aa r X i v : . [ m a t h . QA ] J u l Non-Levi closed conjugacy classes of
S O q ( N ) Andrey Mudrov
Department of Mathematics,University of Leicester,University Road, LE1 7RH Leicester, UKe-mail: [email protected]
Abstract
We construct explicit quantization of semisimple conjugacy classes of the complexorthogonal group SO ( N ) with non-Levi isotropy subgroups through an operator real-ization on highest weight modules of the quantum group U q (cid:0) so ( N ) (cid:1) .Mathematics Subject Classifications: 81R50, 81R60, 17B37.Key words: Quantum groups, deformation quantization, conjugacy classes, representation the-ory. This is a continuation of our recent works [1, 2] on quantization of closed conjugacy classes ofsimple complex algebraic groups with non-Levi stabilizers. There, we extended the methodsthat had been developed in [3] for classes with Levi isotropy subgroups to the non-Levi case,with the focus on the symplectic groups. In this paper, we apply those ideas to the orthogonalgroups. This solves the quantization problem for all non-Levi conjugacy classes of simplecomplex matrix groups. Along with [1]–[4], this result yields quantization of all semisimpleclasses of symplectic and special linear groups and ”almost all” classes of orthogonal groups.A ”thin” family of orthogonal classes with Levi stabilizer is left beyond our scope. Thisfamily can be called ”borderline” as it shares some properties of non-Levi classes. We givea special consideration to this case in a separate publication, [5]. Then the quantization1roblem will be closed for all semisimple conjugacy classes of the four classical series ofsimple groups.Observe that semisimple conjugacy classes of simple complex groups fall into two familiesdistinguished by the type of their isotropy subgroup: whether it is Levi or not. With regardto the classical matrix series, the second type appears only in the symplectic and orthogonalcases. In the orthogonal case, the isotropy subgroup of a given class is not Levi if and onlyif the eigenvalues ± − N , the multiplicity of 1 is at least 4 too. This is what we assume in this paper.If both eigenvalues ± − SO (2) rotating this eigenspace is isomorphic to GL (1). Quantizationof these classes methodologically lies in between of [3] and the present work. We postponethis case to a separate study, in order to simplify the current presentation.Recall that closed conjugacy classes are affine subvarieties of the algebraic group G ofcomplex orthogonal N × N -matrices, [6]. We consider them as Poisson homogeneous spacesover the Poisson group G equipped with the Drinfeld-Sklyanin bracket. Their Poisson struc-ture restricts from a Poisson structure on G , which is different from the Drinfeld-Sklyaninbracket. The group itself and the conjugacy classes are Poisson manifolds over the Pois-son group G with respect to the adjoint action. We are searching for quantization of theaffine coordinate ring of a class as a quotient of the quantized algebra, C ~ [ G ], of polynomialfunctions on G .The algebra C ~ [ G ] should not be confused with the restricted dual to the quantizeduniversal enveloping algebra. As above said, they are quantizations of different Poissonstructures on G . Rather, the algebra C ~ [ G ] is closer to U q ( g ) than to its dual and can berealized as a subalgebra in U q ( g ). Therefore, C ~ [ G ] is represented on all U q ( g )-modules. Wefind a U q ( g )-module of highest weight such that the quotient of C ~ [ G ] by the annihilator isa deformation of the polynomial ring of a non-Levi conjugacy class. Contrary to the Levicase, it is not a parabolic Verma module.The key step of our approach is finding an appropriate submodule in an auxiliaryparabolic Verma module ˆ M λ and pass to the quotient module M λ , to realize the quan-tized coordinate ring of the class G/K by linear operators from End( M λ ). The module ˆ M λ is associated with the quantum universal enveloping subalgebra U q ( l ) ⊂ U q ( g ) of a certainauxiliary Levi subgroup L ⊂ G . The subgroup L is maximal among those contained in thestabilizer K . We obtain L by reducing the orthogonal block SO (2 m ) ⊂ K , which rotatesthe eigenspace of −
