Singular modules for affine Lie algebras, and applications to irregular WZNW conformal blocks
aa r X i v : . [ m a t h . QA ] D ec SINGULAR MODULES FOR AFFINE LIE ALGEBRAS, ANDAPPLICATIONS TO IRREGULAR WZNW CONFORMAL BLOCKS
GIOVANNI FELDER AND GABRIELE REMBADOA bstract . We give a mathematical definition of irregular conformal blocks inthe genus-zero WZNW model for any simple Lie algebra, using modules foraffine Lie algebras whose parameters match up with those of moduli spaces ofirregular meromorphic connections. The Segal–Sugawara representation of theVirasoro algebra is used to show that the spaces of irregular conformal blocksassemble into a flat vector bundle over the space of tame isomonodromy times,and we provide a universal version of the resulting flat connection. Finally wegive a generalisation of the dynamical Knizhnik–Zamolodchikov connection ofFelder–Markov–Tarasov–Varchenko. C ontents Introduction and main results 2Outlook 6Layout of the paper 6Notation/conventions 71. Setup 82. Relation with (irregular) meromorphic connections 123. Bases, gradings and filtrations 134. Dual modules 185. Segal–Sugawara operators 216. Irregular conformal blocks: first version 227. Conformal blocks in terms of finite modules: first version 268. Irregular conformal blocks: second version 309. Connection on the irregular conformal blocks bundle 3210. Universal connections 3611. On conformal transformations 3912. Different dynamical term from infinity 41Acknowledgments 42Appendix A. Computations 42References 43
Key words and phrases.
Conformal field theory, irregular meromorphic connections, integrablequantum systems, flat connections. I ntroduction and main results In this paper we pursue the viewpoint that a natural mathematical formulationof conformal field theory (CFT) lies within the geometry of moduli spaces ofmeromorphic connections, and we take a step in this direction.The prototype are the Knizhnik–Zamolodchikov equations (KZ) [29], in thegenus-zero Wess–Zumino–Novikov–Witten model (WZNW) for 2-dimensionalCFT [43, 44, 34]. They were originally introduced as the partial differential equa-tions satisfied by n -point correlators, and mathematically they amount to a flatconnection on a vector bundle over the space of configurations of n -tuples ofpoints in the complex plane [15].The construction of the flat connection relies on representation-theoretic con-structions for affine Lie algebras, and on the Segal–Sugawara representation ofthe Virasoro algebra on affine-Lie-algebra modules [30]. An alternative derivationis possible via deformation quantisation of the Hamiltonian system controllingisomonodromic deformations of Fuchsian systems on the Riemann sphere [37,22], the Schlesinger system [38]. In particular the vector bundle where the KZconnection is defined comes from the quantisation of moduli spaces of meromor-phic connections with tame/regular singularities (simple poles).In this paper we develop a representation-theoretic setup for any simple finite-dimensional complex Lie algebra g , in order to go beyond the case of regularsingularities and allow for irregular/wild ones. We will thus define a family ofmodules for g and for the affine Lie algebra b g associated to g , which we call “sin-gular” modules, whose parameters match up with those of symplectic modulispaces of (possibly irregular) meromorphic connections on the sphere, generalis-ing Verma modules.Indeed the regular case will correspond to “tame” modules V λ ⊆ b V λ , whichare standard affine Verma modules for g and b g respectively, whose defining rep-resentations depend on character b + → C and b b + → C for a Borel subalgebra b + ⊆ g —corresponding to the root system for a Cartan subalgebra h ⊆ b + . Suchcharacters are encoded by linear maps λ ∈ h ∨ , which in turn match up with localnormal forms for (germs of) meromorphic connections at a simple pole via thenatural residue-pairing L g d z ⊗ L g → C , where L g = g ⊗ C (( z )) is the (formal)loop algebra of g . Moreover, if G is the connected simply-connected Lie groupintegrating g , then the G -action on the coadjoint G -orbit O ⊆ g ∨ of the charactercorresponds to a gauge action on meromorphic connections on a trivial princi-pal G -bundle, and repeating this construction at n > M ∗ dR ⊆ M dR of the de Rham space, that enters into the nonabelian Hodge cor-respondence on complex curves. The full de Rham space M dR is obtained byremoving the requirement that the bundle be holomorphically trivial (rather justtopologically trivial [6, Rem. 2.1]).Hence classically there is a complex symplectic reduction of a product of coad-joint G -orbits O i ⊆ g ∨ , the moduli space M ∗ dR = (cid:16) Q i O i (cid:17) (cid:12) G , whose quantumcounterpart is the vector space H = H g of g -coinvariants of the tensor product INGULAR MODULES AND APPLICATIONS 3 H = N i V λ i of tame modules: the space of WZNW conformal blocks . Variationsof this construction use dual modules, or integrable highest-weight modules atspecific levels; others use g -invariants instead of coinvariants, but in any case onevery important feature are deformations.Namely as the positions of the simple poles vary the moduli spaces assembleinto a symplectic fibre bundle e M ∗ dR → Conf n ( C ) over the space Conf n ( C ) ⊆ C n ofconfigurations of the noncoalescing simple poles, equipped with a flat symplectic(nonlinear) Ehresmann connection, the isomonodromy connection. Leaves of thisconnection are isomonodromic families of meromorphic connections, viz. con-nections sharing the same monodromy data, hence classically we find a flat sym-plectic fibre bundle. On the quantum side one thus looks for a (linear) flat con-nection on the conformal block bundle, to yield identifications of different fibresup to the braiding of the marked points, analogously to the symplectomorphismsdefined by the nonlinear isomonodromy connection. This natural flat connectionis precisely the KZ connection, which is intrinsically defined via the slot-wise ac-tion of the Sugawara operator L − ∈ b U (cid:0)b g (cid:1) on the tensor product b H = N i b V λ i oftame modules for the affine Lie algebra. The action is compatible with that of theLie algebra of g -valued meromorphic functions on the punctured sphere, henceinduces a well defined connection on the bundle of coinvariants, which finally isidentified with the conformal blocks bundle H × Conf n ( C ) → Conf n ( C ) .This is the picture that we wish to generalise on the side of the representationtheory of affine Lie algebras. Namely to define generalisations of Verma moduleswe look at the symplectic geometry of moduli spaces of irregular meromorphicconnections, which has been studied in much greater generality: for arbitrarygenus, complex reductive structure groups, arbitrary polar divisor, twisted irreg-ular types and resonant residues, all in intrinsic terms allowing for the definitionof symplectic local systems [6, 7, 8, 12, 14], entering the wild nonabelian Hodgecorrespondence on complex curves [5] (we concern ourselves here with the case ofgenus zero, of a simple complex group, and untwisted irregular types; see [10, 13]for terminology and motivation).Hence the open de Rham spaces M ∗ dR are still defined. Importantly one nowconsiders isomorphism classes of connections with higher-order poles, whichhave local moduli parametrising the whole of the principal part—beyond theresidue term. This may be formalised in terms of “deeper” coadjoint orbits of thedual Lie algebra g ∨ p , where g p = g J z K (cid:14) z p g J z K ≃ p − M i = g ⊗ z i ,which is a Lie algebra of truncated g -currents, holomorphic at z =
0. Indeed theresidue-pairing matches g p up with a space of meromorphic g -valued 1-forms,which we see as principal parts of (germs of) meromorphic connections on atrivial principal G -bundle at a wild singularity, and the upshot is that one stillhas the description M ∗ dR = (cid:16) Q i O i (cid:17) (cid:12) G : now however one considers coadjoint G p -orbit O i ⊆ g ∨ p , where G p = G (cid:0) C J z K (cid:14) z p C J z K (cid:1) is the group of ( p − ) -jets of The same G -conjugacy class of monodromy representation of the fundamental group of thepunctured sphere, with the poles removed G. FELDER AND G. REMBADO bundle automorphisms of the trivial principal G -bundle on a (formal) disc. Thediagonal G -action corresponds to a change of global trivialisation of the bundle,as in the tame case (see the proof of [6, Prop. 2.1]).Hence we will define modules W ( p ) χ ⊆ c W ( p ) χ (at depth p >
1) for g p and b g re-spectively, whose defining representations depend on elements of h ∨ p ⊆ g ∨ p in theform of the characters for a Lie subalgebra generalising the positive Borels, andso that for p = M ∗ dR .For example (9) has a more general scope than the ”confluent Verma modules“of [33], since we allow for an arbitrary simple Lie algebra and for arbitrary irreg-ular singularities (of arbitrary Poincaré rank). Also we do not work in Liouvilletheory, i.e. we do not consider modules for the Virasoro algebra as in [32]. Ourapproach is closer to the ”level subalgebra“ of [16], or rather to one of its ”morereasonable“ variants (see Rem. 4 of [16, p. 5]); the other variant is used in [17,§ 2.8] to define modules ” I m , χ “: in this setup the natural pairing (13) matches theparameter of the modules with half of principal parts of irregular meromorphicconnections, contrary to (9) —which is one of the motivations behind Def. 1.1(see also § 12). In addition to the parameters of the modules (9) matching up withthose of the open de Rham spaces, the other two important differences with [17]is that we work at noncritical level to define flat connections on a bundle of irreg-ular conformal blocks, and that our g p -modules are highest-weight—leading tofinite-dimensional spaces of coinvariants.The singular modules enjoy several natural generalisations of the standardproperties of tame modules, some of which we gather here. We will refer to”affine“ modules when b g is involved, and to ”finite“ modules when g p is. Theorem 1. • The singular modules admit explicit PBW-generators (Cor. 3.1 and Cor. 3.2). • The singular modules are smooth (Lem. 3.2). • The singular modules are h -semisimple (Prop. 3.1), and the finite singular mod-ules have finite-dimensional h -weight spaces (Prop. 3.2). • The finite singular modules are highest-weight g p -modules (Lem. 3.4). • The singular modules are cyclically generated by a common eigenvector for theSugawara operators { L n } n > p − (Prop. 5.1), which is an ”irregular vector oforder p − “ [21] . We also give a formula for the (finite) dimension of h -weight spaces of finitemodules, generalising the usual Weyl characters of Verma modules, in (24). Thecombinatorial complexity still lies in the positive root lattice, so in the archetypalcase of g = sl( C ) there is a simple solution (see (25)).After establishing these properties we consider tensor products of singularmodules labeled by marked points on the Riemann sphere, and study their space The viewpoint of [17] on meromorphic connections is different: at critical level κ = − h ∨ oneidentifies quotients of the ”universal Gaudin algebra“ with algebras of functions on spaces of op-ers with prescribed singularities for the Langlands dual group L G of G , with a view towards thegeometric Langlands correspondence for loop groups [19]. Recall a g J z K -module is smooth if every vector is annihilated by z N g J z K ⊆ g J z K for N ≫ INGULAR MODULES AND APPLICATIONS 5 H of coinvariants for the action of g -valued meromorphic functions with polesat the marked points. Using a generalisation of the standard filtrations/gradingsof tame modules we prove the following. Theorem 2. • The space H is canonically identified with the space of g -coinvariants for thetensor product of finite modules (Props. 7.1, 7.2 and 8.1). • The space H is finite-dimensional if one module is tame (Cor. 7.1). • The space H is nontrivial by passing to a contragradient representation at onemarked point (Prop. 8.2), or by restricting to the subalgebra of rational functionswhich vanish at an unmarked one (Rem. 7.1). Finally we consider deformations of the marked points, i.e. variations of the tame isomonodromy times. This is not the full set of isomonodromy times, asin the most general setup one may also vary the irregular types and give non-linear differential equations for the invariance of Stokes data along the deforma-tion, generalising conjugacy classes of monodromy representations (and general-ising the isomonodromy times of [28], but also going beyond the ”generic“ caseof [26]). We briefly discuss one natural setup to introduce a space of irregularisomonodromy times in § 6.4, and we plan to pursue its quantum version in fu-ture work, which should be more closely related to to [16, 17] (see the ”Outlook“below).Allowing for variations of marked points at finite distance on the sphere, weuse the Sugawara operators to define a flat connection on the space of coinvari-ants, and show that it is compatible with the action of rational functions on thepunctured sphere (with the same proof of the tame case). Hence the spaces ofcoinvariants assemble into a flat vector bundle over the space of tame isomon-odromy times, so in particular their dimension is a deformation-invariant—whenfinite.Using the above results it is possible to give descriptions of the flat connectionson the space H of coinvariants. Considering all possible cases of our setup werecover as expected:(1) the KZ connection [29] (§ 9.2.1);(2) a variation of the Cartan term of the dynamical KZ connection [18] (§ 9.2.2),and the very same Cartan term with a slightly different setup (§ 12);(3) the general case of [37] (§ 9.2.3), which generalises the KZ connection;(4) a generalisation of [37] with nontrivial action on the module associated tothe point at infinity (§ 9.2.4).In particular the semiclassical limit of the flat connections indeed yields isomon-odromy systems for irregular meromorphic connections on the sphere [37].Note the last two items in principle descend from a more general setup, wherethe point at infinity is not fixed, provided one can show how horizontal sectionstransform under the pull-back diagonal PSL ( C ) -action. Going in this direction,in § 11 we prove that horizontal sections of the bundle of coinvariants are nat-urally equivariant under the action of the subgroup of affine transformations ofthe complex plane, with the explicit transformation (59). G. FELDER AND G. REMBADO
Finally we abstract the formulæ for the reduced connections on the space ofcoinvariants in order to define a family of universal connection: these are connec-tions ∇ p on the trivial vector bundle with fibre U (g p ) ⊗ n for p >
1, over the spaceof tame isomonodromy times, which induce the above connections on H by tak-ing representations. Since all induced connections are flat and well defined on g -coinvariants, it is natural to conjecture that the same holds for the universalconnections before taking representations. Theorem 3 (Thms. 10.1 and 10.2, and Prop. 10.1) . The connection ∇ p is flat, anddescends to a connection on g -coinvariants of the tensor power U (g p ) ⊗ n . The above results show that the singular modules provide a solid mathematicalnotion of irregular conformal blocks in the genus-zero WNZW model [20, 21,45], which are central objects in the recent literature on the asymptotically freeextension of the Alday–Gaiotto–Tachikawa correspondence [1].O utlook
Two viable viewpoints to introduce variations of irregular types (i.e. variationsof ”wild“ Riemann surface structures on the sphere) are:(1) the quantisation of the full irregular isomonodromy connections, in thespirit of [7, 35], generalising the simply-laced quantum connections (whichquantise the simply-laced isomonodromy systems [11]);(2) considering quantum symmetries: the quantum/Howe duality [3] wasused in [41] to relate KZ and the ”Casimir“ connection of De Conciniand Millson–Toledano Laredo (DMT) [41, 31], and at the level of isomon-odromy systems corresponds to the Harnad duality [22]. An analogousquantisation of the Fourier–Laplace transform may be taken here in orderto turn the variations of marked points into variations of irregular types,extending the viewpoint of [9, 36].Beside the extension to the irregular isomonodromy times, another natural di-rection to pursue is the higher genus case, noting in that case the moduli spaces ofconnections on holomorphically trivial bundles have positive codimension insidethe full de Rham space.Finally one may try to introduce integrality conditions, and lift this Lie-algebrarepresentation setup to a Lie groups, with a view towards the geometric quanti-sation of coadjoint G p -orbits [39, 40].L ayout of the paper In § 1 we consider a depth p > S ( p ) ⊆ b g ,singular characters χ : S ( p ) → C , and affine/finite induced singular modules W ( p ) χ ⊆ c W χ .In § 2 we explicitly match up the data ( p , χ ) with the local moduli for theisomorphism class of (the germ of) an irregular meromorphic connection.In § 3 we introduce countable PBW-bases B W ⊆ W of the finite singular mod-ules, as well as gradings and filtrations on the finite and affine singular modules: Note the connection of [37] is given in universal terms: g p -modules and coinvariants are not dis-cussed; neither are the moduli of irregular meromorphic connections or the irregular vectors of [21]. INGULAR MODULES AND APPLICATIONS 7 notably gradings F + • and b F ±• for the degree in the variable ” z “, their associatedfiltrations, and then h -weight gradings.In § 4 we introduce left-module structures on (restricted) dual vector spaces c W ∗ ։ W ∗ .In § 5 we introduce the Sugawara operators L n for n ∈ Z , and prove that thecyclic vector w ∈ W ⊆ c W is a common eigenvector for n > p −
1. This concludesproving the properties of Thm. 1.In § 6 we define the spaces of irregular conformal blocks H . They are quotientsof tensor products H ⊆ b H of finite/affine singular modules labeled by markedpoints on the Riemann sphere with respect to the action of g -valued meromorphicfunctions (and we globalise the action introducing suitable sheaves over the spaceof tame isomonodromy times).In §§ 7 and 8 we study coinvariants, and we prove Thm. 2 using the materialof §§ 3 and 4.In § 9 we introduce the flat connection on the bundle of irregular conformalblocks, using the Sugawara operator L − and fixing the point at infinity. In § 9.2we give explicit formulæ for the reduced connection on the tensor product H ; fur-ther whenever possible we give formulæ for sections with values in the subspace H ′ ⊆ H , where the (tame) module at infinity is trivialised.In § 10 we introduce the universal connection ∇ p at depth p >
1, on the trivialvector bundle with fibre U (g p ) ⊗ n over the (restricted) space of tame isomon-odromy times, and we prove Thm. 3.In § 11 we introduce the action of Möbius transformations on horizontal sec-tions of the bundle of irregular conformal blocks, and establish equivariance un-der affine transformations.Finaly in § 12 we slightly modify the setup of § 1 to generalise the dynamicalKZ connection, i.e. [18, Eq. 3].Lengthy computations are gathered in § A.N otation / conventions Unless otherwise specified affine spaces, vector spaces, vector bundles, asso-ciative/Lie algebras and tensor products are defined over C .The end of a remark is signaled by a “ △ ”. Duals.
