On Singular Localization of g -modules
aa r X i v : . [ m a t h . R T ] S e p Singular Localization of g -modules andapplications to representation theory Erik Backelin and Kobi Kremnizer
Accepted for publication in Journal of the EMS
Abstract
We prove a singular version of Beilinson-Bernstein localization for a complexsemi-simple Lie algebra following ideas from the positive characteristic case doneby [BMR06]. We apply this theory to translation functors, singular blocks in theBernstein-Gelfand-Gelfand category O and Whittaker modules. Keywords.
Lie algebra, Beilinson-Bernstein localization, category O Let g be a semi-simple complex Lie algebra with enveloping algebra U and center Z ⊂ U .Let h ⊂ g be a Cartan subalgebra and B be the flag manifold of g . Let λ ∈ h ∗ be regularand dominant and I λ ⊂ Z be the corresponding maximal ideal determined by the HarishChandra homomorphism. Put U λ := U / ( I λ ) . Let D λ B be the sheaf of λ -twisted differentialoperators on B . The celebrated localization theorem of Beilinson and Bernstein, [BB81],states that the global section functor gives an equivalence Mod( D λ B ) ∼ = Mod(U λ ) . Forapplications and more information, see [HTT08].A localization theory for singular λ was much later found in positive characteristic byBezrukavnikov, Mirkovi´c and Rumynin, [BMR06]. Let us sketch their basic construction(which makes sense in all characteristics):Let G be a semi-simple algebraic group such that Lie G = g . Instead of B considera parabolic flag manifold P = G/P , where P ⊆ G is a parabolic subgroup whose Erik Backelin: Departamento de Matem´aticas, Universidad de los Andes, Carrera 1 N. 18A - 10, Bo-got´a, Colombia; e-mail: [email protected] Kremnizer: Mathematical Institute, University of Oxford, 2429 St Giles’ Oxford OX1 3LB, UK;e-mail: [email protected]
Mathematics Subject Classification (2010):
Primary 17B10, 14-XX
Accepted for publication in Journal of the EMS parabolic roots coincide with the singular roots of λ . Replace the sheaf D λ B by a sheaf D λ P := π ∗ ( D G/R ) L modulo a certain ideal defined by λ . Here L is the Levi factor and R isthe unipotent radical of P and π : G/R → P is the projection. The L -invariants are takenwith respect to the right L -action on G/R . The sheaf π ∗ ( D G/R ) L is locally isomorphic to D P ⊗ U( l ) , where l = Lie L . When P = B we have D λ P = D λ B and when P = G wearrive at a tautological solution: D λ P = U λ ⊗ “sheaf of differential operators on a point” = U λ .We use this construction to prove a singular localization theorem in characteristic zero,Theorem 5.1. This is probably well known to the experts but it isn’t in the literature. Ourproof is similar to the original proof of [BB81], though parabolicity leads to some newcomplications. For instance, [BB81] introduced the method of tensoring a D B -modulewith a trivial bundle and then to filter this bundle with G -equivariant line bundles assubquotients. In the parabolic setting the subquotients will necessarily be vector bundles- which are harder to control - since irreducible representations of P are generally notone-dimensional.In Theorem 4.10 we show that global section Γ( D λ P ) equals U λ by passing to theassociated graded level, i.e. to the level of a parabolic Springer resolution. That this workswe deduce from the usual Springer resolution, Lemma 3.2.Our localization theorem gives an equivalence at the level of abelian categories justlike [BB81] does. This is different from positive characteristic where the localizationtheorem only holds at the level of derived categories. Our principal motivation comes from quantum groups. We do not wish to get into de-tails here, but let us at least mention that we will need a singular localization theory forquantum groups in order to establish quantum analogs of fundamental constructions from[BMR08, BMR06, BM10] that relate modular representation theory to (commutative)algebraic geometry. By our previous work, [BK08], we know that the derived representa-tion categories of quantum groups at roots of unity are equivalent to derived categories ofcoherent sheaves on Springer fibers in T ∗ B .To extend this to the level of abelian categories we must transport the tautological t -structure on the representation theoretical derived category to a t -structure on the coherentsheaf side. It so happens that to describe this so called exotic t -structure (see also [Bez06])a family of singular localizations is needed (even for a regular block).We showed in [BK06] that a localization theory for quantum groups can be neatly for-mulated in terms of equivariant sheaves. The “space” G/B doesn’t admit a quantization.However, one can quantize function algebras O ( G ) and O ( B ) and thus the category ofingularLocalizationof g -modules 3 B -equivariant (= O ( B ) -coequivariant) O ( G ) -modules. This is just the category of qua-sicoherent sheaves on G/B . Therefore, to prepare for the quantum case we have takenthorough care to write down our results in an equivariant categorical language and at thesame time to explain what is going on geometrically while this is still possible.
The theory of singular localization of g -modules clarifies many aspects of representationtheory and will have many applications in its own right. Here we discuss a few of them.It is a basic principle in representation theory that understanding of representationsat singular central characters enhances the understanding also at regular central char-acters. This is illustrated by our D -module interpretation of translation functors (Sec-tion 6). Using regular localization only, such a theory was developed by Beilinson andGinzburg, [BG99]. Singular localization simplifies their picture for the plain reason thatwall-crossing functors between regular blocks factors through a singular block. We shallalso need these results in our work on quantum groups.The localization theorem implies that a (perhaps singular) block O λ in category O corresponds to certain bi-equivariant D -modules on G (Section 7). From this we directlyretrieve Bernstein and Gelfand’s, [BerGel81] , classic result that O λ is equivalent to acategory of Harish-Chandra bimodules, Corollary 7.4.Singular localization also leads to the useful observation that one should study Harish-Chandra g - l -bimodules, where l is the Levi factor of p = Lie P , rather than g - g -bimodules(as well as the only proof we know that such bimodules are equivalent to O λ .) For in-stance, Theorem 8.1 gives this way a very short proof for Miliˇci´c and Soergel’s equiva-lence between O λ and a block in the category of Whittaker modules, [MS97], and Corol-lary 8.6 gives one for its parabolic generalization due to Webster, [W09]. These Whittakercategories have encountered recent interest because they are equivalent to modules overfinite W -algebras, e.g. [W09]. It is probably well worth the effort to further investigatethe relationship between singular localization and finite W -algebras; in particular so inthe affine case.We also retrieve and generalize some other known equivalences between representa-tion categories, e.g. [Soe86]. An interesting task will be to develop a theory for “holonomic” D λ P -modules. Those whichare “smooth along the Bruhat stratification of P ” and have “regular singularities” will cor-respond to O λ . One should then establish a “Riemann-Hilbert correspondence” betweenholonomic D λ P -modules with regular singularities and a suitable category of constructible ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS sheaves on P . Ideally the latter category would be accessible to the machinery of Hodgetheory. This would further strengthen the interplay between representation theory and al-gebraic topology. Because of the simple local description of D λ P we believe that all thiscan be done and is a good starting point for generalizing D -module theory. We shall returnto this topic later on.Another topic we would like to approach via singular localization is the singular-parabolic Koszul duality for O of [BGS96]. Here we fix notations and collect mostly well known results that we shall need.
