk ∞ ( A,A ,ε ) W = W. This implies the second claim of this theorem.By the formula above we also haveHom A ♯ ( Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W ′ , Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W ) == Hom Hk ∞ ( A,A ,ε ) ( W ′ , Hom A ♯ ( Q ∞ ε , Q ∞ ε ⊗ Hk ∞ ( A,A ,ε ) W )) == Hom Hk ∞ ( A,A ,ε ) ( W ′ , W ) , and hence A ♯ − mod ε, ∞ A ♯ is a full subcategory in the category ( A ♯ − mod) . (cid:3) We call the categorical equivalence established in Theorem 2 a generalized semi–infinite Heckeequivalence.Now we apply the above obtained results in the case of affine Lie algebras and their envelopingalgebras. Let g be a complex semisimple Lie algebra, b g = g [ z, z − ] · + C K the non–twisted affine Liealgebra corresponding to g . Recall that b g is the central extension of the loop algebra g [ z, z − ] withthe help of the standard two–cocycle ω st , ω st ( x ( z ) , y ( z )) = Res h x ( z ) , y ( z ) i dzz , where h· , ·i is the standard invariant normalized bilinear form of the Lie algebra g .Let n ⊂ g be a maximal nilpotent subalgebra in g and ˜ n = n [ z, z − ] the loop algebra of thenilpotent Lie subalgebra n . Note that ˜ n ⊂ b g is a Lie subalgebra in b g because the standard cocycle ω st vanishes when restricted to the subalgebra ˜ n = n [ z, z − ] ⊂ g [ z, z − ]. We denote by U ( b g ) and U (˜ n ) the universal enveloping algebras of b g and ˜ n , respectively.Let χ be a character of n which takes non–zero values on all simple root vectors of n . χ hasa unique extension to a character b χ of ˜ n = n [ z, z − ], such that b χ vanishes on the complement z − n [ z − ] + z n [ z ] of n in n [ z, z − ]. We denote by C b χ the left one–dimensional U (˜ n )–module thatcorresponds to b χ .Let U ( b g ) k be the quotient of the algebra U ( b g ) by the two–sided ideal generated by K − k, k ∈ C .Note that for any k ∈ C U (˜ n ) is a subalgebra in U ( b g ) k because the standard cocycle ω st vanisheswhen restricted to the subalgebra ˜ n ⊂ g [ z, z − ]. A. SEVOSTYANOV
Next observe that the algebras U ( b g ) k and U (˜ n ) inherit Z –gradings from the natural Z –gradings of b g and ˜ n by degrees of the parameter z , and satisfy conditions (i)–(viii), with the natural triangulardecompositions U ( b g ) k = U ( b g + ) k ⊗ U ( b g − ) k and U (˜ n ) = U (˜ n + ) ⊗ U (˜ n − ) provided by the decomposi-tions b g = b g − + b g + , ˜ n = ˜ n − + ˜ n + , where b g − = g [ z − ] + C K , b g + = z g [ z ], ˜ n ± = ˜ n ∩ b g ± . Hence one candefine the algebras U ( b g ) ♯k , U (˜ n ) ♯ .The algebra U ( b g ) ♯ is explicitly described in the following proposition. Proposition 2. ( [1] , Proposition 4.6.7)
The algebra U ( b g ) ♯k is isomorphic to U ( b g ) − h ∨ − k , where h ∨ is the dual Coxeter number of g . The algebra U (˜ n ) ♯ is isomorphic to U (˜ n ) . We shall identify the algebra U (˜ n ) ♯ with U (˜ n ) and U ( b g ) ♯k with U ( b g ) − h ∨ − k .Next observe that the algebra U ( b g ) k and the graded subalgebra U (˜ n ) ⊂ U ( b g ) k satisfy the compati-bility conditions (i)–(viii) under which the semi–infinite Hecke algebra of the triple ( U ( b g ) k , U (˜ n ) , C b χ )may be defined. Definition 2.