The (algebraic) dual of a vector space W is W ∨ = Hom ( W , C ) , and thenatural pairing W ∨ ⊗ W → C is written α ⊗ w
7→ h α , w i . If V ⊆ W is a subspace,its perpendicular is V ⊥ := (cid:10) ψ ∈ W ∨ (cid:12)(cid:12)(cid:12) h ψ , V i = ( ) (cid:11) ⊆ W ∨ . If I is a set and W = L i F i ( W ) an I -graded vector space then the restriction map W ∨ ։ F i ( W ) ∨ restricts to a canonical isomorphism T j ∈ I \{ i } F j ( W ) ⊥ → F i ( W ) ∨ , which definesan embedding F i ( W ) ∨ ⊆ W ∨ . Then the restricted/graded dual of ( W , F • ) is the I -graded vector space W ∗ := L i ∈ I F i ( W ) ∨ ⊆ Q i ∈ I F i ( W ) ∨ ≃ W ∨ . Gradings and filtrations. If ( I , ) is a totally ordered set and W = L i ∈ I F i ( W ) an I -graded vector space, the associated I -filtration on W is defined by the sub-spaces F i := L j i F j ( W ) . G. FELDER AND G. REMBADO If I and J are sets and W j = L i ∈ I F ( j ) i ( W j ) a J -family of I -graded vector spaces,the tensor product I J -grading on W = N j ∈ J W j is defined by the subspaces F i := O j ∈ J F ( j ) i ( j ) ! , for i : J → I .If further ( I , ) is a totally ordered Z -module then the tensor product I -filtrationon W is defined by the subspaces F i := M P j ∈ J i ( j ) i F i , for i ∈ I . Lie-algebraic constructions.
Let L be a Lie algebra. The abelianisation of L is theabelian Lie algebra L ab := L (cid:14)(cid:2) L , L (cid:3) , and the opposite of L is the Lie algebra L op onthe same vector space, with bracket (cid:2) X , Y (cid:3) L op := (cid:2) Y , X (cid:3) L for X , Y ∈ L .If p > z ” a variable then the associate Lie algebra of depth p is L p := L J z K (cid:14) z p L J z K ≃ L ⊗ (cid:0) C J z K (cid:14) z p C J z K (cid:1) ,coming with a projection L p ։ L = L . There is then a canonical vector spaceisomorphism L p ≃ L p − i = L ⊗ z i , which can be upgraded to an isomorphism ofLie algebras if one defines a Lie bracket on the direct sum by truncating terms ofdegree greater than p − W is a left L -module then the space of L - coinvariants is W L := W (cid:14) LW , where LW := P X ∈ L XW ⊆ W —in particular L ab is the space of ad L -coinvariants.1. S etup Let g be a finite-dimensional simple Lie algebra, and h ⊆ g a Cartan subalge-bra. Let then R + ⊆ R ⊆ h ∨ be a choice of positive roots within the root system R = R (cid:0) g , h (cid:1) , and R − := − R + the subset of negative roots. Then there is a triangu-lar decomposition g = n − ⊕ h ⊕ n + , where n ± is the maximal positive/negativenilpotent subalgebra defined by the subset of positive/negative roots: n ± := M α ∈ R ± g α , g α := (cid:10) X ∈ g (cid:12)(cid:12)(cid:12) (cid:0) ad H − α ( H ) (cid:1) X = H ∈ h (cid:11) .Equip g with the minimal nondegenerate ad g -invariant symmetric bilinearform ( · | · ) : g ⊗ g → C , and let “ z ” a variable. Consider then the (formal) loopalgebra L g = g(( z )) := g ⊗ C (( z )) , and let b g ( · | · ) = b g ≃ L g ⊕ C K be the associatedaffine Lie algebra. The Lie bracket of b g is defined by K ∈ Z( b g) and (cid:2) X ⊗ f , Y ⊗ g (cid:3) b g = (cid:2) X , Y (cid:3) g ⊗ fg + c ( X ⊗ f , Y ⊗ g ) K , for f , g ∈ C (( z )) , X , Y ∈ g , (1)where c : L g ∧ L g → C is the Lie-algebra cocycle defined by c ( X ⊗ f , Y ⊗ g ) := ( X | Y ) · Res z = ( gdf ) , (2)and where in turn Res z = (cid:0) ω (cid:1) := f − for ω = P i f i z i d z ∈ C (( z )) d z .Then there is an analogous decomposition b g = b n − ⊕ b h ⊕ b n + , where b n + := (n + ⊗ ) ⊕ z g J z K , b n − := z − g (cid:2) z − (cid:3) ⊕ (n − ⊗ ) , b h := (h ⊗ ) ⊕ C K . INGULAR MODULES AND APPLICATIONS 9
Finally let b ± := h ⊕ n ± be the positive/negative Borel subalgebras associatedto the sets of positive/negative roots, and b b ± := (b ± ⊗ ) ⊕ z g J z K ⊕ C K .Hereafter we drop the “ ⊗
1“ from the notation for vector subspaces of theconstant part g ⊆ L g , and the subscripts from the Lie brackets. Remark.
The dual Coxeter number h ∨ of the quadratic Lie algebra (cid:0) g , ( · | · ) (cid:1) ishalf of the eigenvalue for the adjoint action of the standard quadratic tensor on g [27].More precisely let ( X k ) k be a basis of g , ( X k ) k the ( · | · ) -dual basis, and define Ω := X k X k ⊗ X k ∈ g ⊗ ,i.e. intrinsically the element corresponding to Id g ∈ g ⊗ g ∨ in the duality g ∨ ≃ g induced by ( · | · ) . The projection of Ω to the universal enveloping algebra is thequadratic Casimir C = X K X k X k ∈ U (g) , (3)which is a central element—by the invariance of ( · | · ) . The adjoint action of C on g is thus a homothety, and we define h ∨ byad C X = X k (cid:2) X k , [ X k , X ] (cid:3) = h ∨ X , for X ∈ g .We will also need a generalisation of the standard quadratic tensor Ω . For m , l ∈ Z define Ω ml := X k X k z m ⊗ X k z l ∈ L g ⊗ , (4)with the shorthand notation Xz i = X ⊗ z i for X ∈ g and i ∈ Z . Then the identity [ C , X ] = P k (cid:2) X k X k , X (cid:3) =
0, valid for all X ∈ g , can be transferred inside the loopalgebra by means of the canonical vector space isomorphisms g ≃ g ⊗ z i . Thisyields X k X k z m · (cid:2) X k , X (cid:3) z l + (cid:2) X k , X (cid:3) z m · X k z l = m , l ∈ Z > . (5) △ Singular modules.
For an integer p > singular Lie subalgebra S ( p ) ⊆ b b + (of depth p ), defined by S ( p ) := b + J z K + z p g J z K ⊕ C K , (6)so that S ( ) = b b + . Lemma 1.1.
There is an identification of abelian Lie algebras S ( p ) ab ≃ h p ⊕ C K . (7) Proof.
We can define a linear surjection π : S ( p ) ։ h p ⊕ C K with kernel (cid:2) S ( p ) , S ( p ) (cid:3) = n + J z K + z p g J z K , (8) by setting p − X i = ( H i + X i ) ⊗ z i + z p f + aK p − X i = H i ⊗ z i + aK ,where f ∈ g J z K , a ∈ C , H i ∈ h , and X i ∈ n + for i ∈ {
0, . . . , p − } . (cid:3) Characters of (6) are coded by linear maps S ( p ) ab → C , i.e. by elements of h ∨ p plus the choice of a level κ ∈ C for the central element—using (7). We split thenotation: for p = λ ∈ h ∨ the linear map, and for p > ( λ , q ) ∈ h ∨ p ,where q = ( a , . . . , a p − ) ∈ (cid:0) h p (cid:14) h (cid:1) ∨ ≃ L p − i = (cid:0) h ⊗ z i (cid:1) ∨ .We will refer to χ = χ ( λ , q , κ ) : S ( p ) → C as a singular character (of depth p ),and we denote C χ the 1-dimensional left U (cid:0) S ( p ) (cid:1) -module defined by it. We alsorefer to λ as the tame part of the singular character, and to q as the wild part. Remark.
This hints to the dictionary with irregular meromorphic connections onthe Riemann sphere: λ corresponds to a semisimple residue at a simple pole (atame/regular singularity), and q to an unramified irregular type at a higher-orderpole (a wild/irregular singularity), see § 2.We will use the uniform notation λ = a when this distinction is not relevant. △ Definition 1.1 (Affine singular modules) . • The affine singular module (of depth p ) for the singular character χ is c W = c W ( p ) χ := Ind U (cid:0) b g (cid:1) U (cid:0) S ( p ) (cid:1) C χ = U (cid:0)b g (cid:1) ⊗ U (cid:0) S ( p ) (cid:1) C χ . (9) • We write b V = b V χ := c W ( ) χ , and call it the tame affine module for thecharacter χ = χ ( λ , κ ) : b b + → C .Note the latter item is the standard definition of an affine Verma module, andby definition we have level- κ modules. Now in general if L ′ ⊆ L are Lie algebras and χ : L ′ → C a character thenthe induced U ( L ) -module contains the cyclic vector [ ⊗ ] ∈ U ( L ) ⊗ U ( L ′ ) C χ .Relations are prescribed by the annihilator ideal:Ann U ( L ) (cid:0) [ ⊗ ] (cid:1) = U ( L ) (cid:2) L ′ , L ′ (cid:3) + X X ∈ e L U ( L ) (cid:0) X − χ ( X ) (cid:1) ⊆ U ( L ) ,using a splitting L ′ = (cid:2) L ′ , L ′ (cid:3) ⊕ e L to interpret L ′ ab ≃ e L as a subspace of L .In our case we can be more explicit. Denoting w = w ( p ) χ ∈ c W the cyclic vector,writing the action of U (cid:0)b g (cid:1) as a left multiplication, and using (7) and (8) yields z p g J z K w = ( ) = n + J z K w , Hz i w = h a i , Hz i i w , for H ∈ h , i ∈ {
0, . . . , p − } , (10)plus Kw = κw . This generalises the relations satisfied by the highest-weightvector in a tame module. Beware a ”regular“ Verma modules is a Verma module defined by a dominant weight λ ∈ h . Thisis why we prefer using ”tame“. INGULAR MODULES AND APPLICATIONS 11
Consider now the subspace c W − := U (cid:0) g (cid:2) z − (cid:3)(cid:1) w ⊆ c W . Because of (10) it equals c W − = U (cid:0)b n − (cid:1) w , so it is naturally a left U (cid:0)b n − (cid:1) -module with cyclic vector w —and it is canonically isomorphic to U (cid:0)b n − (cid:1) as vector space. Further matchingup cyclic vectors yields an isomorphism c W − ≃ b V of left U (cid:0) g (cid:2) z − (cid:3)(cid:1) -modules,regardless of p > q ∈ (cid:0) h p (cid:14) h (cid:1) ∨ . Note we implicitly use a C -basis of U (cid:0)b g (cid:1) as provided by the Poincaré–Birkhoff–Witt theorem (PBW) for countable-dimensional Lie algebras.Consider then the subspace W := U (cid:0) g J z K (cid:1) w ⊆ c W , which is naturally a left U (cid:0) g J z K (cid:1) -module and which will play a more important role. An inductive proofon the length of monomials—with base (10)—shows that z p g J z K W = ( ) , so the g J z K -action factorises through the finite-dimensional quotient g J z K ։ g p and wenaturally have a left U (g p ) -module. Further W = U (cid:0) n − p (cid:1) w since b + p w = C w , soin particular W ≃ U (n − p ) as vector spaces, independently of χ . Remark.
Here we use the triangular decomposition g p = n − p ⊕ h p ⊕ n + p and theinclusion b + p = n + p ⊕ h p ⊆ g p . Beware these are not Lie subalgebras of L g ,although their underlying vector spaces are naturally subspaces of L g . △ One has n + p = (cid:2) b + p , b + p (cid:3) and (cid:0) b + p (cid:1) ab ≃ h p , so by (10) there is a canonical identi-fication W ≃ Ind U (g p ) U (b + p ) C χ = U (g p ) ⊗ U (b + p ) C χ , (11)where we keep the notation χ : b + p → C for the character defined by ( λ , q ) ∈ h ∨ p —the level κ is lost. Definition 1.2 (Finite singular modules) . • We call W = W ( p ) χ ⊆ c W the finite singular module (of depth p ) for thesingular character χ . • We write V = V χ = W ( ) χ , and call it the tame finite singular module forthe character χ = χ ( λ ) : b + → C .The latter item is the standard definition of a finite Verma module. Analo-gously to the above, the finite tame module is canonically embedded as a U (g) -submodule, namely as the subspace c W − ∩ W = U (g) w ⊆ W .On the whole there there is an identification of left U (cid:0)b n − (cid:1) -modules c W ≃ U (cid:0)b n − (cid:1) ⊗ U (n − ) U (n − p ) , (12)independent of χ .1.2. Algebricity.
We used the completion g J z K ≃ lim ←− p g p of g[ z ] for the z -adictopology—induced from the Krull topology of the maximal ideal z C [ z ] ⊆ C [ z ] .However by Lem. 3.2 the structure of c W as left-module is controlled by algebraicelements, not by arbitrary formal power series.More precisely define L g alg = g (cid:2) z ± (cid:3) := g ⊗ C (cid:2) z ± (cid:3) ⊆ L g and b g alg ։ L g alg using the restriction of the cocycle (2). These are the algebraic loop algebra andthe algebraic affine Lie algebra of g , respectively. Replacing “ g J z K ” by “ g[ z ] ” in (9)then yields left U (cid:0)b g alg (cid:1) -modules, temporarily denoted c W alg , generated by a cyclicvector w alg ∈ c W alg . On the other hand the modules c W are left U (cid:0)b g alg (cid:1) -modules via the inclusion U (cid:0)b g alg (cid:1) ֒ → U (cid:0)b g (cid:1) , and composing with the canonical projection U (cid:0)b g (cid:1) ։ c W ≃ U (cid:0)b g (cid:1)(cid:14) Ann U (cid:0) b g (cid:1) ( w ) yields a linear map ι : U (cid:0)b g alg (cid:1) → c W . Lemma 1.2.