We work over C . Unless stated otherwise, ⊗ = ⊗ C . Let X be an algebraic variety, O X the sheaf of regular functions on X and O ( X ) its global sections. Mod( O X ) denotes thecategory of quasi-coherent sheaves on X and Γ := Γ X : Mod( O X ) → Mod( O ( X )) isthe global section functor. If Y is another variety π YX will denote the obvious projection X → Y if there is a such.For A a sheaf of algebras on X such that O X ⊆ A (e.g., an algebra if X = pt ) weabbreviate an A -module for a sheaf of A -modules that is quasi-coherent over O X . We de-note by Mod( A ) the category of A -modules. More generally, we will encounter categoriessuch as Mod( A , additional data ) that consists of A -modules with some additional data .We will then denote by mod( A , additional data ) its full subcategory of noetherian ob-jects.Throughout this paper G will denote a semi-simple complex linear algebraic group.We have assumed semi-simplictly to simplify notations; all our results can be straightfor-wardly extended to the case that G is reductive. We remark on this fact in those proofsthat reduce to (reductive) Levi subgroups of G . Let B ⊂ G be a Borel subgroup of our semi-simple group G and let T ⊂ B be a maximaltorus. Let h ⊂ b ⊂ g be their respective Lie algebras. For any parabolic subgroup P of G containing B , denote by R = R P its unipotent radical and by L := L P its Levi subgroupand by p = Lie P , r = r P = Lie R and l = l P = Lie L their Lie algebras. We denote by B := G/B the flag manifold and by P := G/P the parabolic flag manifold correspondingto P .ingularLocalizationof g -modules 5Let Λ be the lattice of integral weights and let Λ r be the root lattice. Let Λ + and Λ r + bethe positive weights and the positive linear combinations of the simple roots, respectively.Let W be the Weyl group of g . Let ∆ be the simple roots and let ∆ P := { α ∈ ∆ : g − α ⊂ p } be the subset of P -parabolic roots. Let W P be the subgroup of W generatedby simple reflections s α , for α ∈ ∆ P . Note that h is a Cartan subalgebra of the reductiveLie algebra l P . Denote by S ( h ) W P the W P -invariants in S ( h ) with respect to the • -action(here w • λ := w ( λ + ρ ) − ρ , for λ ∈ h ∗ , w ∈ W , ρ is the half sum of the positive roots ).Let Z( l ) be the center of U( l ) and put Z := Z( g ) . We have the Harish-Chandra homo-morphism S ( h ) W P ∼ = Z( l ) (thus S ( h ) W ∼ = Z ).Put ∆ λ := { α ∈ ∆; λ ( H α ) = − } , λ ∈ h ∗ , where H α ∈ h is the coroot correspondingto α . Let χ l ,λ : Z( l ) → C be the character such that I l ,λ := Ker χ l ,λ annihilates the Vermamodule M λ (for U( l ) ) with highest weight λ . Thus, χ l ,λ = χ l ,µ ⇐⇒ µ ∈ W P • λ . Wehave λ = χ h ,λ and we write χ λ := χ g ,λ and I λ := Ker χ λ .Let U := U( g ) be the enveloping algebra of g and e U := U ⊗ Z S ( h ) the extendedenveloping algebra; thus e U has a natural W -action such that the invariant ring e U W iscanonically isomorphic to U . Let U λ := U / ( I λ ) . We say that • λ ∈ h ∗ is P -dominant if λ ( H α ) / ∈ {− , − , − , . . . } , for α ∈ ∆ P ; λ is dominant ifit is G -dominant. • λ is P - regular if ∆ λ ⊆ ∆ P . λ is regular if it is B -regular, that is if w • λ = λ = ⇒ w = e , for w ∈ W . • λ is a P - character if it extends to a character of P ; thus λ is a P -character iff λ isintegral and λ | ∆ P = 0 .Suppose now that λ ∈ h ∗ is integral and P -dominant. Then there is an irreducible finitedimensional P -representation V P ( λ ) with highest weight λ . Note that V L ( λ ) := V P ( λ ) isan irreducible representation for L . Of course, dim V P ( λ ) = 1 ⇐⇒ λ is a P -character.The following is well-known: Lemma 2.1.
Let λ ∈ h ∗ . Then λ is dominant iff for all µ ∈ Λ r + \ { } we have χ λ + µ = χ λ . We also have
Lemma 2.2.
Let λ ∈ h ∗ be P -regular and dominant. Let µ be a P -character and let V be the finite dimensional irreducible representation of g with extremal weight µ . Then forany weight ψ of V , ψ = µ , we have χ λ + µ = χ λ + ψ .Proof. This is well known for P = B . We reduce to that case as follows: Let g ′ be thesemi-simple Lie subalgebra of g generated by X ± α , α ∈ ∆ \ ∆ P . Let h ′ := g ′ ∩ h bethe Cartan subalgebra of g ′ . The inclusion h ′ ֒ → h gives the projection p : h ∗ → h ′∗ . ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
Consider the restriction V | g ′ of V to g ′ and let V ′ denote the irreducible g ′ -module withhighest weight p ( µ ) ; V ′ is a direct summand in V | g ′ . Let Λ( V ) denote the set of weightsof V . Then p (Λ( V )) = Λ ′ ( V | g ′ ) , the weights of V | g ′ . By the assumption that µ is a P -character, it follows that p (Λ( V )) is contained in the convex hull Λ ′ ( V ′ ) of Λ ′ ( V ′ ) . Since p ( λ ) is regular and dominant it is well known that p ( λ + µ ) / ∈ W ′ ( p ( λ ) + Λ( V ′ )) . Butthen it follows that p ( λ + µ ) / ∈ W ′ ( p ( λ ) + Λ( V ′ )) . Now W ′ = p ( W ) , so it follows that λ + µ / ∈ W ( λ + Λ( V )) . O -modules and induction See [Jan83] for details on this material.Let K be a linear algebraic group and J a closed algebraic subgroup. For X an alge-braic variety equipped with a right (or left) action of K we denote by Mod( O X , K ) thecategory of K -equivariant sheaves of (quasi-coherent) O X -modules. For M ∈ Mod( O X , K ) there is the sheaf ( π X/KX ∗ M ) K on X/K of K -invariant local sections in the direct image π X/KX ∗ M . If the K -action is free and the quotient is nice we have the equivalence [ π X/KX ∗ ( )] K : Mod( O X , K ) → Mod( O X/K ) : π X/K ∗ X . We denote by Γ ( K,J ) the global section functor on Mod( O K , J ) that corresponds to Γ K/J under the equivalence
Mod( O K , J ) ∼ = Mod( O K/J ) . Then Γ ( K,J ) ( M ) = M J , for M ∈ Mod( O K , J ) .Let Rep ( K ) denote the category of algebraic representations of K . We have O ( K ) ∈ Rep ( K ) , via ( gf )( x ) := f ( g − x ) , for g, x ∈ K and f ∈ O ( K ) . We shall also considerthe left J -action on O ( K ) given by ( kf )( x ) := f ( xk ) , for k ∈ J, x ∈ K and f ∈ O ( K ) .These actions commute.For V ∈ Rep ( J ) we consider the diagonal left J -action on e V := O ( K ) ⊗ V . The left K -action on O ( K ) defines a left K -action on e V that commutes with the J -action and themultiplication map O ( K ) ⊗ e V → e V is K - and J -linear. Thus e V belongs to the category Mod( K, O ( K ) , J ) of K - J bi-equivaraint O ( K ) -modules. This gives the functor p ∗ : Rep ( J ) → Mod( K, O ( K ) , J ) , V e V (induced bundle of a representation, p symbolizes projection from K to pt/J ).Let Ind KJ V := e V J ∈ Rep ( K ) .We have the factorization Ind KJ = ( ) J ◦ p ∗ . One can show that R ( ) J ◦ p ∗ ∼ = RInd KJ where R ( ) J and RInd KJ are computed in suitable derived categories. An important for-mula is the tensor identity RInd KJ ( V ⊗ W ) ∼ = RInd KJ ( V ) ⊗ W, f or V ∈ Rep ( J ) , W ∈ Rep ( K ) . (2.1)(In particular RInd KJ ( W ) ∼ = W ⊗ RInd KJ ( C ) , for W ∈ Rep ( K ) and C the trivial repre-sentations.)ingularLocalizationof g -modules 7 In order to treat sheaves of extended differential operators on parabolic flag varieties inthe next section we will here gather information about their associated graded objects.This is encoded in the geometry of the parabolic Grothendieck-Springer resolution.