The algebra W k ( b g ) defined by (17) W k ( b g ) = Hom U ( b g ) − k − h ∨ ( S U ( b g ) k ⊗ U (˜ n + ) U (˜ n − ) C b χ , S U ( b g ) k ⊗ U (˜ n + ) U (˜ n − ) C b χ ) opp == Hk ∞ ( U ( b g ) k , U (˜ n ) , C b χ ) . is called the W –algebra associated to the affine Lie algebra b g of level k . As it is shown in [21], Proposition 3.2.2 the definition of the algebra Hk ∞ , • ( U ( b g ) k , U (˜ n ) , C b χ ) agreeswith the definition of the W –algebra as the cohomology of the BRST complex which appears in [9].This result implies that the completion W k ( b g ) = Hk ∞ ( U ( b g ) k , U (˜ n ) , C b χ ) of Hk ∞ , • ( U ( b g ) k , U (˜ n ) , C b χ )is the completed enveloping algebra, introduced in [5], Section 4.3.1, of the W –algebra defined in [9]as a vertex operator algebra.From Theorem 2 we immediately obtain the following result which can be regarded as a counter-part of the Skryabin equivalence for affine Lie algebras. Theorem 3. U ( b g ) − k − h ∨ − mod b χ, ∞ U (˜ n ) is a full subcategory in the category ( U ( b g ) − k − h ∨ − mod) , andthe functors Hom U ( b g ) − k − h ∨ ( Q ∞ b χ , · ) and Q ∞ b χ ⊗ Hk ∞ ( U ( b g ) k ,U (˜ n ) , C b χ ) · yield mutually inverse equivalencesof the categories, (18) U ( b g ) − k − h ∨ − mod b χ, ∞ U (˜ n ) ≃ (Hk ∞ ( U ( b g ) k , U (˜ n ) , C b χ ) − mod) fg . Now recall that at the critical value of the parameter k , k = − h ∨ , the restricted completion b U ( b g ) − h ∨ of the algebra U ( b g ) − h ∨ has a large center. This center is canonically isomorphic to theW-algebra W − h ∨ ( g ) (see [9], Proposition 6, [10], Proposition 4.3.4 and [4], Theorem 3.7.7), Z ( b U ( b g ) − h ∨ ) ≃ W − h ∨ ( g ) . Thus we obtain a canonical algebraic isomorphism(19) Z ( b U ( b g ) − h ∨ ) ≃ Hom U ( b g ) − h ∨ ( S U ( b g ) − h ∨ ⊗ U (˜ n + ) U (˜ n − ) C b χ , S U ( b g ) − h ∨ ⊗ U (˜ n + ) U (˜ n − ) C b χ ) opp == Hk ∞ ( U ( b g ) − h ∨ , U (˜ n ) , C b χ ) . Here using Proposition 2 we replaced the algebra U ( b g ) ♯ − h ∨ with U ( b g ) − h ∨ (We note that at the criticallevel of the parameter k , k = − h ∨ the algebra U ( b g ) − h ∨ is “self–dual” in the sense that the algebra U ( b g ) ♯ − h ∨ is isomorphic to U ( b g ) − h ∨ ).Description (19) of the center Z ( b U ( b g ) − h ∨ ) is similar to realization (4) of the center Z ( U ( g )) ofthe algebra U ( g ) obtained by Kostant in [15]. EMI–INFINITE HECKE EQUIVALENCE 9
As a corollary of Theorem 3 and of the last observation we have the following statement whichcan be regarded as an affine Lie algebra analogue of Kostant’s classification of Whittaker modules.
Theorem 4. U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) is a full subcategory in the category ( U ( b g ) − h ∨ − mod) , and thefunctors Hom U ( b g ) ♯k ( Q ∞ b χ , · ) and Q ∞ b χ ⊗ Z ( b U ( b g ) − h ∨ ) · yield mutually inverse equivalences of the categories, U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) ≃ (Hk ∞ ( U ( b g ) − h ∨ , U (˜ n ) , C b χ ) − mod) fg ≃ ( Z ( b U ( b g ) − h ∨ ) − mod) fg . Note that the algebra Z ( b U ( b g ) − h ∨ ) is canonically isomorphic to the topological algebra of functionsFun(Op L G ( D × )) on the space Op L G ( D × ) of L G –opers on the punctured disc D × = Spec C (( t )),where L G is the Langlands dual group to the semisimple algebraic group G of adjoint type withLie algebra g (see e.g. [10], Theorem 4.3.6). Thus we obtain the following corollary of the previoustheorem. Corollary 1.