The map ι induces an isomorphism c W alg ≃ c W of left U (cid:0)b g alg (cid:1) -modules.Proof. By (10) the map ι is surjective, since c W is generated by the cyclic vectorover U (cid:0) L g alg (cid:1) . Its kernel isKer ( ι ) = Ann U (cid:0) b g (cid:1) ( w ) ∩ U (cid:0)b g alg (cid:1) = Ann U (cid:0) b g alg (cid:1) ( w alg ) . (cid:3) Hence the action of meromorphic g -valued functions on the singular modulesis given by Laurent polynomials only. We will drop the subscript “alg” from thenotation when we think of the affine singular modules over L g alg .2. R elation with ( irregular ) meromorphic connections There are canonical vector space isomorphisms (cid:0) g ⊗ z i (cid:1) ∨ ≃ g ⊗ z −( i + ) d z , for i ∈ Z . They are induced from the nondegenerate L G -invariant residue-pairing L g d z × L g −→ C , ( X ⊗ ω , Y ⊗ g ) ( X | Y ) · Res z = ( gω ) , (13)where L g d z := g ⊗ C (( z )) d z , G is a connected simply-connected (simple) Liegroup with Lie algebra g , and L G the associated loop group.Thus after fixing a level κ ∈ C the families of singular modules (9) and (11) areboth naturally parametrised by elements A = d Q + Λ d zz ∈ z − h (cid:2) z − (cid:3) d z . (14)Namely the residue term Λz − d z ∈ h ⊗ z − d z corresponds to the tame part λ ∈ h ∨ of a singular character, and the irregular type Q = p − X i = A i z i ∈ h(( z )) (cid:14) h J z K , with A i ∈ h for all i ,is such that d ( A i z − i ) = − iA i z − i − d z ∈ h ⊗ z − i − d z corresponds to the wildpart a i ∈ (cid:0) h ⊗ z i (cid:1) ∨ . The meromorphic 1-forms (14) should be thought of asprincipal parts of germs of meromorphic connections at a point on a Riemannsurface (with semisimple formal residue and untwisted irregular type; here weare considering “very good” orbits in the terminology of [13]).As mentioned in the introduction, the crucial facts are:(1) g p = Lie ( G p ) , where G p := G (cid:0) C J z K (cid:14) z p C J z K (cid:1) is the group of ( p − ) -jetsof bundle automorphisms for the trivial principal G -bundle on a (formal)disc; INGULAR MODULES AND APPLICATIONS 13 (2) the level-zero complex symplectic reduction for the diagonal G -action—on products of coadjoint G p -orbits—yields a description of an open deRham space M ∗ dR , viz. a moduli spaces of isomorphism classes of irregular meromorphic connections on a holomorphically trivial principal bundleover the Riemann sphere (with prescribed positions of poles and irregulartypes [8, § 5]; see [6] for G = GL m ( C ) ).Moreover the diagonal G -action will correspond to taking g -coinvariants forthe tensor product of finite singular modules, generalising the tame case (see §§ 7and 8). Remark . Consider the subgroup B p ⊆ G p ofelements with constant term 1. Then G acts on B p by conjugation, and there arenatural identification G p ≃ G ⋉ B p and g p ≃ g ⋉ b p , where b p = Lie ( B p ) . Thisyields a vector space decomposition g p ≃ g ∨ ⊕ b ∨ p by setting b p := Lie ( B p ) : inthe duality (13) the former summand corresponds to principal parts with zeroirregular type, and the latter to irregular types with zero residue (so in particular q ∈ b ∨ p ). △
3. B ases , gradings and filtrations Denote Π ⊆ R + the set of simple roots identified by the set of positive roots,and fix orders Π = ( θ , . . . , θ r ) and R + = ( α , . . . , α s ) for the set of simple andpositive roots—so rk (g) = r = | Π | (cid:12)(cid:12) R + (cid:12)(cid:12) = s , and we may assume θ i = α i for i ∈ {
1, . . . , r } . Let then ( F α ) α ∈ R + and ( E α ) α ∈ R + be bases of n − and n + with ( F α , E α ) ∈ g − α ⊕ g α , and such that ( F α , H α := [ E α , F α ] , E α ) is an sl -triplet. (Wemay at times write E − α := F α for the sake of a uniform notation.)In particular ( H θ ) θ ∈ Π is a basis of h , and we get a Cartan–Weyl basis of g (withgiven order): ( X , . . . , X s + r ) := ( F α , . . . , F α s , H θ , . . . , H θ r , E α , . . . , E α s ) . (15)For a multi-index n ∈ Z s + r > define X n := X n · · · X n s + r s + r ∈ U (g) .By the PBW theorem these monomials provide a C -basis of U (g) .3.1. PBW-bases of singular modules.
Let β = ( β i ) i > be a sequence of non-negative integers with finite support, and consider another sequence with valuesin the index set of the Cartan–Weyl basis (15), i.e. k = ( k i ) i > ∈ {
1, . . . , r + s } Z > .Then define X k z β := Y i ∈ β − (cid:0) Z > (cid:1) X k i z β i ∈ U (cid:0) L g alg (cid:1) . Lemma 3.1 (PBW-basis of algebraic affine enveloping algebras) . A C -basis of U (cid:0) L g alg (cid:1) is given by B := (cid:10) X k ′ z − β ′ · X n · X k z β (cid:11) k ′ , β ′ , n , k , β , (16) Beware to distinguish the positive/negative deeper Borel subalgebra b ± p from the Birkhoff subal-gebra b p . where β ′ is nonincreasing, β is nondecreasing, and k ′ j k ′ j + (resp. k j k j + ) if β j = β j + (resp. β ′ j = β ′ j + ). This is one statement of the PBW theorem for the countable-dimensional Liealgebra L g alg = g ⊗ C (cid:2) z ± (cid:3) —we have monomials over a totally ordered basis. Corollary 3.1 (PBW-basis of affine singular modules) . A C -basis of the affine singular module c W can be extracted from B c W := (cid:10) X k ′ z − β ′ · X n · X k z β w (cid:11) k ′ , β ′ , n , k , β , (17) where β ′ , k ′ , n , k , and β are as above.Proof. The family generates over C since U (cid:0) L g alg (cid:1) w = c W , and using Lem. 3.1. (cid:3) Remark.
In (17) one may take β bounded above by p −
1, as z p g J z K w = ( ) . △ Using this set of generators we can prove smoothness.
Lemma 3.2.
The singular modules are smooth.Proof.
This is clear in the finite case, as z p g J z K W = ( ) .In the affine case choose X ∈ g and an element b w = X k ′ z − β ′ X n X k z β w of (17).Then the vanishing Xz N b w = N > p + X i > β ′ i ∈ Z > ,and the conclusion follows since (17) is a set of generators. (cid:3) Lemma 3.3 (PBW-basis of depth- p finite enveloping algebras) . A C -basis of U (cid:0) g p (cid:1) is given by B := (cid:10) X n · X k z β (cid:11) n , k , β , (18) where n , k and β are as above, with the condition of Rem. 3.1. Moreover restricting to X i , X k j ∈ n − for i ∈ {
1, . . . , 2 s + r } and j > yields a C -basis of U (n − p ) . This is one statement of the PBW theorem for the finite-dimensional Lie alge-bras g p and n − p . Corollary 3.2 (PBW-basis of finite singular modules) . A C -basis of the finite singular module W ⊆ c W is given by B W := (cid:10) X n · X k z β w (cid:11) n , k , β , (19) where all conditions of Lem. 3.3 apply.Proof. The family generates since W = U (n − p ) w , and using Lem. 3.3 (the generat-ing part). But U (n − p ) has trivial intersection with the annihilator of w , hence thefamily is free by Lem. 3.3 (the linear independence part). (cid:3) INGULAR MODULES AND APPLICATIONS 15
Gradings for z-degree.
We first define two positive Z -gradings on c W . Definition 3.1.
Choose k ∈ Z . Then: • the subspace b F − k = b F − k (cid:0)c W (cid:1) ⊆ c W is the C -span of the vectors of (17) with P i β ′ i = k ; • the subspace b F + k = b F + k (cid:0)c W (cid:1) ⊆ c W is the C -span of the vectors of (17) with P i β i = k .By definition one has b F − = W , b F + = c W − , and g ⊗ z − i (cid:0)b F − k (cid:1) = b F − k + i , for i > (cid:0)c W , b F − • (cid:1) is a Z -graded g (cid:2) z − (cid:3) -module, where g (cid:2) z − (cid:3) is a Z -gradedLie algebra with grading defined by deg (g ⊗ z − i ) = i .For the other grading instead one does not find a graded module. To obtain itone may induce a positive grading on W ⊆ c W . Definition 3.2.
For k ∈ Z set F + k := b F + k ∩ W .It follows that F + = U (g) w ⊆ W , and n − ⊗ z i (cid:0) F + k (cid:1) ⊆ F + k + i , for k , i > (cid:0) W , F + • (cid:1) is a Z -graded n − J z K -module, where n − q z K is a Z -gradedLie algebra with grading defined by deg (n ⊗ z i ) = i .3.3. Filtrations.
We consider the filtration b F − • on c W associated to the grading ofDef. 3.1 for the negative z -degree. It follows from (20) that b F − k + = X m + l = k g ⊗ z − m − (cid:0)b F − l (cid:1) , g ⊗ z i (cid:0)b F − k (cid:1) ⊆ b F − k , (22)for k , i > U (g) w = U (n − ) w ⊆ W the natural filtration E • inducedfrom that of U (n − ) , so that E = C w . Note n − (cid:0) E k (cid:1) + E k = E k + . (23)3.4. Weight gradings.
For µ ∈ h ∨ define as customary b F µ (cid:0)c W (cid:1) = b F µ := (cid:10) b w ∈ c W (cid:12)(cid:12)(cid:12) H b w = µ ( H ) b w for H ∈ h (cid:11) ⊆ c W ,and analogously F µ ( W ) = F µ := W ∩ b F µ ⊆ W . Proposition 3.1.
The singular modules are h -semisimple, i.e. c W = M µ ∈ h ∨ b F µ , W = M µ ∈ h ∨ F µ . Proof.
This follows from the fact that all elements of (17) and (19) are h -weightvectors, which in turn is proven recursively using the identities H · X α z i b w = h µ + α , H i X α z i b w , H · H ′ z i b w = h µ , H i · H ′ z i b w ,for α ∈ R , H , H ′ ∈ h , i ∈ Z and b w ∈ b F µ . (cid:3) Remark.
In the finite case one may define the h p -weight spaces, i.e. the subspacesof vectors b w ∈ W such that Hz i b w = h µ i , Hz i i b w for µ = ( µ , . . . , µ p − ) ∈ h ∨ p .However the very first recursion fails for p >
2: if H ∈ h is such that h α , H i 6 = Hz · X − α w = h a , Hz i X − α w − h α , H i X − α z · w C ( X − α w ) ,where w is the cyclic vector, so the finite singular modules are not h p -semisimple. △ The proof of Prop. 3.1 implies all weights are contained inside λ + Q ⊆ h ∨ ,where Q := Z R is the root lattice. Moreover since the families (17), (19) generatethe singular modules one has E α z i b F µ ⊆ L g α b F µ = b F µ + α , E α z i F µ ⊆ g α J z K F µ = F µ + α ,for α ∈ R and i ∈ Z —with i > Remark.
Consider the z -linear extension of the adjoint action h → gl(g) on L g .Decomposing L g = L α ∈ R L g α ⊕ L h we see L g is naturally a h ∨ -graded Liealgebra (with nontrivial weights still given by R ∪ { } ), and the proof of Prop. 3.1shows the singular modules are h ∨ -graded. △ In the finite case one can go further recovering the standard notion of positivity.Namely (cid:0) h ∨ , (cid:22) (cid:1) is a poset by defining µ ′ (cid:22) µ by µ − µ ′ ∈ Q + , where Q + ⊆ Q isthe positive root lattice, defined by Q + := Z > R + . Lemma 3.4.
One has F λ = C w and W = L µ (cid:22) λ F µ .Proof. It follows from the fact that W is generated over U (n − p ) by a h p -weightvector annihilated by n + p : it is a highest-weight g p -module. (cid:3) In particular (19) consists of weight vectors, and the line C w ⊆ W has thehighest weight.In view of Lem. 3.4 the weight spaces are naturally parametrised by elements ν ∈ Q + , via F ν := F λ − ν . Now for an element ν ∈ h ∨ denoteMult R + ( ν ) := m = ( m α ) α ∈ Z R + > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X α ∈ R + m α · α = ν ⊆ Z R + > ,so that the cardinality of Mult R + ( ν ) is the finite number of ways of expressing ν as a Z > -linear combination of positive roots. In particular Mult R + ( ) = { } ,and Mult R + ( ν ) = ∅ for ν Q + .Finally for m ∈ Z R + > denoteWComp p ( m ) := ϕ = ( ϕ α ) α (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ϕ α : {
0, . . . , p − } → Z > , p − X i = ϕ α ( i ) = m α ,which is the finite set of weak p -compositions of the integers m α > In partic-ular WComp ( m ) is a singleton containing the element ϕ with ϕ α ( ) = m α forall α ∈ R + . A composition of m α is a sequence of positive integers summing to m α ; it is a p -composition ifthe sequence has finite length p >
1; and it is weak if zero is allowed.
INGULAR MODULES AND APPLICATIONS 17
Proposition 3.2.
For ν ∈ h ∨ one has dim (cid:0) F ν (cid:1) = X m ∈ Mult R + ( ν ) (cid:18) m + p − m (cid:19) < ∞ , (24) where (cid:0) m + p − m (cid:1) := Q α ∈ R + (cid:0) m α + p − m α (cid:1) .Proof. Choose µ ∈ h ∨ and set ν = λ − µ . Then for m ∈ Mult R + ( ν ) and for ϕ ∈ WComp p ( m ) consider the vector w ϕ := p − Y i = Y α ∈ R + (cid:0) X − α z i (cid:1) ϕ α ( i ) w ! ∈ B W .The family { w ϕ } ϕ is free since it consists of distinct elements extracted from (19)(beware of the ordering in the product), and by construction X ϕ ∈ F µ .Hence we have an injective map ϕ w ϕ : a m ∈ Mult R + ( ν ) WComp p ( m ) −→ W ( ν ) .Conversely the vectors X ϕ exhaust (19), so the dimension is equal to the cardi-nality of the disjoint union since a basis of F ν can be extracted from (19).The conclusion follows from standard combinatorial identities. (cid:3) Thus Prop. 3.2 strengthen Lem. 3.4: the given sum is empty for ν Q + ,and WComp p ( ) is a singleton containing the element ϕ with ϕ α ( i ) = i ∈ {
0, . . . , p − } .As expected (24) generalises the standard fact that dim (cid:0) F ν (cid:1) = (cid:12)(cid:12) Mult R + ( ν ) (cid:12)(cid:12) forVerma modules, i.e. it generalises the character of Verma modules. The differencein the general case is that one must also specify a z -degree for each occurrence ofa positive root. Remark.