The parabolic flag variety P has a natural left G -action. There is a bijection betweenrepresentations of P and G -equivariant vector bundles on P ; a representation V of P correspond to the induced bundle G × P V on P . We denote by O ( V ) := O P ( V ) thecorresponding locally free sheaf on P which hence has a left G -equivariant structure.Let λ ∈ h ∗ be a P -character and write O ( λ ) := O ( V P ( λ )) for the line-bundle corre-sponding to the one-dimensional P -representation V P ( λ ) . We have P ic ( P ) = P ic G ( P ) ∼ = group of P -characters, (but note that not all vector bundles on P are G -equivariant). Theample line bundles O ( − µ ) are given by P -characters µ such that µ ( H α ) > for all α ∈ ∆ \ ∆ P .Next we define the parabolic Grothendieck resolution: Definition 3.1. e g P := { ( P ′ , x ) : P ′ ∈ P , x ∈ g ∗ , x | r P ′ = 0 } Note that e g P = G × P ( g / r P ) ∗ . Recall that L = L P is the Levi factor of P , U = U P itsunipotent radical and l = l P , r = r P their Lie algebras. We have a commutative square: e g P l ∗ /L = h ∗ / W P g ∗ h ∗ / W ✲❄ ❄✲ (3.1)where the top map sends ( P ′ , x ) to x | l P ′ /L P ′ ∈ l ∗ P ′ /L P ′ ∼ = l ∗ /L . Note that the isomor-phism l ∗ P ′ /L P ′ ∼ = l ∗ /L is canonical. (We can call l ∗ /L the universal coadjoint quotient ofthe Levi Lie subalgebra.)This induces a map: π P : e g P → g ∗ × h ∗ / W h ∗ / W P . (3.2) Lemma 3.2. Rπ P∗ O e g P = O g ∗ × h ∗ / W h ∗ / W P .Proof. We shall reduce to the well known case of the ordinary Grothendieck resolutionfor P = B . It states that Rπ B∗ O e g B = O g ∗ × h ∗ / W h ∗ . (3.3) ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
Translating this to the equivariant language it reads:
RInd GB ( S ( g / n )) = S ( g ) ⊗ S ( h ) W S ( h ) . (3.4)where n := [ b , b ] . To see this, observe first that, since g ∗ × h ∗ / W h ∗ is affine, the equality3.3 is after taking global sections equivalent to the equality R Γ( O e g B ) = O ( g ∗ × h ∗ / W h ∗ ) = S ( g ) ⊗ S ( h ) W S ( h ) of G -modules. Moreover, since the bundle projection p : e g B → B with fiber ( g / n ) ∗ isaffine, p ∗ is exact and hence R Γ( O e g B ) = R Γ( p ∗ ( O e g B )) . Under the identification Mod( O B ) =Mod( O G , B ) we have that p ∗ ( O e g B ) corresponds to S ( g / n ) ⊗ O ( G ) so its derived globalsections are given by RInd GB ( S ( g / n )) as stated. This proves 3.4.By a similar argument the statement of the lemma is equivalent to proving that RInd GP ( S ( g / r )) = S ( g ) ⊗ S ( h ) W S ( h ) W P . (3.5)For any M ∈ Mod( B ) we have an equality of P -modules RInd PB ( M ) = RInd LL ∩ B ( M ) . (3.6)where the R -module structure on the RHS is defined by ( xf )( g ) := g − xg · f ( g ) for f ∈ M or ( L, M ) L ∩ B ∼ = Ind LL ∩ B ( M ) , x ∈ U , g ∈ L . Together with the given L -actionthis makes the RHS a P -module. In particular we have RInd PB ( S ( g / n )) = RInd LL ∩ B ( S ( g / n )) . (3.7)We have a decomposition g = r P ⊕ l ⊕ r , where r P is the image of r under the Chevalleyinvolution of g ; thus g / n = l / ( l ∩ n ) ⊕ r P . Thus RInd LL ∩ B ( S ( g / n )) = RInd LL ∩ B ( S ( l / l ∩ n ) ⊗ S ( r P )) = (3.8) RInd LL ∩ B ( S ( l / l ∩ n )) ⊗ S ( r P ) = S ( g / r ) ⊗ S ( h ) W P S ( h ) where the last equality is given by 3.4 applied to G replaced by L and the second equalityis the tensor identity which applies since S ( r P ) is an L -module. Since RInd GB = RInd GP ◦ RInd PB we get from 3.4, 3.7 and 3.8 that S ( g ) ⊗ S ( h ) W S ( h ) = RInd GP ( S ( g / r ) ⊗ S ( h ) W P S ( h )) = RInd GP ( S ( g / r )) ⊗ S ( h ) W P S ( h ) . Since S ( h ) is faithfully flat over S ( h ) W P this implies 3.5.Let P ⊂ Q be two parabolic subgroups. The projection π QP : P → Q induces a map e π QP : e g P → e g Q that fits into the following commutative square:ingularLocalizationof g -modules 9 e g P l ∗ /L = h ∗ / W P e g Q l ∗ Q /L Q = h ∗ / W Q ✲❄ e π QP ❄✲ (3.9)With similar arguments as in the proof of Lemma 3.2 one can prove Lemma 3.3. R e π QP∗ O e g P = O e g Q × h ∗ / W Q h ∗ / W P . We observe that e g P is an L -torsor over T ∗ P . We put Definition 3.4. e g λ P = e g P × h ∗ / W P λ , for λ ∈ h ∗ .We would like to view e g λ P as the classical Hamiltonian of T ∗ ( G/R ) with respect to the(right) L -action. We have a moment map µ : T ∗ ( G/R ) → l ∗ . Recall that we can take theHamiltonian reduction with respect to any subset of l ∗ stable under the coadjoint action.Let N λ ⊂ l ∗ be the preimage of λ/ W P ∈ h ∗ / W P ∼ = l ∗ P /L under the quotient map. Then T ∗ ( G/R ) // N λ L = µ − ( N λ ) /L = e g λ P . (3.10)Note that we could also reduce with respect to λ ∈ ( l ∗ ) L in which case we would gettwisted cotangent bundles. P In this section we construct the sheaf of extended differential operators on a parabolic flagmanifold and describe its global sections.
Let X be an algebraic variety equipped with a free right action of a linear algebraic group K and let p : X → X/K be the projection. We assume that X , locally in the Zariskitopology, is of the form Y × K , for some variety Y , and p is first projection. Such X iscalled an K -torsor. We get induced right K -actions on the sheaf D X of regular differentialoperators on X and on the direct image sheaf p ∗ ( D X ) . Denote by e D X/K := p ∗ ( D X ) K thesheaf on X/K of K -invariant local sections of p ∗ ( D X ) .Let k := Lie K . The infinitesimal K -action gives algebra homomorphisms ˆ ǫ : U ( k ) →D X and ˜ ǫ : U ( k ) → p ∗ D X , which are injective since the K -action is free. It follows fromthe definition of differentiating a group action that [˜ ǫ ( U ( k )) , e D X/K ] = 0 .0 ErikBackelin and KobiKremnizer
Accepted for publication in Journal of the EMS
Notice that ˜ ǫ (U( k )) * e D X/K , unless K is abelian, but ˜ ǫ (Z( k )) ⊆ e D X/K . We denoteby ǫ : Z( k ) → e D X/K the restriction of ˜ ǫ to Z( k ) . By the discussion above it is a centralembedding.Now, using that p is locally trivial we can give a local description of e D X/K . Let Y × K be a Zariski open subset of X over which p is trivial. Then D X | Y × K = D Y ⊗ D K and e D X/K | Y = D Y ⊗ U( k ) , where U( k ) is identified with the algebra of right K -invariantdifferential operators D KK on K .Note that ˜ ǫ (U( k )) | Y × K = 1 ⊗ K D K is the algebra of left K -invariant differentialoperators on Y × K , with respect to the natural left K -action on Y × K , that are constantalong Y . Since Z( K D K ) = Z( D KK ) we get that ǫ is locally given by the embedding Z( k ) ֒ → U( k ) ∼ = 1 ⊗ U( k ) ֒ → D Y ⊗ U( k ) . This implies that ǫ (Z( k )) = Z( e D X/K ) .Denote by Mod( D X , K ) the category of weakly equivariant ( D X , K ) -modules. In or-der to simplify the description of this category we assume henceforth that X is quasi-affine. Its object M is then a left D X -module equipped with an algebraic right action ρ := { ρ U } , where ρ U : K → Aut C U ( M ( U )) op are homomorphism compatible withthe restriction maps in M , for each Zariski-open K -invariant subset R of X . We requirethat D X ⊗ M → M is K -linear (over K -invariant open sets) with respect to the diago-nal K -action on a tensor. (For a general X , ρ must be replaced by a given isomorphism pr ∗ M ∼ = act ∗ M satisfying a cocycle condition, where pr and act : X × K → X areprojection and the action map, respectively.)Denote by Mod( D X , K, k ) the category of strongly equivariant ( D K , K ) -modules. Itsobject ( M, ρ ) is a weakly equivariant ( D X , K ) -module such that dρ ( x ) m = ˆ ǫ ( x ) m for x ∈ k and m ∈ M .For M ∈ Mod( D X , K ) we consider the sheaf ( p ∗ M ) K of K -invariant local sections in p ∗ M ; it has a natural e D X/K -module structure. Thus we get a functor p ∗ whose right adjointis p ∗ (the pullback in the category of O -modules with its natural equivariant structure).The following is standard (see [BB93]): Lemma 4.1.