The category U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) is equivalent to the category Coh(Op L G ( D × )) ofcoherent sheaves on Op L G ( D × ) , (20) U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) ≃ Coh(Op L G ( D × )) , and irreducible objects in the category U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) are parametrized by L G –opers on thepunctured disc. Equivalence (20) can be viewed as a version of the local geometric Langlands correspondence.In conclusion we briefly compare this correspondence and correspondences of a similar kind estab-lished in [11, 12]. Let ( U ( b g ) − h ∨ − mod) reg be the full subcategory of the category of ( U ( b g ) − h ∨ − mod) with objects being modules on which the action of the center Z ( b U ( b g ) − h ∨ ) factors through the ho-momorphism Z ( b U ( b g ) − h ∨ ) ≃ Fun(Op L G ( D × )) → Fun(Op L G ( D )) , where D is the disk D = Spec C [[ t ]]. Let ( U ( b g ) − h ∨ − mod) G [[ z ]] reg be the full subcategory in ( U ( b g ) − h ∨ − mod) reg with objects being modules on which the action of the Lie subalgebra g [[ z ]] ⊂ b U ( b g ) − h ∨ isintegrated to the action of the group G [[ z ]]. According to [11], Theorem 6.3 (see also [4], Section 8)there is an equivalence of categories,(21) ( U ( b g ) − h ∨ − mod) G [[ z ]] reg ≃ QCoh(Op L G ( D )) , where QCoh(Op L G ( D )) is the category of quasicoherent sheaves on Op L G ( D ).In [12], Section 1.3, Main Theorem, a categorical equivalence of similar kind was established forthe category QCoh(Op unr L G ( D × )) of quasicoherent sheaves on the space Op unr L G ( D × ) of opers on D × that are unramified as local systems.Note that the objects of the category Coh(Op L G ( D × )) associated to the punctured disk D × ex-haust all finitely generated representations of the center Z ( b U ( b g ) − h ∨ ) ≃ Fun(Op L G ( D × )), while theobjects of the category QCoh(Op L G ( D )) associated to the disk D form a special class of represen-tations of it.Theorem 4 and geometric Langlands correspondence (20) look as very natural affine Lie algebracounterparts of Kostant’s result in [15] on the classification of Whittaker modules. Although, thecategory U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) which appears in (20) is quite different from the category ( U ( b g ) − h ∨ − mod) G [[ z ]] reg in (21). Namely, the action of the positively graded Lie subalgebra z g [[ z ]] ⊂ b U ( b g ) − h ∨ onobjects of the category U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) is integrated to the action of the corresponding congruencesubgroup G ( z C [[ z ]]) ⊂ G [[ z ]] since U ( b g ) − h ∨ − mod b χ, ∞ U (˜ n ) is a subcategory in ( U ( b g ) − h ∨ − mod) . Butthis is not true for the Lie subalgebra g [[ z ]] ⊂ b U ( b g ) − h ∨ . This agrees with the properties of the Whittaker modules which are objects of the category U ( g ) − mod χU ( n ) . The action of the Lie subalgebra n on them is not locally nilpotent. However,in the theory of Whittaker modules over g developed in [15] integrable, i.e. finite–dimensional,representations of g also appear. For instance, let M ′ λ be the contragredient (full dual) moduleto a Verma module M λ . In the proof of Theorem 4.6 in [15] it is observed that the subspace V ⊂ M ′ λ which consists of elements on which x − χ ( x ) acts locally nilpotently for any x ∈ n is asubmodule which is in fact an irreducible Whittaker module. Clearly, M ′ λ contains an irreduciblefinite–dimensional submodule, and hence one can associate such modules to irreducible Whittakermodules.More generally, Whittaker vectors and Whittaker representations appear in the study of Whit-taker models for principal series representations of Lie groups (see [13, 15]). It would be natural to in-troduce and explore similar objects related to the modules from the category U ( b g ) − h ∨ − k − mod b χ, ∞ U (˜ n ) . References [1] Arkhipov, S. M., Semi–infinite cohomology of associative algebras and bar duality,