This notion of positivity is lost with the (finite) modules of § 12: inparticular they have infinite -dimensional weight spaces and are less suited to yieldirregular versions of irregular conformal blocks. △ For example consider the case where ν = θ ∈ Π is a simple root. One hasMult R + ( θ ) = (cid:10) m θ (cid:11) , with m θα := δ θ , α . Also WComp p ( m θ ) = (cid:10) ϕ θ , i (cid:11) i , where ϕ θ , iα ( j ) = δ α , θ δ ij for i , j ∈ {
0, . . . , p − } . Hence X ϕ θ , i = X − θ z i , so we recoverdim (cid:0) F θ (cid:1) = p with F θ = span C (cid:10) X − θ w , . . . , X − θ z p − w (cid:11) . Remark.
It follows from the above that U (cid:0) n + J z K (cid:1) F ν = M (cid:22) ν ′ (cid:22) ν F ν ′ , for ν ∈ Q + .Hence the module W is locally n + J z K -finite, i.e. the vector spaces U (cid:0) n + J z K (cid:1) b w ⊆ W are finite-dimensional for all b w ∈ W .One is tempted to say that W lies in a “Bernstein–Gelfand–Gelfand category O J z K ” [25]—of h -semisimple finitely generated left U (cid:0) g J z K ) -modules which arelocally n + J z K -finite. △ Archetypal case.
One may get to the end of this story when g = sl( C ) withthe standard basis ( F , H , E ) and the standard A -root system R = { ± α } , where α is positive and h α , H i =
2. Then Q + = Z > α , so simply Mult R + ( ν ) = { m } forelements ν = mα with m ∈ Z > .Thus (24) reduces todim (cid:0) F mα (cid:1) = (cid:12)(cid:12)(cid:12) WComp p ( m ) (cid:12)(cid:12)(cid:12) = (cid:18) m + p − m (cid:19) . (25)In the tame case one recovers the line generated by F m v , whereas in the generalcase a basis is given by w ϕ = p − Y i = (cid:0) Fz i (cid:1) ϕ ( i ) · v , for ϕ ∈ WComp p ( m ) . (26)4. D ual modules In view of Prop. 3.1 we consider the restricted duals of the h ∨ -graded singularmodules, i.e. the h ∨ -graded vector spaces c W ∗ := M µ ∈ h ∨ b F ∨ µ ⊆ c W ∨ , W ∗ := M µ ∈ h ∨ F ∨ µ ⊆ W ∨ . (27)They are naturally equipped with a right U (cid:0)b g (cid:1) - and U (cid:0) g p (cid:1) -module structure (re-spectively), namely D b ψXz i , b w E = D b ψ , Xz i b w E , b ψK = κ b ψ , for i ∈ Z , X ∈ g , b ψ ∈ c W ∗ , b w ∈ c W ,and analogously in the finite case.To get a left action compose with a Lie algebra morphism b g → b g op (resp. g p → g op p ) , or rather with the induced ring morphism U (cid:0)b g (cid:1) → U (cid:0)b g op (cid:1) = U (cid:0)b g (cid:1) op (resp. U (g p ) → U (g p ) op ). In particular a Lie algebra morphism θ : g → g op hasa unique Z -graded extension b θ : L g → L (cid:0) g op (cid:1) = (cid:0) L g (cid:1) op : in the finite case onecan then consider the restriction b θ : g J z K → g J z K op (which is compatible with theprojections g J z K ։ g p and g J z K op ։ g op p ); in the affine case one may further askthat θ is ( · | · ) -orthogonal—and extend the definition by b θ ( K ) := − K .In what follows we only consider morphisms of this type. Definition 4.1 (Dual singular modules) . The affine (resp. finite) θ -dual singular module c W ∗ θ (resp. W ∗ θ ) is the left U (cid:0)b g (cid:1) -module (resp. U (g p ) -module) defined by the morphism θ : g → g op .The U (cid:0) g J z K ) -linear inclusion map W ֒ → c W then dually correspond to U (cid:0) g J z K ) -linear restriction maps c W ∗ θ ։ W ∗ θ . Remark.
Basic examples of morphisms θ : g → g op preserving ( · | · ) are the tauto-logical θ = − Id g , and the transposition θ , defined by θ ( E α ) = E − α , θ (cid:12)(cid:12) h = Id h , for α ∈ R .We refer to θ -duals simply as dual modules, and to θ -duals as contragradient modules. △ INGULAR MODULES AND APPLICATIONS 19
Consider then the element ψ ∈ W ∗ dual to the cyclic vector in the basis (19),i.e. h ψ , w i = ψ vanishes on all other vectors of (19)—whence F ∨ λ = C ψ .Assume hereafter that θ (h) = h op (up to conjugating θ by an inner auto-morphism of g ), and canonically identify h ≃ h op and their duals. Then wehave a well defined pull-back map θ ∗ ∈ GL (h ∨ ) , which we extend z -linearly to (cid:0) h ⊗ z i (cid:1) ∨ ≃ h ∨ ⊗ z i . Moreover by orthogonality the subspace n + ⊕ n − ⊆ g is θ -stable. Lemma 4.1.
The vector ψ ∈ W ∗ satifies the relations z p g J z K ψ = ( ) = θ − (n − ) J z K , Hz i ψ = h θ ∗ a i , Hz i i ψ , for H ∈ h , i ∈ {
0, . . . , p − } . (28) Proof.
Use (10), (21), z p g J z K W = ( ) , and the fact that b θ : g J z K → g J z K op preservesthe z -grading of Def. 3.2. (cid:3) In particular n − J z K ψ = ( ) in the dual case, and n + J z K ψ = ( ) in the contragra-dient case.4.1. Dual weight grading.
Denote θ ∗ := (cid:0) θ ∗ (cid:1) − = (cid:0) θ − (cid:1) ∗ , and introduce thenotation b F ∗ µ ⊆ c W ∗ θ and F ∗ µ ⊆ W ∗ θ for the h -weight spaces. Lemma 4.2.
One has b F ∨ µ = b F ∗ θ ∗ µ and E α z i b F ∨ µ ⊆ b F ∨ µ + θ ∗ α , for µ ∈ h ∨ , α ∈ R , i ∈ Z ,and analogously in the finite case—restricting to i ∈ Z > .Proof. Let b I ν : c W → c W be the idempotent for the direct summand b F µ ⊆ c W ,viz. the endomorphism such that b I µ (cid:12)(cid:12) c W ( µ ′ ) = δ µ , µ ′ Id c W ( µ ′ ) . Then by definition b ψ ∈ b F ∨ µ means b ψ = b ψ ◦ b I µ , and by construction θ ( H ) b I µ = b I µ θ ( H ) = h θ ∗ µ , H i b I µ ∈ End (cid:0)c W (cid:1) , for µ ∈ h ∨ , H ∈ h .Hence for b w ∈ c W one has D H b ψ , b w E = D b ψ , b I µ (cid:0) θ ( H ) b w (cid:1)E = h θ ∗ µ , H i D b ψ , b w E ,whence the inclusion b F ∨ µ ⊆ b F ∗ θ ∗ µ , and the equality follows from (27).The latter inclusion follows from θ ( E α ) z i b F µ ⊆ b F µ − θ ∗ α for α ∈ R , which is astraightforward computation using (1).The same pair of arguments applies verbatim to the finite case. (cid:3) Hence (27) is the h -weight decomposition of θ -dual singular modules, and theweights are contained inside θ ∗ ( λ + Q ) ⊆ h ∨ (resp. θ ∗ ( λ + Q + ) ) in the affine(resp. finite) case. By Lem. 3.4 we conclude that ψ ∈ W ∗ θ is a lowest -weightvector of lowest weight θ ∗ λ = − λ , whereas ψ ∈ W ∗ θ is a highest-weight vector ofhighest weight θ ∗ λ = λ .In particular in the contragradient case matching the cyclic vector with itsdual yields a canonical morphism Φ : W → W ∗ θ , hence a generalisation of theShapovalov form S : W ⊗ W −→ C , b w ⊗ b w ′ Φ (cid:0) b w (cid:1) , b w ′ i .This may be degenerate, particularly since the image of the canonical morphismis the submodule W ′ θ := U (cid:0) g J z K (cid:1) ψ ⊆ W ∗ θ , which in general is a proper submodule (e.g. in the finite dual tame case for g = sl( C ) and λ = ψ to generate the θ -dual module. To give anecessary condition consider the vector b w = E − α z p − w ∈ F λ − α , for α ∈ R + . ByLem. 4.2 a linear form b ψ ∈ W ′ θ that vanishes on B W \ { b w } must lie in the span of (cid:10) θ − ( E α ) ψ , . . . , θ − ( E α ) z p − ψ (cid:11) ⊆ F ∨ λ − α , so consider a generic element b ψ = b ψ ( b , . . . , b p − ) = p − X j = b j θ − ( E α ) z j ψ , with b j ∈ C .Using z p g J z K W = ( ) = n + J z K w and h ψ , w i = D b ψ , b w E = b p − h a p − , H α z p − i ,so we need the highest irregular part of the singular part to be regular (cf. § 6).Conversely we have the following. Proposition 4.1.
One has W ′ θ = W ∗ θ for parameters ( λ , q ) in a (classically) densesubspace of the affine space h ∨ p .Proof. Clearly F ∨ λ ⊆ W ′ θ , and then we reason recursively on the h ∨ -weight spacedecomposition of W .Choose b w ∈ B W ∩ F µ , and consider the vectors b w α ( k ) := E − α z k b w ∈ F µ − α ,for α ∈ R + and k ∈ {
0, . . . , p − } . As b w , α and k vary, the vectors b w α ( k ) exhaust B W ∩ F µ − α , so we must find coefficients b ij ∈ C such that D b ψ α ( i ) , b w α ( k ) E = δ ik ,where b ψ α ( i ) = p − X j = b ij θ − ( E α ) z j b ψ ∈ F ∨ µ − α , for i ∈ {
0, . . . , p − } ,and where b ψ ∈ F ∨ µ is the dual of b w —lying in W ′ θ by the recursive hypothesis.Now one has D b ψ α ( i ) , b w α ( k ) E = p − X j = b ij D b ψ , E α z j E − α z k b w E ,hence the given condition means BM = Id C p , where B and M are the p -by- p matrices with coefficients B ij = b ij and M jk = D b ψ , E α z j E − α z k b w E , respectively(the latter selects the component of E α z j E − α z k b w ∈ F µ along the line C b w , in thebasis (19)). A solution exists if and only if det ( M ) = M = M ( b w , α ) is a degree- p polynomial whose coeffi-cients depend polynomially on ( λ , q ) , hence it amounts to a polynomial function h ∨ p → C . Thus W ′ θ = W ∗ θ by taking ( λ , q ) in a countable intersection of opendense subsets—which is dense in the Baire space h ∨ p . (cid:3) Finally we can choose a complementary subspace to W inside c W , and extend ψ : W → C by zero to the whole of c W —e.g. extract a PBW-basis from (17). Thenone can consider the module c W ′ θ ⊆ c W ∗ θ generated by this extension over L g , anddefine gradings/filtrations on c W ′ θ ։ W ′ θ analogously to §§ 3.2 and 3.3, using INGULAR MODULES AND APPLICATIONS 21 the generating set (16), the basis (18), and the standard filtration of U (cid:0) θ − (n + ) (cid:1) .These satisfy the analogous identities of (20)–(23).5. S egal –S ugawara operators For n ∈ Z define L n := (cid:0) κ + h ∨ (cid:1) X j ∈ Z X k : X k z − j · X k z n + j : ! , (29)where ( X k ) k and ( X k ) k are ( · | · ) -dual bases of g , κ = − h ∨ is a noncritical level,and in the normal-ordered product one puts elements of g J z K ⊆ L g to the right.The Sugawara operators (29) (due to Segal in this particular form) are well-defined elements of the completion b U (cid:0)b g (cid:1) of U (cid:0)b g (cid:1) with respect to the inversesystem of left ideals U (cid:0)b g (cid:1) z • + g J z K ։ U (cid:0)b g (cid:1) z • g J z K . If follows from Lem. 3.2 thatthere are well-defined actions of (29) on the modules W ⊆ c W .5.1. Cyclic vector as Sugawara eigenvector.
The cyclic vector w ∈ c W is a com-mon eigenvector for the Sugawara operators when n ≫
0. To get explicit formulæfor the eigenvalues we recall further euclidean properties of the Cartan–Weyl ba-sis (15).
Remark . Since (g α | g β ) = α + β =
0, and since 2 ( E α | E − α ) = ( H α | H α ) , we con-clude that E ± α = ( H α | H α ) E ∓ α . Using the pairing ( · | · ) : h ∨ ⊗ h ∨ → C inducedby the minimal-form duality h ≃ h ∨ this can be written E ± α = ( α | α ) E ∓ α . (Recallthe minimal form is defined so that ( α | α ) = highest root.)Then we replace the simple-root basis of h with a ( · | · ) -orthonormal basis,denoted ( H k ) k —i.e. we “divide” by the Cartan matrix—, and for i ∈ Z wetransfer the basis and the pairings to g ⊗ z i and (cid:0) g ⊗ z i (cid:1) ∨ ≃ g ∨ ⊗ z i using thecanonical vector space isomorphisms g ≃ g ⊗ z i . Then one has the tautologicalbasis-independent identity ( µ | µ ′ ) = r X k = h µ , H k z i ih µ ′ , H k z j i for µ ∈ h ∨ ⊗ z i , µ ′ ∈ h ∨ ⊗ z j .Finally denote as customary ρ := P α ∈ R + α ∈ h ∨ the half-sum of positiveroots. △ Proposition 5.1.
The cyclic vector w is a common eigenvector for the operators (29) with n > p − . If n > ( p − ) then L n w = , else L n w = l n w with l n := (cid:0) κ + h ∨ (cid:1) p − X j = − p + n ( a j | a n − j ) , for n ∈ (cid:8) p , . . . , 2 ( p − ) (cid:9) , (30) and l p − := (cid:0) κ + h ∨ (cid:1) p − X j = ( a j | a p − − j ) + p ( ρ | a p − ) ! . (31) Proof.
Postponed to § A.1. (cid:3)
Hence the cyclic vector is an “irregular vector of order p −
1” in the WZNWmodel—instead of the Liouville model [21].
Remark . This generalises the standard fact that L n v = n >
0, and that v isan L -eigenvector, with nonzero eigenvalue for generic values of λ ∈ h ∨ . Namelyif p = L v = ∆ λ v , ∆ λ = ( λ | λ + ρ ) ( κ + h ∨ ) ,reverting to the notation λ = a , which recovers the conformal weight correspond-ing to the action of the quadratic Casimir (3). △ Action on finite modules.
We will use the action of the operator L − on thefinite module W ⊆ c W .Using z p g J z K W = ( ) we see nonvanishing terms arise for 1 − p j p in (29).Resolving the ordered product yields L − b w = κ + h ∨ p X j = X k X k z − j X k z j − ! b w , for b w ∈ W . (32)As expected L − b w W , and it can be put back into the finite module usingthe action of the loop algebra (see § 7). Remark.