The functors i ) p ∗ ( ) K : Mod( D X , K ) ⇆ Mod( e D X/K ) : p ∗ and ii ) p ∗ ( ) K :Mod( D X , K, k ) ⇆ Mod( D X/K ) : p ∗ are mutually inverse equivalences of categories. On G/R we shall always consider the right L -action ( g, h ) gh , for g ∈ G and h ∈ L .Thus, G/R is an L -torsor. We put Definition 4.2. e D P := π P G/R ∗ ( D G/R ) L .ingularLocalizationof g -modules 11By the results of the previous section we have that locally on P , e D P ∼ = D P ⊗ U( l ) ,and we have the central algebra embedding ǫ : Z( l ) → e D P .For λ ∈ h ∗ we define: Definition 4.3. D λ P := e D P ⊗ ǫ (Z( l )) C λ . For any Z( l ) -algebra S and λ ∈ h ∗ let Mod b λ ( S ) be the category of left S -modules whichare locally annihilated by some power of I l ,λ .We shall give equivariant descriptions on G and on G/R of the category
Mod( e D P ) and its subcategories Mod( D λ P ) and Mod b λ ( e D P ) . It is best to work on G . We start with G/R as an intermediate step.By Lemma 4.1 we have mutually inverse equivalences π P G/R ∗ ( ) L : Mod( D G/R , L ) ⇆ Mod( e D P ) : π P∗ G/R . (4.1)Differentiating the right L -action on G/R gives an algebra embedding U( l ) ֒ → D G/R .This allows us to consider Z( l ) ⊆ U( l ) as a subalgebra of D G/R . Transporting conditionsfrom the right-hand side to the left-hand side of 4.1 we see that
Mod( D λ P ) is equivalentto the full subcategory Mod( D G/R , L, λ ) of Mod( D G/R , L ) whose object M satisfy I l ,λ · M L = 0 . Similarly, Mod b λ ( e D P ) is equivalent to the full subcategory Mod( D G/R , L, b λ ) of Mod( D G/R , L ) whose object M satisfies that I l ,λ is locally nilpotent on M L .Now we pass to G . Let us introduce some notations:We have a left and right actions µ l and µ r of G on O ( G ) defined by µ l ( g ) f ( h ) := f ( g − h ) and µ r ( g ) f ( h ) := f ( hg − ) , for f ∈ O ( G ) , g, h ∈ G , respectively. Differen-tiating µ l , resp., µ r , gives an injective algebra homomorphism ǫ l : U → D G , resp., ananti-homomorphism ǫ r : U → D G . We have that ǫ l (U) = D GG consists of right invari-ant differential operators on G and ǫ r (U) = G D G consists of left invariant differentialoperators on G , Z = ǫ l (U) ∩ ǫ r (U) and ǫ l | Z = ǫ r | Z .The actions µ l and µ r induce left and right actions of G on D G that we denote by thesame symbols.Let Mod( D G , P, r ) be the category whose objects are ( M, ρ ) where(1) M is a left D G -module.(2) ρ is a right algebraic P -action on M such that D G ⊗ M → M is P -linear, with respectto the right P -action µ r | P on D G and the diagonal P -action on the tensor product.(3) dρ | r = ǫ r | r on M .2 ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
In particular, by (3) the action ǫ r | r is integrable, i.e. this r -action is locally nilpotent. By4.1 and Lemma 4.1 ii) (applied to X = G and K = R ) we have an equivalence π P G ∗ ( ) P : Mod( D G , P, r ) ⇆ Mod( e D P ) : π P∗ G . (4.2)Note that the functor on the left hand side (that corresponds to) the global section functoris the functor of taking P -invariants.Let f M P := U / U · r be a sort of “ P -universal” Verma module for U and equip it withthe P -action that is induced from the right adjoint action of P on U . Note that the object O G ⊗ ǫ r ( f M P ) ∈ Mod( D G , P, r ) represents global sections and therefore corresponds to e D P ∈ Mod( e D P ) .Our next task is to describe the (full) subcategories Mod( D G , P, r , λ ) and Mod( D G , P, r , b λ ) of Mod( D G , P, r ) corresponding to the subcategories Mod( D λ P ) and Mod b λ ( e D P ) of Mod( e D P ) ,respectively.Let us consider the smash product D G ∗ U( l ) of D G and U( l ) with respect to the adjointaction of l on g . Thus, D G ∗ U( l ) = D G ⊗ U( l ) as a vector space and its (associative)multiplication is defined by y ⊗ x · y ′ ⊗ x ′ := y [ ǫ r ( x ) , y ′ ] ⊗ x ′ + yy ′ ⊗ xx ′ , x ∈ l , x ′ ∈ U( l ) , y, y ′ ∈ D G . Observe that a ( D G , L ) -module is the same thing as a D G ∗ U( l ) -module on which theaction of ⊗ l is integrable (i.e. its the differential of the given L -action). We have analgebra isomorphism D G ⊗ U( l ) ∼ −→ D G ∗ U( l ) , y ⊗ y ⊗ , ⊗ x ⊗ x − ǫ r ( x ) ⊗ , y ∈ D G , x ∈ l . This restricts to the algebra homomorphism α l : U( l ) → D G ∗ U( l ) , ⊗ x ⊗ x − ǫ r ( x ) ⊗ , x ∈ l . (4.3)Note that the algebra anti -isomorphism ∗ : U( l ) ∼ −→ U( l ) , x
7→ − x , for x ∈ l , restricts toan isomorphism ∗ : Z( l ) ∼ −→ Z( l ) . Proposition 4.4. i) Let M ∈ Mod( e D P ) and z ∈ Z( l ) . Since ǫ l ( z ) ∈ Z( l ) = Z( e D P ) itdefines a morphism ǫ l ( z ) : M → M . By functoriality we get a morphism π P∗ G ( ǫ l ( z )) : π P∗ G ( M ) → π P∗ G ( M ) . We have π P∗ G ( ǫ l ( z )) = α l ( z ∗ ) | π P∗ G ( M ) . ii) Let M ∈ Mod( D G , P, r ) . Then M ∈ Mod( D G , P, r , λ ) iff the following holds:(4) ( α l ( z ∗ ) − χ l ,λ ( z )) m = 0 , m ∈ M, z ∈ Z ( l ) . iii) Let M ∈ Mod( D G , P, r ) . Then M ∈ Mod( D G , P, r , b λ ) iff the following holds:( b ) α l ( z ) − χ l ,λ ( z ) is locally nilpotent on M, for z ∈ Z ( l ) . ingularLocalizationof g -modules 13 Proof. i) . We have π P∗ G ( M ) = O G ⊗ π P− G ( O P ) π P− G ( M ) . Let f ∈ O G and m ∈ π P− G ( M ) .Then for x ∈ l we have dρ ( x ) m = 0 and consequently α l ( − x )( f ⊗ m ) = ( ǫ r ( x ) − dρ ( x ))( f ⊗ m ) = f ⊗ ǫ r ( x ) m. Since α l is an algebra homomorphism we get for z ∈ Z( l ) that α l ( z ∗ )( f ⊗ m ) = f ⊗ ǫ r ( z ) m = π P∗ G ( ǫ l ( z ))( f ⊗ m ) . This proves i) . ii) follows from i) . iii) is similar to ii) and left to the reader.Let M P,λ := U / U · ( r + Ker χ l ,λ ) be a left U -module equipped with the right P -actionthat is induced from the adjoint action of P on U . Note that the object O G ⊗ ǫ r ( M P,λ ) of Mod( D G , P, r , λ ) represents global sections (= taking P -invariants) and therefore corre-sponds to D λ P ∈ Mod( D λ P ) . Remark 4.5.
Note that when l = h condition (4) becomes the traditional condition of[BB93]: ǫ r ( x ) m − dρ ( x ) m = λ ( x ) m , for x ∈ h , m ∈ M . Remark 4.6.
Assume that M ∈ Mod( D G , P, r ) . Then condition (4) holds for M ⇐⇒ ( ′ ) ( ǫ r ( z ) − χ l ,λ ( z )) m = 0 , for m ∈ M L , z ∈ Z ( l ) .(Because (4 ′ ) is obviously equivalent to ( π P G ∗ M ) L ∈ Mod( D λ P ) .)If we consider M L as a sheaf on G/L it global sections equal Γ G ( M ) L , where Γ G ( M ) is the O ( G ) -module corresponding to the O G -module M . Since L is reductive G/L isaffine, [Mat60], and therefore we may replace M L by Γ G ( M ) L in (4 ′ ) .However, condition (4) is better to work with then (4 ′ ) , particularly while consideringmodules with an additional equivariance condition from the left side, see Section 7. Example 4.7.
Let us consider the simplest case when P = G . Then r = 0 and we write Mod( D G , G, λ ) := Mod( D G , G, r G , λ ) for simplicity.The equivalence Mod( C ) ∼ = Mod( O G , G ) , V
7→ O G ⊗ V , induces for any λ ∈ h ∗ theequivalence Mod(U λ ) ∼ = Mod( D G , G, λ ) given by V
7→ O G ⊗ V where ( O G ⊗ V ) G = V is a left module for ǫ l (U) λ . Similarly with χ λ replaced by c χ λ . Example 4.8.