Internat. Math. Res. Notices , , 833–863.[2] Arkhipov, S. M., Semi–infinite cohomology of quantum groups, Comm. Math. Phys. , (1997), 379–405.[3] Arkhipov, S. M., Semi–infinite cohomology of quantum groups II, in: Topics in quantum groups and finite-typeinvariants, Amer. Math. Soc. Translations, Ser. 2 , AMS, Providence, RI (1998), 3–42.[4] Beilinson, A., Drinfeld V., Quantization of Hitchin’s integrable system and Hecke eigensheaves,http://math.uchicago.edu/ ∼ drinfeld/langlands/QuantizationHitchin.pdf[5] Ben-Zvi, D., Frenkel, E., Vertex Algebras and Algebraic Curves, American Mathematical Society (2001).[6] De Boer, J., Tjin, T., Quantization and representation theory of finite W–algebras, Comm. math. Phys. , (1993), 485–516.[7] De Boer, J., and Tjin, T., The relation between quantum W–algebras and Lie algebras, Comm. Math. Phys. (1994) 317–332.[8] De Sole, A., Kac, V. G., Finite vs affine W -algebras, Jpn. J. Math. (2006), 137–261.[9] Feigin, B., Frenkel, E., Affine Kac–Moody algebras at the critical level and Gelfand–Dikii algebras, Int. J. Mod.Phys. , A7 , Suppl. 1A (1992), 197-215.[10] Frenkel, E., Langlands correspondence for loop groups, Cambridge Univ. Press, Cambridge (2007).[11] Frenkel, E., Gaitsgory, D., D-modules on the affine Grassmannian and representations of affine Kac-Moodyalgebras, Duke Math. J. (2004), 279–327.[12] Frenkel, E., Gaitsgory, D., Local Geometric Langlands Correspondence: The Spherical Case, in: Algebraicanalysis and around: In honor of Professor Masaki Kashiwara’s 60th Birthday (Tetsuji Miwa, Atsushi Matsuo,Toshiki Nakashima, Yoshihisa Saito eds.),
Advanced Studies in Pure Mathematics , Tokyo, MathematicalSociety of Japan (2009) 167–186.[13] Goodman, R., Wallach, N., Whittaker Vectors and Conical Vectors, J. of Func. Anal. (1980), 199–279.[14] Iwahori, N., On the structure of a Hecke ring of a Chevalley group over a finite field, J. Fac, Sci. Uni. Tokyo ,Sect.
IA 10 (1964), 215–236.[15] Kostant, B., On Whittaker vectors and representation theory,
Invent. Math. (1978), 101-184.[16] Lynch, T.E., Generalized Whittaker vectors and representation theory. PhD thesis, MIT (1979).[17] Positselski, L., Homological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebraof Associative Algebraic Structures, Monografie Matematyczne , Birkh¨auser (2010).[18] Premet, A., Special transverse slices and their enveloping algebras. With an appendix by Serge Skryabin, Adv.Math. (2002), 1–55.[19] Sevostyanov, A., The Whittaker model of the center of the quantum group and Hecke algebras, Ph.D. thesis,Uppsala (1999).[20] Sevostyanov, A., Reduction of quantum systems with arbitrary first class constraints and Hecke algebras,
Comm.Math. Phys. , (1999), 137.[21] Sevostyanov, A., Semi–infinite cohomology and Hecke algebras, Adv. Math (2001), 83–141.[22] Sevostyanov, A., Q-W-algebras, Zhelobenko operators and a proof of De Concini-Kac-Procesi conjecture,arXiv:2102.03208.[23] Voronov, A., Semi–infinite homological algebra,