We see (32) generalises the usual formula from the tame case: L − b v = κ + h ∨ X k X k z − X k b v , for b v ∈ V . (33) △
6. I rregular conformal blocks : first version Consider the Riemann sphere Σ := C P , choose an integer n > p , . . . , p n ∈ Σ . Denote J = {
1, . . . , n } the set of labels for the points—withthe canonical order—and denote p = ( p , . . . , p n ) the ordered set of points.Let O Σ be the structure sheaf of regular functions on Σ , seen as a (smooth)complex projective curve. Then consider the stalks O j = O Σ , p j at the markedpoints, their (unique) maximal ideals M j = M p j ⊆ O j of germs of functionsvanishing at p j , the completions b O j := lim ←− n O j (cid:14) M nj , and their field of fractions b O j ֒ → c K j . Remark. If z j is a local coordinate on Σ vanishing at p j then O j ≃ C [ z j ] , M j = z j C [ z j ] , b O j ≃ C J z j K , c K j ≃ C (( z j )) . △ Then consider the loop algebras ( L g) j := g ⊗ c K j and the associated affine Liealgebras b g j ։ ( L g) j . There are canonical isomorphisms b g i ≃ b g j for i , j ∈ J , andthe subscripts distinguish the local picture at the marked points.Now for j ∈ J further choose an integer r j >
1, and set up singular modulesas in § 1. Hence consider the Lie subalgebras S ( r j ) ⊆ b g j , a common level κ ∈ C for the central elements, and singular characters χ j = χ ( λ j , q j , κ ) , where λ j ∈ h ∨ and q j = (cid:0) ( a j ) , . . . , ( a j ) r j − (cid:1) ∈ b ∨ r j (with the notation of Rem. 2.1), and further INGULAR MODULES AND APPLICATIONS 23 ( a j ) i ∈ (cid:0) h ⊗ z i (cid:1) ∨ . This yields singular modules W ( r j ) χ j =: W j ⊆ c W j := c W ( r j ) χ j , andwe consider the vector spaces b H = b H p , χ := O j ∈ J c W j , H = H p , χ := O j ∈ J W j , (34)where χ = ( χ j ) j ∈ J . Clearly H ⊆ b H , and the dependence on the choice of markedpoints is void (it becomes relevant after considering the action of g -valued mero-morphic functions in § 6.2).The spaces (34) are endowed with natural structures of left modules for theassociative algebras U (cid:0)b g (cid:1) ⊗ n ≃ N j ∈ J U (cid:0)b g j (cid:1) and N j ∈ J U (cid:0) g r j (cid:1) , respectively.Moreover for indices i = j ∈ J denote ι ( ij ) : U ( L g) ⊗ → U ( L g) ⊗ n the naturalinclusion on the i -th and j -th slot, defined on pure tensors by X ⊗ Y ⊗ i − ⊗ X ⊗ ⊗ j − i − ⊗ Y ⊗ ⊗ n − j , (35)for i < j , and analogously for i > j . Finally define ι ( ii ) : U ( L g) ⊗ → U ( L g) ⊗ n by X ⊗ Y ⊗ i − ⊗ XY ⊗ ⊗ n − i . This yields an action of quadratic loop-algebratensors on (34).6.1. Tame isomonodromy times.
We now vary part of the parameters definingthe spaces (34), namely the marked points. An admissible deformation is onewhere they do not coalesce, so marked points vary inside the configuration space C n := Conf n ( Σ ) ⊆ Σ n of ordered n -tuples of (labeled) points on Σ .The space C n is the space of tame isomonodromy times. It is a complex mani-fold of dimension n . Remark.
The terminology points again to meromorphic connections on the sphere.Namely the positions of the poles and the irregular types together controlStokes data of irregular meromorphic G -connections over the sphere. RecallStokes data generalise the conjugacy class of the monodromy representation ν : π (cid:0) Σ ◦ , b (cid:1) → G , where Σ ◦ := Σ \ (cid:8) p j (cid:9) j ∈ J is the punctured sphere with thepoles removed and b ∈ Σ ◦ a base point [7].Then one may consider admissible deformations of the connections alongwhich Stokes data do not vary, which yields by definition isomonodromic deforma-tions. This can be set up as a system of nonlinear differential equations where thepositions of the poles and the irregular types are precisely the independent vari-ables, hence they become the “times” of isomonodromic deformations: the posi-tions of the poles are the tame/regular times, and the rest are the wild/irregularones.Geometrically these differential equations constitute a nonlinear flat/integrablesymplectic connection in the local system of moduli spaces M ∗ dR of meromorphicconnections, as the marked points and the irregular types vary (i.e. as the wild Riemann surface structure on the sphere varies [12]). △ Remark . Let Σ ⊇ U z −→ C be a local affine chart on Σ —so Σ ≃ C ∪ { ∞ } .Then coordinates on the open subset C n ( U ) := Conf n ( U ) ⊆ C n are given by t : C n ( U ) → C n , where t = ( t j ) j ∈ J and t j ( p ) := z ( p j ) —so C n ( U ) ≃ Conf n ( C ) ,and C n ≃ Conf n ( C ) ∪ Conf n − ( C ) . This yields an atlas on the configurationspace. △ Now for a J -tuple χ of singular characters we consider the vector bundles b H = b H • , χ → C n and H = H • , χ → C n , whose fibres over p ∈ C n are thespaces (34), respectively. We have inclusions H ⊆ b H of vector bundles, andglobal vector bundle trivialisations: b H ≃ O J ∈ J U (cid:0)b n − (cid:1) ⊗ U (n − ) U (cid:0) n − r j (cid:1) × C n −→ C n ,by (12), and the simpler H ≃ O j ∈ J U (cid:0) n − r j (cid:1) × C n −→ C n ,by W j ≃ U (cid:0) n − r j (cid:1) . Importantly both vector space isomorphisms do not depend onthe choice of marked points (nor on the character, cf. 6.4).6.2. Action of meromorphic functions: punctual version.
Given marked points p j ∈ Σ consider the effective divisor D := P j ∈ J (cid:2) p j (cid:3) on Σ , and denote as custom-ary O ∗ D ( Σ ) = O Σ , ∗ D ( Σ ) the vector space of meromorphic functions on Σ withpoles at most on (the support of) D . Then let g ∗ D ( Σ ) := g ⊗ O ∗ D ( Σ ) be the Liealgebra of g -valued such meromorphic functions, with constant bracket comingfrom g .Taking Laurent expansions at p j yields a linear map τ j : O ∗ D ( Σ ) → c K j , andtensoring with g a linear map g ∗ D ( Σ ) → L g j ⊆ b g j . Remark. If z j is a local coordinate on Σ vanishing at p j , and f ∈ O ∗ D ( Σ ) , thenthere are coefficients f i ∈ C such that τ j ( f ) = f ( z j ) = X i > − ord pj ( f ) f i z ij ∈ C (( z j )) ,where ord p ( f ) > p ∈ Σ as a pole of f . △ Thus there is an arrow τ : g ∗ D ( Σ ) −→ End (cid:0) b H (cid:1) , τ ( X ⊗ f ) := X j ∈ J (cid:0) X ⊗ τ j ( f ) (cid:1) ( j ) . (36)Using (1), and the fact that the sum of the residues of a meromorphic 1-form on Σ vanishes, shows that (36) is a morphism of Lie algebras.Then the action τ : g ∗ D ( Σ ) → gl (cid:0) b H (cid:1) endows b H with a left g ∗ D ( Σ ) -modulestructure. Definition 6.1 (Irregular conformal blocks space, first version) . The space of irregular conformal block at the pair ( p , χ ) is the space of coinvari-ants of the g ∗ D ( Σ ) -module b H : H := b H g ∗ D = b H p , χ (cid:14) g ∗ D ( Σ ) b H p , χ . (37) Remark.
In our terminology (37) would be better called the space of singular con-formal blocks, and be irregular/wild when r j > j ∈ J . △ By (36), the fundamental identity inside the space of irregular conformal blocksis h(cid:0) X ⊗ τ i ( f ) (cid:1) ( i ) b w i = − X j ∈ J \{ i } h(cid:0) X ⊗ τ j ( f ) (cid:1) ( j ) b w i , (38) INGULAR MODULES AND APPLICATIONS 25 for i ∈ J , X ∈ g , f ∈ O ∗ D ( Σ ) and b w ∈ b H , where square brackets denote equiva-lence classes modulo g ∗ D ( Σ ) b H p , χ .6.3. Action of meromorphic functions: global version.
Now we want to glob-alise the action (36) over the space of configurations of n -tuples of points on thesphere, i.e. we want a map of sheaves of Lie algebras on C n .To define the domain sheaf consider the projection π Σ : Σ n + −→ Σ n , ( p , p , . . . , p n ) ( p , . . . , p n ) .Then set Y := π − Σ ( C n ) = (cid:8) ( p , p , . . . , p n ) (cid:12)(cid:12) p i = p j for i = j (cid:9) ⊆ Σ n + ,so that π Σ : Y ։ C n is the universal family of n -marked spheres—whose quotientby the diagonal PSL ( C ) -action on the marked points yields the Riemann modulispace M n Now for j ∈ J define the hyperplane P j := (cid:8) p = p j (cid:9) ⊆ Σ n + , consider theeffective divisor D := P j ∈ J (cid:2) Y ∩ P j (cid:3) on Y , and let O ∗ D = O Y , ∗ D be the sheaf ofmeromorphic functions on Y with poles at most along (the support of) D . Thenwe have the push-forward sheaf ( π Σ ) ∗ O ∗ D on C n , and by tensoring we obtainthe sheaf of Lie algebras g ∗ D := g ⊗ ( π Σ ) ∗ O ∗ D . Remark. If U ′ ⊆ C n is open then g ∗ D ( U ′ ) is then the Lie algebra of g -valuedmeromorphic functions on Σ × U ′ , such that the restriction to Σ × { p } ≃ Σ haspoles at most at the set (cid:8) p j (cid:9) j ∈ J for all p ∈ U ′ , as wanted. △ Now for U ′ ⊆ C n open we consider the Laurent expansion τ j ( U ′ )( f ) of func-tions f ∈ O ∗ D (cid:0) π − Σ ( U ′ ) (cid:1) along the divisor Y ∩ P j . Then tensoring with g yields amap of sheaves τ j : g ∗ D −→ O C n ⊗ L g j ⊆ O C n ⊗ b g j ,where O C n is the structure sheaf on the configuration space. Remark. If z j is a local coordinate on Σ vanishing at p j , U ′ = Conf n ( U ) for U ⊆ Σ an open affine subset, and f ∈ ( π Σ ) ∗ O ∗ D ( U ′ ) , then there are suitable functions f i : U ′ → C such that τ j ( U ′ )( f ) = f ( z j , t , . . . , t n ) = X i f i ( t , . . . , t n ) z ij ∈ O C n ( U ′ ) ⊗ C (( z j )) ,using the local coordinates ( t j ) j ∈ J on U ′ ⊆ C n of Rem. 6.1. By definition thefunctions f i may have poles on the hyperplanes (cid:8) t i = t j (cid:9) ⊆ C n . △ Finally summing the action over each slot of the tensor product we have asheaf-theoretic analogue of (36), acting on sections of b H .6.4. Irregular isomonodromy times.
One may add the other possible deforma-tions, e.g. with the following setup.Recall the regular parts of the Cartan subalgebra and its dual are the comple-ments of (co)root hyperplanes: h reg := h \ [ α ∈ R Ker ( α ) , h ∨ reg := h ∨ \ [ α ∈ R Ker (cid:0) ev H α (cid:1) ,and analogously for h ⊗ z i and its dual. Then consider irregular parts q j ∈ b ∨ r j such that the most irregular coefficient ( a j ) r j − is regular, and define an admissible deformation of as one in which themost irregular coefficient does not cross coroot hyperplanes. Remark.
This is the analogous condition as for the marked points: the open charts C n ( C ) ⊆ C n are regular parts for Cartan subalgebras of rank- n type-A simpleLie algebras. △ Doing so we get to the space of isomonodromy times B = C n × Y j ∈ J (cid:0) h ∨ r j (cid:1) reg , (39)with the notation (cid:0) h ∨ r j (cid:1) reg = r j − Y i = (cid:0) h ⊗ z i (cid:1) ∨ × (cid:0) h ⊗ z r j − (cid:1) ∨ reg .The space (39) is a complex manifold of dimension d = n + r P j ∈ J ( r j − ) ,where r = rk (g) . Ax expected it coincides with the space of tame isomonodromytimes if r j = j ∈ J . Remark.
If there is just one irregular module W j with r j = B = C n × (cid:0) h ⊗ z (cid:1) ∨ reg ,and one recovers the base space for the FMTV connection [18]—up to the canon-ical vector space isomorphism h ⊗ z ≃ h . If further the variations of markedpoints are neglected then (39) becomes the base space for the DMT/Casimir con-nection [31, 41]. △ Then in (34) one can let both p ∈ C n and χ ∈ Q j ∈ J (cid:0) h ∨ r j (cid:1) reg vary, getting avector bundle over the base space (39). This also comes with a canonical vectorbundle trivialisation, reasoning in the same way as for H ⊆ b H (namely (12) isalso independent of χ ).Finally one may extend the sheaf g ∗ D trivially along the Cartan directions.Namely the projection π C n : B ։ C n is open, so one may take the naïf pullbacksheaf: π ∗ C n g ∗ D ( U ) = g ∗ D (cid:0) π C n ( U ) (cid:1) , for U ⊆ B open .7. C onformal blocks in terms of finite modules : first version Throughout this section fix a pair ( p , χ ) to define the spaces H ⊆ b H as in (34).Compose the inclusion H ֒ → b H with the canonical projection π H : b H ։ H toobtain a map ι : H → H .To study the image of ι consider the tensor product filtration b F − • := O j ∈ J (cid:0)b F − j (cid:1) • , (40)where (cid:0)b F − j (cid:1) • is the filtration defined in § 3.3 on c W j . By definition b F − = H ,and we push (40) forward to a filtration c F − • on H , along the surjection π H .Note c F − • is exhaustive, since b FFF − • is. INGULAR MODULES AND APPLICATIONS 27
Proposition 7.1.
The map ι is surjective,Proof. We will show that c F − k lies in the image of ι by induction on k >
0. Thebase is given by c F − = π H (cid:0) H (cid:1) .Now we use (38) for a function f i ∈ O ∗ D ( Σ ) with a pole at p i , and only there.Such a function is e.g. defined by f i ( z ) = ( z − t i ) − m , with the notations ofRem. 6.1, working in a local chart containing p .Hence τ j (cid:0) f i (cid:1) ∈ b O j for j = i , and if b w ∈ b F − k the rightmost identity of (22)shows that the right-hand side of (38) lies in c F − k . Then by induction the image of ι contains c F − k and all the vectors on the left-hand side of (38), and the conclusionfollows from the leftmost identity of (22). (cid:3) Proposition 7.2.
One has
Ker ( ι ) = g H ⊆ H .Proof. Consider an element b w = τ ( X ⊗ f ) b u with b u = N j ∈ J b u j ∈ b H , f ∈ g ∗ D ( Σ ) and X ∈ g .If the function f is noncostant then it has a pole, say at p j ∈ Σ . It follows that τ j ( f ) b O j , whence X ⊗ τ j ( f ) ( j ) b u j (cid:0)b F + j (cid:1) by (20), and b w H = b F + .Thus to have element of the kernel we must restrict to f ∈ C . Then using (20)again we see that X ⊗ f = X ⊗ τ j ( f ) ∈ g preserves the grading (cid:0)b F − j (cid:1) • on c W j , so ( X ⊗ f ) b u j ∈ W j = (cid:0)b F − j (cid:1) implies b u j ∈ W j .Conversely g H ⊆ g ∗ D ( Σ ) b H ∩ H lies in the kernel. (cid:3) Hence there is an identification H ≃ H g = H (cid:14) g H , generalising the analogousstandard fact for the tame case. Remark.
The moduli space of meromorphic connections—with given positions ofpoles and irregular types—is the quotient of a product of deeper orbits by theaction of the constant subgroup of gauge transformations, so this also matchesup. △ To go further one may appeal to the tensor product of the weight grading of§ 3.4, which is a (cid:0) h ∨ (cid:1) J -grading on H . Namely we consider the subspaces F µ = F µ ( H ) := O j ∈ J F µ j ( W j ) ⊆ H , for µ = ( µ j ) j ∈ J ∈ (cid:0) h ∨ (cid:1) J .By (36) the subspace F µ lies inside the weight space of weight | µ | := P j µ j ∈ h ∨ for the tensor product h -action.If | µ | = F µ ⊆ h F µ is annihilated by π H , so we still have a surjectivemap H ⊇ M | µ | = F µ π H −−−→ H , (41)and the h -action is now trivialised. Remark.