Let P = B . Let λ ∈ h ∗ and let M λ be the Verma module for ǫ r (U) withhighest weight λ . Let µ ∈ h ∗ be integral. Consider the algebraic B -action ρ on M λ whichafter differentiation satisfies dρ ( x ) m = ( x − λ ( x ) + µ ( x )) m, m ∈ M λ , x ∈ b . Accepted for publication in Journal of the EMS
Denote by M λ,µ the Verma module M λ equipped with this B -action. Then we have that O G ⊗ M λ,µ ∈ Mod( D G , B, n , λ − µ ) . For µ = 0 we have mentioned that the functor Hom
Mod( D G ,B, n ,λ ) ( O G ⊗ M λ, , ) is natu-rally equivalent to the global section functor on Mod( D G , B, n , λ ) , so that O G ⊗ M λ, ∼ = π G ∗B D λ B . This implies End
Mod( D G ,B, n ,λ ) ( O G ⊗ M λ ) = Γ( D λ B ) = U λ . (4.4)To get an idea of a general O G ⊗ M λ,µ assume for instance that µ ≥ . Then there is aninjective map f : O G ⊗ M λ,µ → O G ⊗ M λ − µ, . (4.5)By the Peter-Weyl theorem O G ∼ = ⊕ φ ∈ Λ + V ∗ G ( φ ) ⊗ V G ( φ ) as a G -bimodule. Let v φ ∈ V G ( φ ) be a highest weight vector. Let λ and λ − µ be highest weight vectors in M λ,µ and M λ − µ, , respectively. We can define f by f (1 ⊗ λ ) := ( v ⊗ v µ ) ⊗ λ − µ where v ∈ V ∗ G ( µ ) isany non-zero vector. f is injective since both sides of 4.5 are free over the integral domain O G ⊗ ǫ r (U( n − )) . Note that f is not an isomorphism (and the two objects of 4.5 must benon-isomorphic) unless µ = 0 . The left G -action on G/R , ( g, g ′ ) gg ′ , commutes with the right L -action and thereforeinduces a homomorphism U → e D P . There is also the map ǫ : S ( h ) W P = Z( l ) → e D P .These maps agree on S ( h ) W and hence induces a map e U W P = U ⊗ Z S ( h ) W P → e D P . This induces a homomorphism U λ = e U W P / ( I l ,λ ) → D λ P .Consider the sheaf of algebras O P ⊗ U on P with multiplication determined by thosein O P and in U and by the requirement that [ A, f ] = ǫ ( A )( f ) for A ∈ g and f ∈ O P .Then we have a surjective algebra homomorphism η : O P ⊗ U → e D P . Its kernel is theideal generated by ξ ∈ O P ⊗ r , ξ ( x ) ∈ p x , for x ∈ P and p x ⊆ g the correspondingparabolic subalgebra.Hence, to define a e D P -module structure on an O P -module M is the same thing asdefining a U -module structure on M such that Ker η vanishes on M and A ( f m ) = f ( Am ) + ǫ ( A )( f ) m , for A ∈ g , f ∈ O P and m ∈ M .Let µ ∈ h ∗ be integral and P -dominant. Recall that V P ( µ ) denotes the correspondingirreducible representation of P with highest weight µ and O ( V P ( µ )) the correspondingleft G -equivariant locally free sheaf on P .ingularLocalizationof g -modules 15Let M ∈ Mod( e D P ) . We shall show that the O P -module M ⊗ O P O ( V P ( µ )) is naturallya e D P -module. We proceed as follows:The G -action on O ( V P ( µ )) differentiates to a left g -action on it, which extends to a g -action on M ⊗ O P O ( V P ( µ )) by Leibniz’s rule. Since V P ( µ ) is an irreducible P -module wehave that R acts trivially on it (recall V P ( µ ) = V L ( µ ) ). Hence, r acts trivially O ( V P ( µ )) and from this it now follows that the compatibilities for being a e D P -module are satisfiedby M ⊗ O P O ( V P ( µ )) .Assume that M ∈ Mod( e D P ) . In the equivariant language on G we see that M and M ⊗ O P O ( V P ( µ )) correspond to π P∗ G M and M V P ( µ ) := ( π P∗ G M ) ⊗ V P ( µ ) ∈ Mod( D G , P, r ) ,respectively. Here, the D G -action on M V P ( µ ) is given by the action on the first factor andthe P -action is diagonal. Again, it is the fact that R acts trivially on V P ( µ ) that shows that M V P ( µ ) is an object of Mod( D G , L, r ) . Lemma 4.9.
Let λ ∈ h ∗ , M ∈ Mod( D λ P ) and µ ∈ h ∗ be integral and P -dominant.Then M ⊗ O P O ( V P ( µ )) ∈ ⊕ ν ∈ Λ( V P ( µ )) Mod [ λ + ν ( e D P ) , where Λ( V P ( µ )) denotes the set ofweights of V P ( µ ) .Proof. In equivariant translation we want to prove that M V P ( µ ) ∈ ⊕ ν ∈ Λ( V P ( µ )) Mod( D G , P, r , [ λ + ν ) . (4.6)We use Proposition 4.4 i) . We have an action e α l : U( l ) → End( M V P ( µ ) ) . We see that thisaction is actually the tensor product of the e α l -action of U( l ) on π P∗ G M and the U( l ) -actionon V P ( µ ) , which is the differential of the given L -action. Now, since for z ∈ Z( l ) , we byassumption have that α l ( z ) = e α l ( z ) acts by χ l ,λ ( z ) on π P∗ G M it follows from [BerGel81]that 4.6 holds. Theorem 4.10. i ) Rπ PB ∗ e D B = e D P ⊗ Z( l ) S ( h ) , ii ) Rπ QP ∗ e D P = e D Q ⊗ Z( l Q ) S ( h ) W P , iii ) R Γ( e D P ) = e U W P and iv ) R Γ( D λ P ) = U λ .Proof. By Lemma 3.2 and Lemma 3.3 the associated graded maps i ) and ii ) are isomor-phisms; hence i ) and ii ) are also isomorphisms. iii ) is a special case of ii ) and iv ) followsfrom iii ) because R Γ commutes with ( ) ⊗ Z( l ) C λ , since e D P is locally free over Z( l ) .The functor Γ : Mod( D λ P ) → Mod(U λ ) has a left adjoint L := D λ P ⊗ U λ ( ) , calledthe localization functor. Also Γ : Mod b λ ( e D P ) → Mod b λ (U) has a left adjoint L :=lim ←− n D P / ( I λ ) n ⊗ U ( ) . Here we prove the singular version of Beilinson-Bernstein localization.6 ErikBackelin and KobiKremnizer
Accepted for publication in Journal of the EMSTheorem 5.1.
Let λ be dominant and P -regular then Γ : Mod( D λ P ) → Mod(U λ ) is anequivalence of categories.Proof. Essentially taken from [BB81]. Since Γ has a left adjoint L which is right exactand since Γ ◦ L (U λ ) = Γ( D λ P ) = U λ , the theorem will follow from the following twoclaims: a ) Let λ be dominant. Then Γ : Mod( D λ P ) → Mod(U λ ) is exact. b ) Let λ be dominant and P -regular and M ∈ Mod( D λ P ) , then if Γ( M ) = 0 it followsthat M = 0 .Let V be a finite dimensional irreducible G -module and let V − ⊂ V ⊂ . . . ⊂ V n = V be a filtration of V by P -submodules, such that V i /V i − ∼ = V P ( µ i ) is an irreducible P -module.We first chose V so that its highest weight µ is a P -character. Thus M ⊗ O O ( V ) = M ( − µ ) and we get an embedding M ( − µ ) ֒ → M ⊗ O O ( V ) , which twists to the em-bedding M ֒ → M ( µ ) ⊗ O O ( V ) ∼ = M ( µ ) dim V . Now, by Lemmas 2.1, 4.9 and Theorem4.10 iii) we get that this inclusion splits on derived global sections, so R Γ( M ) is a di-rect summand of R Γ( M ( µ )) dim V . Now, for µ big enough and if M is O -coherent wehave R > Γ( M ( µ )) = 0 (since O ( µ ) is very ample). Hence, R > Γ( M ) = 0 in this case.A general M is the union of coherent submodules and by a standard limit-argument itfollows that R > Γ( M ) = 0 . This proves a ) .Now, for b ) we assume instead that the lowest weight µ n of V is a P -character. Thenwe have a surjection M dim V ∼ = M ⊗ O O ( V ) → M ( − µ n ) . Applying global sections andusing Lemmas 2.2, 4.9 and Theorem 4.10 iv) we get that Γ( M ( − µ n )) is a direct summandof Γ( M ) dim V . For µ n small enough we get that Γ( M ( − µ n )) = 0 . Hence, Γ( M ) = 0 . Thisproves b ) .Assume that λ is P -regular. Then the projection h ∗ / W P → h ∗ / W is unramified at λ and from this one deduces, see [BG99], that restriction defines an equivalence of cate-gories Mod b λ ( e U W P ) ∼ −→ Mod b λ (U) . Theorem 5.2.
Let λ be dominant and P -regular then Γ : Mod b λ ( e D P ) → Mod b λ ( e U W P ) ∼ =Mod b λ (U) is an equivalence of categories.Proof. This follows from Theorem 5.1 and a simple devissage.
We geometrically describe translations functors on g -modules in the context of singularlocalization. For regular localization this was worked out in [BG99]. Singular localizationingularLocalizationof g -modules 17clarifies the picture. We get one-one correspondences between translation functors andgeometric functors and all global section functors can be made to take values in Mod(U) .Thus ramified coverings of the form h ∗ / W λ → h ∗ / W µ will not complicate the picture asthey appeared to do in [BG99]. For any Z( l ) -algebra S let Mod Z( l ) - fin ( S ) be the category of S -modules that are locallyfinite over Z( l ) . Thus Mod Z( l ) - fin ( S ) = ⊕ µ ∈ h ∗ Mod b λ ( S ) and we have exact projections pr l , b µ : Mod Z( l ) - fin ( S ) → Mod b µ ( S ) . We put pr b µ := pr g , b µ .Assume λ, µ ∈ h ∗ satisfy λ − µ is integral. Then there is the translation functor T µ l ,λ : Mod b λ (U( l )) → Mod b µ (U( l )) , M pr l , b µ ( M ⊗ E ) where E is an irreducible finite dimensional representation of l with extremal weight µ − λ . Again, put T µλ := T µ g ,λ . See [BerGel81] for further information about translationfunctors.We shall give a D -module interpretation of these functors. We use the language of e D P -modules; it is a simple task to pass to an equivariant description on G . Define for anyparabolic subgroup P ⊂ G a geometric translation functor T µP,λ : Mod b λ ( e D P ) → Mod b µ ( e D P ) , M pr l , b µ ( M ⊗ O P O ( E )) for M ∈ Mod b λ ( e D P ) , where E is an irreducible P -representation with highest weight in W P ( µ − λ ) .Note that if µ − λ is a P -character then O P ( E ) = O P ( µ − λ ) and in this case T µP,λ =( ) ⊗ O P O ( µ − λ ) is an equivalence with inverse given by T λP,ν = ( ) ⊗ O P O ( λ − µ ) . Inparticular, for P = B we have T µB,λ = ( ) ⊗ O B O ( µ − λ ) for any µ and λ .Let Q ⊂ G be another parabolic subgroup with P ⊂ Q . We have Lemma 6.1.
The diagram
Mod b λ ( e D P ) Mod b µ ( e D P )Mod b λ ( e D Q ) Mod b µ ( e D Q ) ✲ T µP,λ ❄ π QP∗ ❄ π QP∗ ✲ T µQ,λ of exact functors commutes up to natural equivalence. In the case of P = B and Q = G this was proved in [BG99].8 ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
Proof.
Let V (resp., V ′ ) be an irreducible finite dimensional representation for Q (resp.,for P ) whose highest weight belongs to W Q ( µ − λ ) (resp., W P ( µ − λ ) ). Let M ∈ Mod b λ ( e D P ) . Then, since V is a Q -representation, we have O P ( V ) = π Q∗P ( O Q ( V )) andtherefore it follows from the projection formula that π QP∗ ( O P ( V ) ⊗ O P M ) = O Q ( V ) ⊗ O Q π QP∗ ( M ) . Thus we get T µQ,λ ◦ π QP∗ ( M ) = pr l Q , b µ ( O Q ( V ) ⊗ O Q π QP∗ ( M )) = pr l Q , b µ ( π QP∗ ( O P ( V ) ⊗ O P M )) = π QP∗ ( pr l , b µ ( O P ( V ) ⊗ O P M )) ( ∗ ) = π QP∗ ( pr l , b µ ( O P ( V ′ ) ⊗ O P M )) = π QP∗ ◦ T µP,λ ( M ) . The equality ( ∗ ) follows from Lemma 2.2 applied to the reductive Lie algebra l Q and itsparabolic subalgebra l Q ∩ p (compare with the proof of the localization theorem).Let us geometrically describe translation to the wall : In this case ∆ λ ( ∆ µ . We assumethat λ and µ are dominant. We choose the parabolic subgroups P ⊂ Q ⊂ G such that theparabolic roots of P equal ∆ λ and the parabolic roots of Q equal ∆ µ . By Theorem 5.2and Lemma 6.1 it follows that the diagram below commutes up to natural equivalence: Mod b λ (U) Mod b λ ( e D P )Mod b λ ( e D Q ) Mod b µ ( e D P )Mod b µ (U) Mod b µ ( e D Q ) ❄ (4) T µλ ✛ (1) Γ ❄ (3) π QP∗ ◗◗◗◗◗◗s (2) T µP,λ ❄ (5) T µQ,λ ✑✑✑✑✑✑✰ (7) π QP∗ ✛ (6) Γ (6.1)Note that (1) and (6) are equivalences by the choices of P and Q and that (2) = ( ) ⊗ O P O ( µ − λ ) is an equivalence, since µ − λ is a P -character.We see that (3) is an equivalence of categories because both the source and the targetcategories are D-affine, since λ is P - and Q -regular, and Γ ◦ π QP∗ = Γ . On the other hand,the functor (7) is not faithful, because µ is not P -regular. (5) is also not faithful. Weremind that all functors involved are exact.Let us now describe translation out of the wall : This is done by taking the diagram ofadjoint functors in the diagram 6.1, so we keep assuming that λ , µ , P and Q are as in 6.1.The left and right adjoint of T µλ is T λµ , the translation out of the wall. The equivalences (1) , (2) , (3) and (6) of course have left and right adjoints that coincide. Also, the left andingularLocalizationof g -modules 19right adjoint of (5) coincide; it is given by T λQ,µ . Finally (7) has the left adjoint π Q∗P ; thus, π Q∗P must also be the right adjoint of (7) . Summing up we have:
Mod b λ (U) Mod b λ ( e D P )Mod b λ ( e D Q ) Mod b µ ( e D P )Mod b µ (U) Mod b µ ( e D Q ) ✲ L ✻ π Q∗P ◗◗◗◗◗◗❦ T λP,µ ✻ T λµ ✲ L ✻ T λQ,µ ✑✑✑✑✑✑✸ π Q∗P (6.2) O and Harish-Chandra (bi-)modules. Singular localization allows us to interpret blocks of category O as bi-equivariant D G -modules which in turn are equivalent to categories of Harish-Chandra (bi-)modules. Aswe mentioned in the introduction, the novelty here is that we are lead to consider g - l -bimodules, which we believe is a better notion. Parabolic (and singular) blocks of O arediscussed in Section 8.2.The material here is related to Section 6 because translation functors restrict to functorsbetween blocks in O . O and generalized twisted Harish-Chandra modules. See [Hum08] for generalities on category O and [Dix77] for generalities on Harish-Chandra modules.We are interested in the Bernstein-Gelfand-Gefand category O of finitely generated left U -modules which are locally finite over U( n ) and semi-simple over h . For λ ∈ h ∗ welet O λ , O b λ ⊂ O be the subcategories of modules with central character, respectively,generalized central character, χ λ . Generalized twisted Harish-Chandra modules.
Let K ⊂ G be a subgroup and let k := Lie K be its Lie algebra. A weak Harish-Chandra ( K, U) -module (or simply a ( K, U) -module) is a left U -module M equipped with an algebraic left action of K such that theaction map U ⊗ M → M is K -equivariant with respect to the adjoint action of K on U . A Harish-Chandra ( K, U) -module (or simply a ( k , K, U) -module) is a weak Harish-Chandra module such that the differential of the K -action coincides with the action of k ⊂ U .Similarly, there are ( K, U λ ) -modules and ( k , K, U λ ) -modules, for λ ∈ h ∗ .0 ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
Let µ ∈ K ∗ . A µ - twisted Harish-Chandra module is a ( K, U) -module M on whichthe action of k ⊂ U minus the differential of the K -action is equal to µ .We shall now give certain generalizations of twisted Harish-Chandra modules in thecase when K = P . Consider the smash-product algebra U ∗ U( l ) with respect to theadjoint action of l on U . Observe that an ( L, U) -module is the same thing as a U ∗ U( l ) -module on which ⊗ l acts semi-simply and ⊗ H α has integral eigenvalues for eachsimple coroot H α . The algebra anti-homomorphism U( l ) → U ∗ U( l ) , defined by x x ⊗ − ⊗ x , for x ∈ l , restricts to a homomorphism α l : Z( l ) → Z( U ( g ) ∗ U( l )) . (7.1)(Compare with the map α l ( z ∗ ) from 4.3.) We define Mod( b λ, r , P, U λ ′ ) to be the categoryof ( P, U λ ′ ) -modules M such that, if ρ denotes the P -action on M , then dρ | r coincideswith the action of r ⊂ U λ ′ on M and for z ∈ Z( l ) we have that α l ( z ) − χ l ,λ ( z ) acts locallynilpotently on M .Similarly, one defines categories Mod b λ ′ ( b λ, r , P, U) and Mod( λ, r , P, U λ ′ ) , etc.We see that if λ, λ ′ ∈ h ∗ , λ − λ ′ is integral then O λ = mod( λ ′ , n , B, U λ ) and O b λ = mod b λ ( λ ′ , n , B, U) are (non-generalized) categories of twisted Harish-Chandra modules. For P = B we liketo think of mod( b λ, r , P, U λ ′ ) and mod( λ, r , P, U λ ′ ) as “non-standard parabolic blocks in O ” although, in reality, they are not even subcategories of O , since the b -action is notlocally finite. The categories of the previous section can be described in terms of Harish-Chandra bi-modules, [BerGel81]. Let e H ( l ) be the category of U - U( l ) -bimodules on which the adjointaction of l is integrable and the left action of r is locally nilpotent. Write e H := e H ( g ) andreplacing g by l we write e H ( l , l ) for the category of U( l ) - U( l ) -bimodules on which theadjoint l -action is integrable.Let H ( l ) ⊂ e H ( l ) be the subcategory of noetherian objects. Note that for M ∈ e H ( l ) we have M ∈ H ( l ) ⇐⇒ M is f.g. as a U - U( l ) -bimodule ⇐⇒ M is f.g. as a left U -module (and in case l = g this holds if and only if M is f.g. as a right U -module). Put Z - fin H ( l ) := { M ∈ H ( l ); Z acts locally finitely on M from the left } , H ( l ) Z( l ) - fin := { M ∈ H ( l ); Z( l ) acts locally finitely on M from the right } and Z - fin H ( l ) Z( l ) - fin := Z - fin H ( l ) ∩ H ( l ) Z( l ) - fin . Observe that Z - fin H = H Z - fin = Z - fin H Z - fin . (7.2)ingularLocalizationof g -modules 21We set λ ′ H ( l ) := { M ∈ H ( l ); I λ ′ M = 0 } , H ( l ) λ := { M ∈ H ( l ); M I l ,λ = 0 } and b µ H ( l ) := { M ∈ H ( l ); I λ ′ acts locally nilpotently on M } , etc. Similarly, we define λ ′ H ( l ) b λ := λ ′ H ( l ) ∩ H ( l ) b λ , e H ( l ) λ , etc. Lemma 7.1.