The condition | µ | = Σ —in the duality (13). △ Auxiliary tame module.
Suppose one of the modules is tame, e.g. the lastone: r n = W n = V n . Then split the notation H = H ′ ⊗ V n , H ′ := O j ∈ J ′ W j ,where J ′ := J \ { n } , and embed H ′ ֒ → H , O j ∈ J ′ b w j O j ∈ J ′ b w j ⊗ v n ,where v n ∈ V n is the cyclic/highest-weight vector. (That is, identify H ′ with H ′ ⊗ C v n ⊆ H .) Proposition 7.3.
One has ι ( H ′ ) = H .Proof. Denote ( E j ) • the filtration on U (g) w j ⊆ W j defined in § 3.3. We willprove by induction on k > ι (cid:0) H ′ (cid:1) contains the classes of all vectors inside H ′ ⊗ ( E n ) k . The base follows from the identity ( E n ) = C v n .For the induction we use the constant version of of (38). For X ∈ g this showsthat the class of X ( n ) b w lies in ι (cid:0) H ′ (cid:1) as soon as that of b w ∈ H ′ ⊗ ( E n ) k does,which is precisely the inductive hypothesis. The conclusion follows from (23). (cid:3) Now C v n = F λ n ( V n ) , so using (41) we still have a surjection: H ′ ⊇ M | µ | =− λ n F ′ µ π H −−−→ H , where µ ∈ (cid:0) h ∨ (cid:1) J ′ ,where F ′ µ ⊆ H ′ is the tensor product of the weight-gradings over J ′ ⊆ J . Fur-ther the subalgebra n − ⊆ g has nonzero action on the associated graded of thetensor product filtration E • = N j ∈ J ( E j ) • , so the residual n + -action yields thecoinvariants.7.1.1. On dimensions.
To go further we use the results of § 3.4; in particular weemploy the notation F ν ( W j ) := F λ j − ν ⊆ W j for ν ∈ Q + —i.e. we parametrise theweights µ j = λ j − ν (cid:22) λ j by ν ∈ Q + .By definition H ′ (− λ n ) is the direct sum of the spaces F ′ ν ⊆ H ′ such that0 = λ n + P j ∈ J ′ ( λ j − ν j ) , so only elements such that | ν | = | λ | ∈ h ∨ will contribute,where | ν | = P j ∈ J ′ ν j . Hence the weight space only depends on the sum of thetame parts of the singular characters, and we ought to change notation: H ′ | λ | := M | ν | = | λ | F ′ ν ⊆ H ′ . (42) Proposition 7.4.
The weight space H ′ | λ | ⊆ H ′ has dimension: dim (cid:0) H ′ | λ | (cid:1) = X | ν | = | λ | Y j ∈ J ′ X m ∈ Mult R + ( ν j ) (cid:18) m + r j − m (cid:19)! < ∞ . (43)We deduce the following. Corollary 7.1.
If one module is tame then the space of irregular conformal blocks ofDef. 6.1 is finite-dimensional for all choices of marked points and singular characters.
INGULAR MODULES AND APPLICATIONS 29
Proof of Prop. 7.4.
It follows from (24), taking the products of the dimensions ofthe weight spaces F ν j ⊆ W .The dimension is finite since for ν ∈ Q + there are finitely many J ′ -tuples ν ∈ (cid:0) Q + (cid:1) J ′ such that | ν | = ν —analogously to (cid:12)(cid:12) Mult R + ( ν ) (cid:12)(cid:12) < ∞ . (cid:3) In particular the weight space is trivial if | λ | Q + , and the simplest nontrivialcase is when | λ | =
0. Then | ν | = ν j = j ∈ J ′ , so H ′ is the linegenerated by the tensor product N i ∈ J ′ w i of the cyclic vectors w i ∈ W i .The next nontrivial example is when | λ | = θ ∈ Π is a simple root. Now | ν | = θ implies ν ∈ (cid:10) ν θ , i (cid:11) i , with ν θ , ij = δ ij θ for i , j ∈ J ′ . Then one finds the singletonMult R + (cid:0) ν θ , ij (cid:1) = (cid:10) δ ij m θ (cid:11) , with m θα = δ αθ . On the whole (43) reduces todim (cid:0) H ′ θ (cid:1) = X i ∈ J ′ Y j ∈ j ′ (cid:18) δ ij m θ + r j − δ ij m θ (cid:19)! = X i ∈ J ′ (cid:18) m θ + r i − m θ (cid:19) = X i ∈ J ′ r i ,independently of the choice of simple root.A basis is given by the pure tensors b w ji := i − O k = w k ⊗ F θ z j w i ⊗ n − O k = i + w k ,for i ∈ J ′ and j ∈ {
0, . . . , r i − } . Remark . One way to ensure coinvariants are nontrivial is the following: for agiven configurations of points p = ( p j ) j ∈ J consider the Lie subalgebra of g -valuedmeromorphic functions with poles at p j , and further with a zero elsewhere, say at p ′ ∈ Σ \ (cid:8) p j (cid:9) j . Then the proof of Prop. 7.1 can easily be adapted working in thechart where p ′ = ∞ —as the function f i ( z ) = ( z − t i ) − m vanishes at infinity.Then there is still a surjection of H on the space of coinvariants, and similarlyto Prop. 7.2 one needs constant functions to get elements of the kernel. Hence thekernel is trivial, and in this setup H = ( ) itself is the space of coinvariants.Another way to ensure nontriviality is to put a θ -dual module in the tensorproduct (introduced in § 4). Further when it is tame then one still has a finite-dimensional space, see § 8. △ Archetypal case.
Consider the same setup of § 3.4.1 for g = sl( C ) . In thiscase | λ | = mα for an integer m > Proposition 7.5.
One has dim (cid:0) H ′ mα (cid:1) = (cid:18) m + R − m (cid:19) , where R := X j ∈ J ′ r j . (44) A basis is provided by the pure tensors b w Φ = O j ∈ J ′ r j − Y i = (cid:0) Fz i (cid:1) Φ ( i , j ) w j ! , where Φ ∈ WComp R ( m ) —identifying {
1, . . . , R } ≃ ` j ∈ J ′ (cid:8)
0, . . . , r j − (cid:9) . Proof.
Fix an integer m > ν ∈ ( Z > α ) J ′ satisfying | ν | = mα . Suchelements are given by weak J ′ -compositions of m , i.e. functions φ : J ′ → Z > satisfying P j ∈ J ′ φ ( j ) = m , with bijection φ ν φ , ν φj := φ ( j ) α .Then by definition Mult R + (cid:0) ν φj (cid:1) = (cid:8) φ ( j ) (cid:9) for j ∈ J ′ , so we need only give ele-ments ϕ j ∈ WComp r j (cid:0) φ ( j ) (cid:1) to allocate the z -degrees of the occurrences of − α ateach slot of the tensor product.The data of φ and ϕ = ( ϕ j ) j is equivalent to that of the weak R -composition Φ : R → Z > defined by Φ ( i , j ) = ϕ j ( i ) , and the result follows. (cid:3) Remark.
In the tame case (44) simplifies todim (cid:0) H ′ mα (cid:1) = (cid:12)(cid:12)(cid:12) WComp J ′ ( m ) (cid:12)(cid:12)(cid:12) = (cid:18) m + (cid:12)(cid:12) J ′ (cid:12)(cid:12) − m (cid:19) ,and a basis is given by the pure tensors b v φ = O j ∈ J ′ F φ ( j ) v j for φ ∈ WComp J ′ ( m ) .This is somehow the opposite of (26): there we had an arbitrary singular module,here we have a tensor product of arbitrarily many tame modules. △
8. I rregular conformal blocks : second version We now vary the setup of § 6 giving a special role to one of the marked points(e.g. the last one): choose a ( · | · ) -orthogonal morphism θ : g → g op and at the lastmarked point put a θ -dual module c W ′ θ ։ W ′ θ , defined in § 4.Hence in this case b H = c W ′ n ⊗ O j ∈ J ′ c W j , H = W ′ n ⊗ O j ∈ J ′ W j , (45)where J ′ = J \ { n } as in § 7.1—and omitting the subscript θ . These are naturallysubspaces of Hom ( c W n , b H ′ ) and Hom ( W n , H ′ ) , respectively, where H ′ is as in§ 7.1 and b H ′ := N j ∈ J ′ c W j . Moreover they still assemble into trivial vector bundles b H ։ H over the space C n = Conf n ( Σ ) —but also over the full space (39) ofisomonodromy times.The Lie algebra of g -valued meromorphic functions on Σ acts on the leftmosttensor product of (45). Thinking in terms of linear maps b ψ : c W n → b H ′ , andusing (36) and the dual actions of § 4, one has the formula D τ ( X ⊗ f ) b ψ , b w E = X j ∈ J ′ (cid:0) X ⊗ τ j ( f ) (cid:1) ( j ) D b ψ , ( θ ( X ) ⊗ τ n ( f )) b w E ∈ b H ′ ,where X ∈ g , f ∈ O ∗ D ( Σ ) and b w ∈ c W n . Taking coinvariants of the resulting leftmodule yields a second version of the space of irregular conformal blocks, stilldenoted H . Moreover the material of § 6 goes through, and there is an action ofthe sheaf of Lie algebras g ∗ D on sections of b H and H . INGULAR MODULES AND APPLICATIONS 31
On coinvariants.
Consider first the natural inclusion ι : c W ′ n ⊗ H ′ ֒ → b H ,which can be composed with the canonical projection π H : b H → H .Reasoning as in Prop. 7.1 (which may be thought of as the case c W ′ n = C )shows this composition is surjective. Then reasoning as in Prop. 7.2 shows thekernel is obtained from the action of meromorphic functions with no poles at (cid:8) p , . . . , p n − (cid:9) ⊆ Σ , but only (at most) at the point p n . Hence there is a vectorspace isomorphism H ≃ c W ′ n ⊗ H ′ (cid:14) g ∗ p n ( Σ ) (cid:0)c W ′ n ⊗ H ′ (cid:1) ,thinking of p n ∈ Σ as a divisor.Now a function with a pole at most at p n is either constant, or its Laurentexpansion at p n lies in z − n g (cid:2) z − n (cid:3) ⊆ ( L g) n , where as usual z n is a local coordinateon Σ vanishing at p n . Hence a coinvariant function is uniquely determined byits restriction to W n ⊆ c W n , and since now poles are not allowed we get thefollowing. Proposition 8.1.
There is a canonical vector space identification H ≃ W ′ n ⊗ H ′ (cid:14) g( W ′ n ⊗ H ′ ) .Thus in this case as well we can reduce the discussion to g -coinvariants for thetensor product of finite modules.Now suppose the dual module is tame, and adapt the discussion of § 7.1. Wesee there is a surjective map H ′ → H , where H ′ = N j ∈ J ′ W j —embedded in H via b w ψ ⊗ b w , where ψ ∈ V ′ θ is the cyclic vector. Now the θ − (n − ) -actioncannot give coinvariant elements, so we are left with the action of h ⊕ θ − (n + ) .In the dual case where θ = θ = − Id g we have θ − (n + ) = n + , so we are es-sentially back to § 7.1.1. The contravariant case where θ = θ (the transposition)instead allows to go further. In this case θ − (n + ) = n − , whence a new identifi-cation H ≃ H ′ b − , and to trivialise the h -action we consider the zero-weight sub-space inside H ′ . This is again (42), whose (finite) dimension is given in Prop. 7.4.Finally in this setup we can recover nontriviality, as follows. Recall that weattach weights λ = ( λ j ) j ∈ J ⊆ (cid:0) h (cid:1) ∨ to the marked points, and that we considerthe sum | λ | = P j ∈ J λ j ∈ h ∨ . Proposition 8.2.
The space of coinvariants is nontrivial for | λ | ∈ Q + —and for anychoice of wild parts. A fortiori then nontriviality holds if the last module is not tame.
Proof.
For i ∈ J ′ and b w ∈ F | λ | ( W i ) consider the pure tensor b w i = i − O k = w k ⊗ b w ⊗ n − O k = i + w k ∈ H ′ | λ | .By (23) one has b w i n − H ′ , so (cid:2) b w i (cid:3) = H . (cid:3) Remark . One may also consider the tensor products of the grading of Def. 3.2,in addition to the h -weight grading—i.e. use the fact that every finite module is a graded n − J z K -module. Namely there is a decomposition H ′ = M k ∈ Z J ′ F + k , where F + k = O j ∈ J ′ F + k j ( W j ) ,which is preserved by the tensor product b − -action, so H ≃ L k ∈ Z J ′ ( F + k ) b − .This is a new feature, as in the tame case the grading in positive z -degree istrivial—concentrated at zero. △
9. C onnection on the irregular conformal blocks bundle
Consider a particular case of the setup of § 6: mark n + Σ , vary the first n > fix singular characters at those points.Thus we work on a closed subspace of Conf n + ( Σ ) , which is naturally identi-fied with the local chart U ′ = Conf n ( U ) ⊆ C n of Rem. 6.1 where p n + = ∞ —whence { p , . . . , p n } ⊆ U ≃ C . The label set becomes J = {
1, . . . , n , ∞ } , and wewrite J ′ := J \ { ∞ } .Then we have two versions of spaces of irregular conformal blocks: eitherwe put a singular module at infinity, or a θ -dual. In any case we consider therestrictions of the vector bundles H ⊆ b H over U ′ ≃ C n ( C ) := Conf n ( C ) , as wellas for the sheaves ( π Σ ) ∗ O ∗ D and g ∗ D on U ′ —and keep the same notation forthem.Then we want to define a connection b ∇ on b H → C n ( C ) which is compatiblewith the action of the sheaf of Lie algebras g ∗ D . In the given trivialisation thiswill be of the form b ∇ = d − b ̟ , where b ̟ is a 1-form on C n ( C ) with values inendomorphisms of the fibres, and with a view towards a generalisation of theKnizhnik–Zamolodchikov connection (KZ) [29] we set h b ̟ , ∂ t i i := L ( i )− , for i ∈ J ′ ,where we use the coordinates t : C n ( C ) → C n of Rem. 6.1 and the Sugawaraoperator (29). This is a translation-invariant 1-form on the parallelisable manifold C n ( C ) , so in particular d b ̟ =
0. Further the actions of L − on different slotscommute, so (cid:2) b ̟ ∧ b ̟ (cid:3) =
0, and the connection b ∇ is (strongly) flat.9.1. Compatibility with the action of meromorphic functions.
We now considera natural connection D on the sheaf g ∗ D —a linear map D : g ∗ D → Ω C n ( C ) ⊗ g ∗ D satisfying Leibnitz’s rule. Namely we set D ( X ⊗ f ) := X ⊗ d f ,where d : Ω C n ( C ) → Ω C n ( C ) is the standard de Rham differential. (This is aconnection because of the identity g ( X ⊗ f ) = X ⊗ fg .) Proposition 9.1.
One has b ∇ (cid:0) τ ( X ⊗ f ) b w (cid:1) = τ (cid:0) D ( X ⊗ f ) (cid:1) b w + τ ( X ⊗ f ) b ∇ b w , (46) where X ∈ g , and f and b w are local sections of ( π Σ ) ∗ O ∗ D and b H , respectively. To prove this we use the following well-known fact.
Lemma 9.1 ([27], Lem. 12.8) . One has (cid:2) L − , Xz m (cid:3) = − mXz m − , for X ∈ g and m ∈ Z . INGULAR MODULES AND APPLICATIONS 33
This identity holds inside b U (cid:0)b g (cid:1) . Proof of Proposition 9.1.