Mod( λ, r , P, U λ ′ ) ∼ = λ ′ H ( l ) λ . and Mod( b λ, r , P, U λ ′ ) ∼ = λ ′ H ( l ) b λ .Proof. A ( P, U λ ′ ) -module is the same thing as a U λ ′ ∗ U( p ) -module such that ⊗ p actsintegrably. Under the algebra isomorphism U λ ′ ∗ U( p ) ∼ −→ U λ ′ ⊗ U( p ) , ⊗ x ⊗ x + x ⊗ , y ⊗ y ⊗ the latter modules are equivalent to the category of U λ ′ ⊗ U( p ) -modules on which theaction of ∆ p is integrable, where ∆ : p → U λ ′ ⊗ U( p ) is given by ∆ x := x ⊗ ⊗ x .The ∆ p -integrability is equivalent to ∆ l -integrability and that ∆ r acts locally nilpo-tently. Thus Mod( r , P, U λ ′ ) is equivalent to the category of U λ ′ ⊗ U( l ) -modules such thatthe action of ∆ l is integrable and r ⊂ U λ ′ acts nilpotently. Thus, using the principalanti-isomorphism of l to identify U λ ′ ⊗ U( l ) -modules with U λ ′ - U( l ) -bimodules, we get Mod( r , P, U λ ′ ) ∼ = λ ′ H ( l ) . From this one deduces the lemma. D -modules and category O We want to describe blocks in category O in terms of bi-equivariant D G -modules. Let λ ∈ h ∗ . Throughout this section we assume that λ ′ ∈ h ∗ is a regular dominant weightsuch that λ − λ ′ is integral.Denote by Mod( λ ′ , n , B, D G , P, r , b λ ) the full subcategory of Mod( D G , P, r , b λ ) whoseobject M satisfies (1) − (3) , ( b from Section 4.2 and is in addition equipped with a left B -action τ : B → Aut( M ) that commutes with ρ : P → Aut( M ) op and satisfies(5) dτ ( x ) m = ( ǫ l ( x ) − λ ′ ( x )) m, for m ∈ M, x ∈ b . (Strictly speaking, Mod( λ ′ , n , B, D G , P, r , b λ ) is obtained from Mod( D G , P, r , b λ ) by addinga B -action, but since this B -action is determined by its differential it identifies with a sub-category of it.) Lemma 7.2.
Assume that λ is P -regular. Then mod( λ ′ , n , B, D G , P, r , b λ ) ∼ = O b λ .Proof. We remind that, since λ is P -regular, restriction defines an equivalence of cate-gories res : Mod b λ ( e U W P ) ∼ −→ Mod b λ (U) . Now ( b , the two lines preceding it and Theo-rem 5.2 give the equivalence Mod( D G , P, r , b λ ) ∼ = Mod b λ (U) , V res ( V P ) . From this we deduce that the full subcategory O b λ = mod b λ ( λ ′ , n , B, U) of Mod b λ (U) isequivalent to mod( λ ′ , n , B, D G , P, r , b λ ) .2 ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS
Using the inversion on G , left B -action and right P -action become right B -actionand left P -action, so mod( λ ′ , n , B, D G , P, r , b λ ) is equivalent to a full subcategory of Mod( D G , B, n , λ ′ ) that we denote by mod( b λ, r , P, D G , B, n , λ ′ ) (7.3)whose definition is obvious. Since λ ′ is dominant and regular we get from Beilinson-Bernstein localization that Mod( D G , B, n , λ ′ ) ∼ = Mod(U λ ′ ) . This induces an equivalencebetween 7.3 and mod( b λ, r , P, U λ ′ ) . (This is not the parabolic-singular Koszul duality of[BGS96].)Similarly, if we don’t pass to global sections on B , we have that 7.3 is equivalent tothe category mod( b λ, r , P, D λ ′ B ) , whose definition is also obvious.Summarizing we get Proposition 7.3. O b λ ∼ = mod( b λ, r , P, U λ ′ ) ∼ = mod( b λ, r , P, D λ ′ B ) , for λ dominant and P -regular. Thus, by Lemma 7.1
Corollary 7.4. O b λ ∼ = λ ′ H ( l ) b λ . Similarly, one shows that O λ ∼ = mod( λ, r , P, U λ ′ ) ∼ = mod( λ, r , P, D λ ′ B ) ∼ = λ ′ H ( l ) λ . Example 7.5.
Let P = B and λ ∈ h ∗ be regular and dominant. Then O b λ ∼ = mod( b λ, n , B, U λ ′ ) ,which is the category of left U λ ′ -modules which are locally finite over b (so the h -actionneed not be semi-simple). This equivalence was first established in [Soe86]. Example 7.6.
Let P = G and λ ∈ h ∗ be any weight. Since r G = 0 we write for simplicity Mod( b λ, G, U λ ′ ) := Mod( b λ, r G , G, U λ ′ ) . Put O [ λ +Λ := ⊕ µ ∈ Λ O [ λ + µ . Then we have O b λ ∼ −→ mod( b λ, G, U λ ′ ) and O [ λ +Λ ∼ −→ mod( G, U λ ′ ) , both given by V ( O G ⊗ V ) B . Thus O b λ ∼ = λ ′ H b λ . See [BerGel81], [Soe86]. Remark 7.7. mod( b λ, r , P, D λ ′ B ) will not consist of holonomic D -modules, unless P = B .For instance, if λ = − ρ , P = G and λ ′ = 0 , then O c − ρ will consist of direct sums of copiesof the simple Verma module M − ρ . Corresponding to M − ρ is a non-holonomic submoduleof the D B -module D B (see 4.5). Let f : U( n ) → C be an algebra homomorphism, ∆ f := { α ∈ ∆; f ( X α ) = 0 } and J f := Ker f . Let e N f := e N ( g ) f be the category of left U -modules on which J f acts locallyingularLocalizationof g -modules 23nilpotently and let N f be its subcategory of modules which are f.g. over U . Objects of N f are called Whittaker modules. Replacing g by l and f by f | U( n ∩ l ) we get the category N f ( l ) . For regular f , i.e. when ∆ f = ∆ , it was studied by Kostant, [K78]; he showed that N f has the exceptionally simple description Mod(Z) ∼ −→ N f , M M ⊗ Z U / U · J f . (8.1)In the other extreme, when f = 0 , N f is O with the h -semi-simplicity condition droppedand it has the same simple objects as O .Our main result here is a new proof of Theorem 8.1 of [MS97]. It enables one tocompute the characters of standard Whittaker modules by means of the Kazhdan-Lusztigconjectures. (For non-integral weights they were computed in [B97].)Throughout this section we assume λ ∈ h ∗ and ∆ P = ∆ f = ∆ λ . N f and of singular O Fix a charcater f : U( n ) → C . For µ ∈ h ∗ we put µ N f := { M ∈ N f ; I µ M = 0 } , b µ N f := { M ∈ N f ; I µ acts locally nilpotently on M } . (Categories µ e N f and b µ e N f are similarly defined.) Our aim is to prove Theorem 8.1.
Assume that λ, λ ′ ∈ Λ satisfies ∆ f = ∆ λ and that λ ′ is regular dominant.Then O b λ ∼ = λ ′ N f . Before proving this we establish some preliminary results.
Lemma 8.2. i) For each µ, λ ∈ h ∗ , µ dominant, such that W µ ⊆ W λ , µ H b λ identifieswith a finite length subcategory of O b λ which is non-zero iff λ − µ is integral (analogousstatements hold with µ and/or λ replaced by b µ and/or b λ ). ii) µ H c − ρ ∼ = mod( C ) and µ e H c − ρ ∼ = Mod( C ) , for µ integral. iii) H Z - fin is a finite length category.Proof. That µ H b λ = 0 if µ − λ is not integral is a consequence of the fact that any G -module is a sum of G -modules with integral central characters.On the other hand, let µ − λ be integral and E be an irreducible G -module with ex-tremal weight µ − λ . For M ∈ H λ we have E ⊗ M ∈ H λ , with respect to the diagonal left U -action and the right U -action on the second factor. Thus, T µλ M = pr b µ ( E ⊗ M ) ∈ b µ H λ .(Similarly, with λ replaced by b λ .)Now U λ ∈ λ H λ with its natural bimodule structure. Since W µ ⊆ W λ it is known that T µλ is faithful. Hence we get = T µλ (U λ ) ∈ b µ H λ . Thus, also µ H λ and µ H b λ are non-zero.We have µ H b λ ∼ = mod( λ, G, U µ ) L −→ mod( λ, G, D G , B, µ ) ∼ = Accepted for publication in Journal of the EMS mod( µ, B, D G , G, b λ ) ∼ = mod b λ ( µ, B, U) = O b λ . Since µ is dominant we have Γ ◦ L = Id . Since O b λ is a finite length category this implies µ H b λ is dito as well. This proves i) . Moreover, the fact that O c − ρ ∼ = mod( C ) now implies µ H c − ρ ∼ = mod( C ) . A similar argument shows µ e H c − ρ ∼ = Mod( C ) . This proves ii) .By 7.2, H Z - fin = Z - fin H Z - fin . Since µ H λ is a finite length category for all µ, λ ∈ h ∗ adevissage implies iii) . Lemma 8.3.
Let µ ∈ Λ . The functors Θ µ := ( ) ⊗ U( n ∩ l ) C f : µ e H ( l , l ) b λ → µ e N ( l ) f and Θ b µ := ( ) ⊗ U( n ∩ l ) C f : b µ e H ( l , l ) b λ → b µ e N ( l ) f are equivalences of categories.Proof. This certainly holds for l = h and from that we immediately reduce to the case g = l , ∆ f = ∆ and λ = − ρ . We must then show that the functor Θ µ : µ e H c − ρ → µ e N f , M M ⊗ U( n ) C f , is an equivalence of categories. It follows from Kostant’s equivalence 8.1 that µ e N f isequivalent to Mod( C ) (for all µ ∈ h ∗ ). By Lemma 8.2 ii) also µ e H c − ρ ∼ = Mod( C ) ; henceit suffices to show that Θ µ takes simples to simples. The Θ µ ’s commutes with translationfunctors, so since U − ρ ∈ − ρ H c − ρ we get Θ µ T µ − ρ (U − ρ ) = T µ − ρ Θ − ρ (U − ρ ) = T µ − ρ ( U − ρ ⊗ U( n ) C f ) . By [K78] the latter is simple. This implies both that T µ − ρ (U − ρ ) is simple generator for µ e H c − ρ and that Θ µ takes simples to simples. Thus Θ µ is an equivalence.A devissage using Lemma 8.4 now shows that Θ b µ is an equivalence. Lemma 8.4.
Each M ∈ e H c − ρ which is countably generated as a left U -module is faithfullyflat as a right U( n ) -module.Proof. Assume first that M is simple. Then it follows from Schur’s lemma that M ∈ µ H c − ρ , for some integral µ ∈ h ∗ . By Lemma 8.2 we know that µ H c − ρ ∼ = mod( C ) . Hence, M ∼ = T µ − ρ (U − ρ ) as this is simple (and hence a simple generator for µ H c − ρ ) by the proofof Lemma 8.3. By an adjunction argument M is projective as a right U − ρ -module. ByKostant’s separation of variables theorem, [K63], U − ρ is free over U( n ) . Hence M isprojective over U( n ) .Assume now that M ∈ H c − ρ is finitely generated. By Lemma 8.2 M has finite lengthand an induction on its length shows that M again is projective as a right U( n ) -module.For arbitrary M choose a filtration M ⊆ M ⊆ M ⊆ . . . ⊆ M of finitely generatedsubmodules. Put M i = M i /M i − . Since all M i and M i are projective we get that M i ∼ = ⊕ j ≤ i M j and thus M = lim −→ M i ∼ = lim −→ ⊕ j ≤ i M j = ⊕ i ∈ N M i ingularLocalizationof g -modules 25is projective, and therefore flat, as a right U( n ) -module.To see that M is faithful over U( n ) , we observe that the above implies that M , as aright U( n ) -module, is a direct sum of modules of the form T µ − ρ (U − ρ ) , so it suffices toshow that T µ − ρ (U − ρ ) is faithful over U( n ) . Let V ∈ Mod(U( n )) be non-zero. We have T µ − ρ ( U − ρ ) ⊗ U( n ) V ∼ = T µ − ρ ( U − ρ ⊗ U( n ) V ) = 0 , since U − ρ ⊗ U( n ) V = 0 and T µ − ρ is faithful (since W µ ⊆ W − ρ ). Lemma 8.5.
Let µ ∈ Λ and M ∈ b µ N f . Then M = ⊕ ν ∈ Λ pr l , b ν M .Proof. Note that M has a filtration M ⊆ M ⊆ . . . ⊆ M n = M such that each subquo-tient M i := M i /M i − is generated over U by a vector v i such that J f · v i = I µ · v i = 0 .Thus each M i is a quotient of a sum of copies of U µ / U µ · J f and by [MS97] the latterhas a filtration with subquotients of the form U µ / U µ ( I l ,w · µ + J f ) , w ∈ W . These are inturn quotients of U µ / U µ · I l ,w · µ . Thus, it is enough to prove that U µ / U µ · I l ,w · µ = ⊕ ν ∈ Λ pr l , b ν U µ / U µ · I l ,w · µ , w ∈ W . Since b ν H ( l , l ) w · µ = 0 , for ν / ∈ w · µ + Λ = Λ , and since U µ / U µ · I l ,w · µ ∈ e H ( l , l ) w · µ = Z( l ) - fin e H ( l , l ) w · µ we are done. Proof of Theorem 8.1.
We have O b λ ∼ = λ ′ H ( l ) b λ , so we need to construct an equivalence Θ : λ ′ H ( l ) b λ ∼ −→ λ ′ N f , M M ⊗ U( n ∩ l ) C f . (8.2)Consider the restriction functor res : λ ′ H ( l ) b λ → e H ( l , l ) b λ . A “reductive version” ofLemma 8.4 applied to l shows that each object of H ( l , l ) b λ is faithfully flat as a right U( n ∩ l ) -module. Hence, Θ is faithful and exact.Denote by Ψ the right adjoint of Θ . Thus Ψ V = Hom C (lim ←− i U( l ) / ( I l ,λ ) i ⊗ U( n ∩ l ) C f , V ) l - int , where ( ) l - int is the functor that assigns a maximal l -integrable sub-object. (The left U -module structure on Ψ V comes from the left U -action on V and its right U( l ) -modulestructure comes from the left U( l ) -action on lim ←− i U( l ) / ( I l ,λ ) i ⊗ U( n ∩ l ) C f .)In order to prove that Θ is an equivalence its enough to show that the natural transfor-mation Θ ◦ Ψ → Id is an isomorphism. Take V ∈ λ ′ N f and put K := Ker { ΘΨ V → V } , C := Coker { ΘΨ V → V } . By Lemma 8.5 we have K = ⊕ ν ∈ Λ pr l , b ν K and C = ⊕ ν ∈ Λ pr l , b ν C . Let Ψ b ν be the rightadjoint of the functor Θ b ν from Lemma 8.3. Note that pr l , b ν V ∈ b ν e N ( l ) f and that pr l , b ν K =Ker { Θ b ν Ψ b ν pr l , b ν V → pr l , b ν V } and pr l , b ν C = Coker { Θ b ν Ψ b ν pr l , b ν V → pr l , b ν V } .Assume ν ∈ Λ . Then Θ b ν is an equivalence of categories, by Lemma 8.3, and hencewe have pr l , b ν K = pr l , b ν C = 0 . Thus K = C = 0 , by Lemma 8.5, and consequently Θ isan equivalence.6 ErikBackelin and KobiKremnizer Accepted for publication in Journal of the EMS