For i ∈ J ′ and for local sections b w and X ⊗ f of b H and g ∗ D —respectively—we must prove that ∂ t i (cid:0) τ ( X ⊗ f ) b w (cid:1) − (cid:2) L ( i )− , τ ( X ⊗ f ) (cid:3) b w = τ (cid:0) X ⊗ ∂ t i f (cid:1) b w + τ ( X ⊗ f ) ∂ t i b w .Now for j ∈ J ′ we have the expansions τ j ( f ) = X k f k ( t , . . . , t n ) z kj ,where f k is a regular function on an open subset of C n ( C ) , and we take the localcoordinate z j = z − t j on Σ —vanishing at p j . Since ∂ t i ( z j ) + δ ij = ∂ t i (cid:0) τ j ( f ) (cid:1) = τ j ( ∂ t i f ) + δ ij (cid:2) L − , τ j ( f ) (cid:3) ,using Lem. 9.1. Hence by (36): ∂ t i (cid:0) τ ( X ⊗ f ) b w (cid:1) = τ (cid:0) X ⊗ ∂ t i f (cid:1) b w + (cid:2) L − , X ⊗ τ i ( f ) (cid:3) ( i ) b w + τ ( X ⊗ f )( ∂ t i b w ) ,and we conclude with (cid:2) L ( i )− , τ ( X ⊗ f ) (cid:3) = (cid:2) L ( i )− , (cid:0) X ⊗ τ i ( f ) (cid:1) ( i ) (cid:3) = (cid:2) L − , X ⊗ τ i ( f ) (cid:3) ( i ) . (cid:3) Thus a reduced connection is well defined on H → C n ( C ) , since b ∇ preservesthe sheaf of sections with values in the subspaces g ∗ D b H p , χ ⊆ b H p , χ , by (46).We conclude the sheaf of irregular conformal blocks has a natural structure offlat vector bundle over the space of tame isomonodromy times. It follows that thedimension of the spaces of irregular conformal blocks is constant along variationsof the marked points—when finite.9.2. Description on finite modules: first version.
By the results of § 7 it is pos-sible to describe the reduction of b ∇ as the g -reduction of a connection ∇ livingon the vector sub-bundle H ⊆ b H , and further as a connections acting on H ′ ⊆ H when the module at infinity is tame.The goal is to find an explicit expression for ∇ . For this we will use the follow-ing elementary fact, where we further set z ∞ := z − —a local coordinate vanishingat infinity. Lemma 9.2 (Expansions at irregular singularities) . For i ∈ J ′ and for an integer m > one has τ j (cid:0) z − mi (cid:1) = P l > (cid:0) m + l − l (cid:1) z lj ( t i − t j ) l ( t j − t i ) m , j ∈ J \ { i } , P l > (cid:0) m + l − l (cid:1) t li z m + l ∞ , j = ∞ . (47)As expected τ j (cid:0) z − mi (cid:1) ∈ C J z j K for i , j ∈ J ′ with i = j , and a pole at a finite pointgives a zero at infinity—of the same order. Tame case.
Suppose r j = j ∈ J . Then using (47) with m = X ⊗ τ j (cid:0) z − i (cid:1)b v j = Xt j − t i b v j , X ⊗ τ ∞ (cid:0) z − i (cid:1)b v ∞ = X ∈ g , i = j ∈ J ′ , b v j ∈ V j and b v ∞ ∈ V ∞ —since z j g J z j K V j = ( ) for j ∈ J . Henceby (38) one has the following identity inside H —with tacit use of π H : (cid:0) X ⊗ z − i (cid:1) ( i ) b v ⊗ b v ∞ = X j ∈ J ′ \{ i } X ( j ) t i − t j b v ⊗ b v ∞ ,where b v = N j ∈ J ′ b v j ∈ H , and linearly extended to non-pure tensors. In particularthe action is trivial on the tame module at infinity.Looking at (33) and writing L ( i )− ( b v ⊗ b v ∞ ) = b v i ⊗ b v ∞ we find b v i = κ + h ∨ X j ∈ J ′ \{ i } X k ( X k ) ( i ) X ( j ) k t i − t j !b v = κ + h ∨ X j ∈ J ′ \{ i } Ω ( ij ) t i − t j b v , (48)writing Ω ( ij ) := ι ( ij ) ( Ω ) for the embedding (35) of the quadratic tensor (4)—with m = l = H ′ | λ | ֒ → H , taking V ∞ as auxiliary tame module.9.2.2. Tame modules in the finite part.
Now allow r ∞ > X ⊗ τ ∞ ( z − i ) b w ∞ = r ∞ − X l = t li Xz l + ∞ · b w ∞ ,for X ∈ g , b w ∞ ∈ W ∞ and i ∈ J ′ , using the case m = r ∞ > H —with tacit use of π H : (cid:0) X ⊗ z − i (cid:1) ( i ) b v ⊗ b w ∞ = X j ∈ J ′ \{ i } X ( j ) t i − t j − r ∞ − X l = t li (cid:0) Xz l + (cid:1) ( ∞ ) b v ⊗ b w ∞ .Thus looking at (33) one finds L ( i )− ( b v ⊗ b w ∞ ) = b v i ⊗ w ∞ + D i ( b v ⊗ b w ) , where b v i isas in (48), and D i ( b v ⊗ b w ∞ ) = κ + h ∨ r ∞ − X l = t li Ω ( i ∞ ) l + ( b v ⊗ b w ∞ ) ,using again the embedding ι ( i ∞ ) ( Ω l + ) of (4) defined by (35). Remark.
E.g. if r ∞ = D i ( b v ⊗ b w ∞ ) = Ω ( i ∞ ) b v ⊗ b w ∞ κ + h ∨ . (49)In this case the reduced connection is close to the dynamical KZ connection,i.e. [18, Eq. 3]. We will recover the very same “dynamical” Cartan term in § 12. △ INGULAR MODULES AND APPLICATIONS 35
Tame module at infinity.
Suppose symmetrically r ∞ =
1, but r j is arbitraryfor j ∈ J ′ . Using the general case of (47) yields X ⊗ τ j ( z − mi ) b w j = r j − X l = (cid:18) m + l − l (cid:19) Xz lj b w j ( t i − t j ) l ( t j − t i ) m , X ⊗ τ ∞ ( z − mi ) b v ∞ = X ∈ g , i = j ∈ J ′ , b w j ∈ W j and b v ∞ ∈ V ∞ —since z ∞ g J z ∞ K V ∞ = = z r j j g J z j K V j .Hence by (38) one has the identity (cid:0) X ⊗ z − mi (cid:1) ( i ) b w ⊗ b v ∞ = b w i , m , X ⊗ b v ∞ inside H , where b w i , m , X = − X j ∈ J ′ \{ i } r j − X l = (cid:18) m + l − l (cid:19) (cid:0) Xz l (cid:1) ( j ) b w ( t i − t j ) l ( t j − t i ) m ! .Then looking at (32) we find L ( i )− b w ⊗ b v ∞ = b w i ⊗ b v ∞ , with b w i = − κ + h ∨ X j ∈ J ′ \{ i } r i − X m = r j − X l = (cid:18) m + ll (cid:19) Ω ( ij ) ml b w ( t i − t j ) l ( t j − t i ) m + ! . (50)One recovers the “generalised KZ connection” [37] acting on H ′ | λ | ֒ → H , taking V ∞ as auxiliary tame module. Remark.
The flat connection (50) is known to admit an isomonodromy systemas semiclassical limit [37]: precisely the irregular isomonodromy system on C P for variations of the positions of the poles (the tame isomonodromy times, asconsidered in [28]).This generalises the same fact from the tame case: the quantisation of theSchlesinger system [38] yields the KZ connection [37, 23]. △ General case.
Finally take r ∞ > X ⊗ τ ∞ ( z − mi ) b w ∞ = r ∞ − m − X l = (cid:18) m + l − l (cid:19) t li Xz m + l ∞ · b w ∞ ,for X ∈ g , b w ∞ ∈ W ∞ and i ∈ J ′ , using the general case of (47). So the action isnontrivial at infinity for r ∞ > m + L ( i )− b w ⊗ b w ∞ = b w i ⊗ b w ∞ + D i ( b w ⊗ b w ∞ ) , with b w i as in (50) and D i ( b w ⊗ b w ∞ ) = κ + h ∨ r i − X m = r ∞ − m − X l = (cid:18) m + ll (cid:19) t li Ω ( i ∞ ) m , m + l + ( b w ⊗ b w ∞ ) . (51)9.3. Description on finite modules: second version.
Finally one may considerthe setup of § 8, i.e. put a θ -dual module W ′ θ at infinity. In the analogue of§§ 9.2.1 and 9.2.3—when the module at infinity is tame—the description of thereduced connection does not change, using (28). In the remaining cases one findsthe action of the same quadratic tensors on the last slot, acting on the θ -dual.Hence in the next section we will introduce a universal versions of the reducedconnection, looking at (50) and (51), to treat the two versions on the same footing.
10. U niversal connections
Fix again a depth p >
1, an integer n >
1, and the finite ordered sets {
1, . . . , n } = J ′ ⊆ J = {
1, . . . , n , ∞ } .Consider then the nonautomous (quantum) Hamiltonian systems b H i = b H ( p ) i : C n ( C ) → U (cid:0) g p (cid:1) ⊗ | J | ,with Hamiltonians b H i = b H ′ i + b H ′′ i for i ∈ J ′ , where b H ′ i ( t ) := − κ + h ∨ X j ∈ J ′ \{ i } p − X m , l = Ω ( ij ) ml (cid:18) m + ll (cid:19) (− ) m ( t i − t j ) − − m − l ! , (52)and b H ′′ i ( t ) := κ + h ∨ p − X m , l = Ω ( i ∞ ) m , m + l + (cid:18) m + ll (cid:19) t li , (53)as suggested by (50) and (51).These Hamiltonians are equivalent to the universal connection (at depth p ): ∇ p = d − ̟ p , ̟ p = ̟ ′ p + ̟ ′′ p := X j ∈ J ′ (cid:0) b H ′ i + b H ′′ i (cid:1) d t i , (54)defined on the trivial vector bundle U ( J , p ) := C n ( C ) × U (cid:0) g p (cid:1) ⊗ | J | → C n ( C ) bymeans of the U (g p ) ⊗ | J | -valued 1-forms ̟ ′ p and ̟ ′′ p on the base space. This gen-eralises [37] with a nontrivial action at infinity.Then for every choice of singular modules labeled by J there is an action of (54)on H for p ≫
0, which reproduces the most general case of § 9.2 (with θ -duals ornot), so in particular there are induced integrable quantum Hamiltonian systems.Hence one expects (54) to be flat before taking representations, as we will show. Remark.
One directly checks that ∂ b H ′ j ∂t i − ∂ b H ′ i ∂t j = ∂ b H ′′ j ∂t i = δ ij , for i , j ∈ J ′ ,so (strong) flatness is equivalent to the commutativity of the quantum Hamilto-nians. △ Flatness at finite distance.
The 1-form defining the Hamiltonians (52) canbe written ̟ ′ p = κ + h ∨ X i = j ∈ J ′ r ( ij ) p ( t i − t j ) d ( t i − t j ) ,where r p : C \ { } → g ⊗ p is the following rational function: r p ( t ) := − p − X m , l = Ω ml ⊗ (− ) m (cid:18) m + ll (cid:19) t − − m − l . (55) Remark.
It is easy to see that r p is skew-symmetric, meaning r ( ij ) p ( t ) + r ( ji ) p (− t ) = t ∈ C \ { } , i , j ∈ J ′ . (56) △ INGULAR MODULES AND APPLICATIONS 37
The study of the connection ∇ ′ p := d − ̟ ′ p is closely related to the theory of theclassical Yang–Baxter equation (CYBE) [4]. In particular flatness (for (cid:12)(cid:12) J ′ (cid:12)(cid:12) >
3) isequivalent to the CYBE for (55) in the Lie algebra g p , i.e. to the following identityinside g ⊗ p : (cid:2) r ( ) p ( t ) , r ( ) p ( t ) (cid:3) + (cid:2) r ( ) p ( t ) , r ( ) p ( t ) (cid:3) + (cid:2) r ( ) p ( t ) , r ( ) p ( t ) (cid:3) = t ij := t i − t j . Theorem 10.1 ([37]) . The rational function (55) is a solution of the CYBE.Proof.
We will reduce the proof to the well-known case p =
1, where g p = g . Inthis case we have the classical result that the rational function r ( t ) = Ωt − is askew-symmetric solution of the CYBE [4], which is an easy consequence of theDrinfeld–Kohno relations (cid:2) Ω ( ij ) , Ω ( ik ) + Ω ( jk ) (cid:3) =
0, and the Arnold relations [2]:1 t ij t jk + t jk t ki + t ki t ij = g ⊗ p ≃ g ⊗ ⊗ A ( p ) ,where A ( n , p ) := C J w , . . . , w n K (cid:14) I p is the quotient of the power-series ring bythe ideal I p = (cid:0) w p , . . . , w pn (cid:1) generated by (cid:10) w p , . . . , w pn (cid:11) . In this identification Ω ml = Ω ⊗ w m w l , and (55) can be written r p ( t ) = Ω ⊗ τ ( p )( ) ( f t ) ∈ g ⊗ p , where f t ( w i , w j ) := t + w i − w j ,and where τ ( p )( ) ( f t ) is the class mod I p of the Taylor expansion of f t at the origin.Then, up to the identification g ⊗ p ≃ g ⊗ ⊗ A ( p ) , the CYBE follows again from(57), with t i replaced by t i − w i , for i ∈ {
1, 2, 3 } . (cid:3) Hence we have an inverse system of classical r -matrices, with respect to thecanonical projections g J z K (cid:14) z • + g J z K ։ g J z K (cid:14) z • g J z K , corresponding to an inversesystem of flat vector bundles (cid:0) U ( n , p ) , ∇ ′ p (cid:1) over the space of configurations of J ′ -tuples of points in the complex plane. The inverse limit of the vector bundlesis naturally identified with the trivial vector bundle with fibre U (cid:0) g J z K (cid:1) b ⊗ | J | , thecompletion of the n -th tensor power of the positive part of the loop algebra. Remark.
The inverse limit r ∞ ( t ) = lim ←− p r p ( t ) ∈ g ⊗ [ t − ] J z , z K is a solution of theCYBE in a completion of g J z K ⊗ ⊗ O C ( C ) (cid:0) C ( C ) (cid:1) .Analogously on the representation-theoretic side one may consider charactersof the Lie subalgebra S ( ∞ ) := \ p > S ( p ) = b + J z K ⊕ C K ⊆ b g ,using (6). Then S ( ∞ ) ab ≃ h J z K ⊕ C K , so the induced non-smooth modules c W ( ∞ ) depend on infinitely many Cartan parameters (and a level κ ), and are generatedover U ( L n − ) by a cyclic vector annihilated by n + J z K . Under (13) the parame-ters of these modules correspond to principal parts of connections with essential singularities. △ Flatness overall.
The 1-form defining the Hamiltonians (53) can be written ̟ ′′ p = κ + h ∨ X i ∈ J ′ s ( i ∞ ) p ( t i ) d t i ,where s p : C \ { } → g ⊗ p is the following rational function: s p ( t ) := p − X m , l = Ω m , m + l + ⊗ (cid:18) m + ll (cid:19) t l . Theorem 10.2.
The universal connection ∇ p is flat for p > .Proof. Reasoning as in the proof of Thm. 10.1 consider the function g t ( w i , w j ) := w j − w j ( t + w i ) .Then one directly checks that the Taylor expansion of g t at the origin satisfies s p ( t ) = Ω ⊗ τ ( p )( ) ( g t ) ,and we can conclude by proving a version of the CYBE in the Lie algebra g p .Namely by Thm. 10.1 the commutator of two Hamiltonians becomes (cid:2) b H i , b H j (cid:3) = (cid:2) r ( ij ) p ( t ij ) , s ( i ∞ ) p ( t i ) (cid:3) + (cid:2) r ( ij ) p ( t ij ) , s ( j ∞ ) p ( t j ) (cid:3) + (cid:2) s ( i ∞ ) p ( t i ) , s ( j ∞ ) p ( t j ) (cid:3) ,using the fact that actions on disjoint pairs of slots commute, and the skew-symmetry (56). Now we can use the standard Drinfeld–Kohno relations to reduceflatness (for all p >
1) to a variation of the Arnold relations (57), namely to thefollowing identity: g t i ( w i , w ∞ ) g t j ( w j , w ∞ ) + f t ij ( w i , w j ) (cid:0) g t i ( w i , w ∞ ) − g t j ( w j , w ∞ ) (cid:1) = f t = f t ( w i , w j ) is as in the proof of Thm. 10.1. (cid:3) Remark . One can give a more symmetric expression of (54), with no specialrole for the marked point at infinity.To this end consider the generating function ϕ ( w i , w j ) := w i − w j , (58)which is a meromorphic function on C with poles along (cid:8) w i = w j (cid:9) ⊆ C —and only there. It can be extended (by zero) to a meromorphic function on thecomplex surface Σ \ (cid:8) ( ∞ , ∞ ) (cid:9) , so we can take Taylor expansions τ ( p i , p j ) ( ϕ ) of ϕ at any pair of distinct points p i , p j ∈ Σ —using the local coordinates w − i and w − j at infinity.Then analogously to the above one checks that τ ( p )( p i , p j ) ( ϕ ) = r p ( t ij ) , τ ( p )( p i , ∞ ) ( ϕ ) = s p ( t i ) ,for points p i , p j ∈ Σ at finite distance of coordinates t i , t j ∈ C , respectively.Hence ( κ + h ∨ ) ̟ p = X i = j ∈ J τ ( p )( p i , p j ) ( ϕ ) d t ij + X i ∈ J ′ τ ( p )( p i , ∞ ) ( ϕ ) d t i ,and all marked points are treated the same. INGULAR MODULES AND APPLICATIONS 39
Then the flatness of (54) for p > generalised Arnold relations,relating the Taylor expansions of (58) at pairs extracted from a triple of distinctpoints on the Riemann sphere. △ Hence we find again an inverse system of flat vector bundles (cid:0) U ( J , p ) , ∇ p (cid:1) ,over the space of configurations of J ′ -tuples of points in the complex plane.10.3. Connection on coinvariants.
The universal connection (54) is well definedfor sections with values in the space of g -coinvariants of U (g p ) ⊗ n .To prove this consider the canonical embedding g ֒ → g p ≃ g ⋉ b p and theuniversal embedding g p ֒ → U (g p ) . Composing them we let g act on U (g p ) inthe regular representation, and there is an induced action on the tensor product(analogous of (36) in the case of constant functions). Then we get a g -action ondifferential forms with values in the flat vector bundle (cid:0) U ( J , p ) , ∇ p (cid:1) . Proposition 10.1.
The g -action is flat for all p > . Note this is a particular case of a compatibility such as (46), for constant sec-tions of the trivial bundle C n ( C ) × g → C n ( C ) , equipped with the trivial connec-tion. Proof.
Postponed to § A.2. (cid:3)
It follows that (54) preserves sections with values in g U (g p ) ⊗ | J | ⊆ U (g p ) ⊗ | J | ,so a reduced (flat) connection is well defined on the space of g -coinvariants of thetensor product. This was to be expected, as it holds for the induced connectionon every tensor product of a J -tuple of singular modules.11. O n conformal transformations Consider the action of Möbius transformations on Σ = P ( C ) , that is g . (cid:2) t : t (cid:3) = (cid:2) at + bt : ct + dt (cid:3) ,for ( t , t ) ∈ C \ { } , with g = g ( a , b , c , d ) given by numbers a , b , c , d ∈ C suchthat ad − bc =
1. In the standard affine chart U = Σ \ { [ : ] } t −→ C we then havethe subgroup of affine transformation of the complex plane, with diagonal actionon C n ( C ) ⊆ C n , and with induced pull-back (right) action on sections of vectorbundles over that base.In particular translations t t + b correspond to a = d = c = E ∈ Lie (cid:0)
PSL ( C ) (cid:1) = sl( C ) , and the associated infinitesimal action reads d (cid:0) c w ◦ γ (cid:1) ( ε ) d ε (cid:12)(cid:12)(cid:12) ε = = d c w ( t + ε ) d ε (cid:12)(cid:12)(cid:12) ε = = X i ∈ J ′ ∂ b w ∂t i ,considering the path γ : ε g ( ε , 0, 1 ) .Analogously dilations correspond to the 1-parameter subgroup generated by H ∈ sl( C ) , and the associated infinitesimal action is given by the Euler vectorfield d (cid:0) c w ◦ γ (cid:1) ( ε ) d ε (cid:12)(cid:12)(cid:12) ε = = d c w (cid:0) ( + ε ) t (cid:1) d ε (cid:12)(cid:12)(cid:12) ε = = X i ∈ J ′ t i ∂ b w ∂t i ,considering the path γ : ε g (cid:0) + ε , 0, 0, ( + ε ) − (cid:1) . These two infinitesimal actions on b ∇ -horizontal sections of the trivial vectorbundle b H × C n ( C ) → C n ( C ) then become b w X i ∈ J ′ L ( i )− b w , and b w X i ∈ J ′ t i L ( i )− b w ,respectively. Proposition 11.1.
Suppose the module at infinity is tame. Then the action of affinetransformations on horizontal sections of the irregular conformal blocks bundle reads b w ( t ′ ) = Y i ∈ J ′ exp (cid:0) aL ( i ) (cid:1) · b w ( t ) , (59) where t ′ = ( t ′ i ) i ∈ J ′ with t ′ i = e a t i + b . In particular horizontal sections are invariantunder translations.Proof. Indeed if b w is a ∇ ′ p -horizontal section of U ( J , p ) → C n ( C ) then E b w = X i ∈ J ′ X j ∈ J ′ \{ i } r ( ij ) p ( t ij ) ! b w ,which vanishes by the skew-symmetry (56), and which implies the statementabout translations after taking g p -modules.As for dilations, in the universal case of a ∇ ′ p -horizontal section one finds H b w = X i ∈ J ′ X j ∈ J ′ \{ i } t i r ( ij ) p ( t ij ) ! b w = X i = j ∈ J ′ t ij r ( ij ) p ( t ij ) b w ,and we must consider the induced action on finite singular modules. Now onecomputes L b w = κ + h ∨ p X j = X k X k z − j X k z j ! b w , for b w ∈ W ,analogously to (32), using a ( · | · ) -orthonormal basis ( x k ) k of g . Then reasoningas in § 9.2.3 the induced slot-wise action on coinvariants is L ( i ) b w = − κ + h ∨ X j ∈ J ′ \{ i } r i − X m = r j − X l = (cid:18) m + ll (cid:19) (− ) m t − m − lij Ω ( ij ) ml ! b w ,with tacit use of the projection π H : H ′ | λ | → H , and on the whole H b w = (cid:0) L − L ( ∞ ) (cid:1) b w = X i ∈ J ′ L ( i ) b w ,by (55). This is the action of an endomorphism on the finite-dimensional vectorspace H , and the statement follows by integrating the resulting linear first-orderdifferential equation. (cid:3) Remark.
As in the tame case, the g -coinvariance implies X i = j ∈ J Ω ( ij ) b w + X k ∈ J Ω ( kk ) b w = INGULAR MODULES AND APPLICATIONS 41 in the space H . The action of Ω ( kk ) is that of the quadratic Casimir (3) on the k -th slot, so this term acts diagonally and can be exponentiated to find the usualconformal weight (cf. Rem. 5.2). The point is that in general the dilation actionhas further nonscalar terms. △
12. D ifferent dynamical term from infinity
In this section we generalise the dynamical KZ connection [18], varying thesetup of § 1.Namely note another natural family of Lie algebras S ( p ) ⊆ S ( p ) ⊆ b g is givenby S ( p ) := h J z K + z p g J z K ⊕ C K .The derived Lie algebra of S ( ) yields the first “level subalgebra” of [16], then thetwo differ for p >
2. One can then define (smooth) induced modules c W as in § 1,where c W = c W ( p ) χ depends on a character χ : S ( p ) → C . However one does notrecover the standard affine Verma module as the starting element of the family,contrary to (9)—which is one motivation behind Def. 1.1.Moreover one has S ( p ) ≃ h p ⊕ C K , analogously to Lem. 1.1, so for p =
1a character is defined by elements λ ∈ h ∨ and by the irregular Cartan term µ ∈ (h ⊗ z ) ∨ (plus the choice of a level κ ). Hence for p = c W with principal parts of meromorphic connections at polesof order two, but in general only poles of even order can be obtained with thisconstruction, contrary to (9)—which is another motivation behind Def. 1.1. Remark.
The fact that the abelianisation of S ( p ) depends on “2 p ” Cartan param-eters is our interpretation of the insightful Rem. 4 of [16, p. 5]. The same dilationis seen in the formulæ of [17, §§ 2.8, 3.4], where one allows for poles “of orders2 m α ”, but only fixes “the m α most singular terms” via the character.One may alternatively shift degrees to consider poles “of order m α ” using thecanonical vector space isomorphisms g ⊗ z i ≃ g for i ∈ Z , but this loses theinformation about the order of the singularity and breaks (13). △ In any case one can put the module c W = c W ( ) χ at infinity in the tensor prod-uct b H , and consider the spaces of coinvariants H as in § 6. The proofs ofProps. 7.1, 7.2 and 7.3 can be adjusted introducing suitable filtrations on c W and W = U (g J z K ) w , where w ∈ c W is the cyclic vector, as well as the whole of § 9.1.Hence in brief one can use W as auxiliary module at infinity, which yields a dif-ferent “dynamical” Cartan term in the reduced connection—with respect to (49).Namely (49) simplifies to D i ( b v ⊗ w ) = κ + h ∨ X k µ k H ( i ) k · b v ⊗ w ,where ( H k ) k is a ( · | · ) -orthonormal basis of h , using (n + ⊕ n − ) ⊗ z ∞ · w = H k z ∞ · w = µ k w , and writing µ k = h µ , H k z ∞ i .We see the reduced connection generalises the dynamical KZ equations, i.e. [18,Eq. 3], and it coincides with it when the modules over finite points are tame. Replace κ + h ∨ ∈ C with “ κ ” and P k µ k H k ∈ h with “ µ ” to retrieve the exact [18, Eq. 3]. So we recover the Felder–Markov–Tarasov–Varchenko connection (FMTV) overvariations of marked points as a particular case of this construction.Note the whole of the FMTV connection also allows for variations of the ir-regular part µ ∈ (h ⊗ z ) ∨ , in addition to the deformations à la Klares consideredhere [28]. In particular when there is only one simple pole the resulting flat con-nection for variations of µ is the DMT connection [31, 41], which is derived froma representation-theoretic setup in [16, § 3.11], and [17, § 3.7] (for the latter seealso [42]). Remark . Just as in the caseof the KZ connection, a different derivation of these flat connections has beenobtained by (filtered) deformation quantisation of isomonodromy systems, thistime importantly for irregular meromorphic connections.Namely [7] derived the DMT connection from the quantisation of a dual ver-sion of the Schlesinger system (related to the usual Schlesinger system by theHarnad duality [22], i.e. the Fourier–Laplace transform). In the same spirit, thewhole of FMTV connection can be obtained by quantising the isomonodromysystem of Jimbo–Miwa–Môri–Sato [26] (see [35, § 11], and [35, 36] for a furthergeneralisation to connections with poles of order three, including all the casesabove). △ A cknowledgments G.F. is supported in part by the National Centre of Competence in ResearchSwissMAP—The Mathematics of Physics—of the Swiss National Science Foun-dation.G.R. is supported by the Deutsche Forschungsgemeinschaft (DFG, GermanResearch Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.We thank Philip Boalch and Leonid Rybnikov for helpful discussions.A ppendix
A. C omputations
A.1.
Proof of Prop. 5.1.
Proof.
Set a ( j ) k := h a j , H k z j i for k ∈ {
0, . . . , r } and j ∈ {
0, . . . , p − } , and further a ( j ) α := h a j , H α z j i for α ∈ R .By (10) we see that : X k z − j X k z n + j : w = − p j p − − n , so n ( p − ) is necessary for nonvanishing terms.Now importantly for n ∈ (cid:8) p −
1, . . . , 2 ( p − ) (cid:9) and j ∈ { − p , . . . , p − − n } one has − j , n + j ∈ { − p + n , . . . , p − } ⊆ {
0, . . . , p − } , so the normal orderedproducts are void in (29). Then for α ∈ R + and i ∈ {
1, . . . , r } one computes H k z − j H k z j + n w = a (− j ) k a ( j + n ) k w , E α z − j E α z n + j w = ( H α | H α ) E α z − j E α z n + j w = H α z n w = δ n , p − a ( n ) α w . INGULAR MODULES AND APPLICATIONS 43
Hence2 (cid:0) κ + h ∨ (cid:1) L n w = p − − n X j = − p r X k = (cid:0) H k z − j H k z j + n (cid:1) + X α ∈ R + (cid:0) E α z − j E α z j + n (cid:1) w = X j , k (cid:0) a (− j ) k a ( j + n ) k (cid:1) + δ n , p − ( p − n − ) X α ∈ R + (cid:16) ( α | α ) a ( n ) α (cid:17) w ,which implies (30) and (31) using ( α | α ) h µ , H α z i i = ( α | µ ) , for µ ∈ h ∨ ⊗ z i . (cid:3) A.2.
Proof of Prop. 10.1.
Proof.
We prove the g -action commutes with ∇ p : Ω (cid:0) U ( n , p ) (cid:1) → Ω (cid:0) U ( n , p ) (cid:1) .Since the g -action is independent of the point on the base space, this is equivalentto ̟ ′ p ⊗ Xψ − X (cid:0) ̟ ′ p ⊗ ψ (cid:1) = = ̟ ′′ p ⊗ Xψ − X (cid:0) ̟ ′′ p ⊗ ψ (cid:1) , for X ∈ g .Now by (52) one has ( κ + h ∨ ) (cid:16) ̟ ′ p ⊗ Xψ − X (cid:0) ̟ ′ p ⊗ ψ (cid:1)(cid:17) = X i = j p − X m , l = (− ) m (cid:18) m + ll (cid:19) t − − m − lij d t ij ⊗ X k ∈ J ′ (cid:2) Ω ( ij ) ml X ( k ) (cid:3) ψ ! ,and analogously by (53) ( κ + h ∨ ) (cid:16) ̟ ′′ p ⊗ Xψ − X (cid:0) ̟ ′′ p ⊗ ψ (cid:1)(cid:17) = X i ∈ J ′ p − X m , l = (cid:18) m + ll (cid:19) t li ⊗ X k ∈ J ′ (cid:2) Ω ( i ∞ ) ml , X ( k ) (cid:3) ψ ! .Hence it is enough to show that X k ∈ J ′ (cid:2) Ω ( ij ) ml , X ( k ) (cid:3) = ∈ U (g p ) ⊗ n ,for all i = j ∈ J and for all m , l ∈ Z . Finally by (4) we have X k ∈ J ′ (cid:2) Ω ( ij ) ml , X ( k ) (cid:3) = X r (cid:16)(cid:2) X r , X (cid:3) z m (cid:17) ( i ) (cid:0) X r z l (cid:1) ( j ) + (cid:0) X r z m (cid:1) ( i ) (cid:16)(cid:2) X r , X (cid:3) z l (cid:17) ( j ) ,where we let ( X r ) r be a ( · | · ) -orthonormal basis of g , which vanishes by (5). (cid:3) R eferences
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