1, to GL ( m ) ⊂ SO (2 m ). 2aving constructed M λ we proceed to the study of the U q ( g )-module C N ⊗ M λ . Wefind the spectrum and the minimal polynomial of the image of an invariant matrix Q ∈
End( C N ) ⊗ U q ( g ) in End( C N ⊗ M λ ) whose entries generate the algebra C ~ [ G/K ]. We startfrom the minimal polynomial on C N ⊗ ˆ M λ , which is known from [3]. Further we analyze thestructure of C N ⊗ ˆ M λ and show that a submodule responsible for a simple divisor becomesinvertible under the projection C N ⊗ ˆ M λ → C N ⊗ M λ and drops from the minimal polynomialof Q . This reduction yields a polynomial identity on Q which determines the conjugacy classin the classical limit.As a result, we obtain an explicit expression of the annihilator of M λ in C ~ [ G ] in terms ofthe ”quantum coordinate matrix” Q . This annihilator is the quantized defining ideal of theclass G/K . This way we obtain an explicit description of C ~ [ G/K ] as a quotient of C ~ [ G ],in terms of generators and relations.Non-Levi conjugacy classes include symmetric spaces SO ( N ) /SO (2 m ) × SO ( N − m ).Their quantum counterparts were studied in connection with in integrable models, [7], andrepresentation theory, [8]–[10]. Contrary to our approach, quantum symmetric spaces wereviewed as subalgebras in the Hopf dual to U ~ ( g ) annihilated by certain coideal subalgebras,the quantum stabilizers, [11]–[14]. That is possible for symmetric classes since they admit”classical points”, where the Poisson bracket turns zero. At the quantum level, classicalpoints give rise to one-dimensional representations of C ~ [ G ]. Other conjugacy classes do notadmit classical points, so our method of quantization remains the most general. Throughout the paper, G designates the algebraic group SO ( N ), N > N = 5, oforthogonal matrices preserving a non-degenerate symmetric bilinear form ( C ij ) Ni,j =1 on thecomplex vector space C N ; the Lie algebra of G will be denoted by g . We choose the realization C ij = δ ij ′ , where δ ij is the Kronecker symbol, and i ′ = N + 1 − i for i = 1 , . . . , N .The polynomial ring C [ G ] is generated by the matrix coordinate functions ( A ij ) Ni,j =1 ,modulo the set of N relations written in the matrix form as ACA t = C. (2.1)Strictly speaking, this equation defines the group O ( N ), but we will ignore this distinction,because the relation det A = 1 will be automatically covered by the defining relations ofconjugacy classes. 3he right conjugacy action of G on itself induces a left action on C [ G ] by duality; thematrix A is invariant as an element of End( C N ) ⊗ C [ G ].The group G is equipped with the Drinfeld-Sklyanin bivector field { A , A } = 12 ( A A r − rA A ) , (2.2)where r ∈ g ⊗ g is a solution of the classical Yang-Baxter equation, [15]. This equation isunderstood in End( C N ) ⊗ End( C N ) ⊗ C [ G ], and the subscripts indicate the natural tensorfactor embeddings of End( C N ) in End( C N ) ⊗ End( C N ), as usual in the literature.The bivector field (2.2) is skew-symmetric when restricted to G and defines a Poissonbracket making G a Poisson group. We fix the standard solution of the classical Yang-Baxterequation: r = N X i =1 ( e ii ⊗ e ii − e ii ⊗ e i ′ i ′ ) + 2 N X i,j =1 i>j ( e ij ⊗ e ji − e ij ⊗ e i ′ j ′ ) . (2.3)At the end of the article, we lift this restriction to include an arbitrary factorizable r-matrix,[17]. This extends our results to arbitrary quasitriangular quantum orthogonal groups.We regard the group G as a G -space under the conjugation action. The object of ourstudy is another Poisson structure on G , { A , A } = 12 ( A r A − A rA + A A r − r A A ) , (2.4)see [16]. It is compatible with the conjugation action and makes G a Poisson space over thePoisson group G equipped with the Drinfeld-Sklyanin bracket (2.2).We reserve n to denote the rank of the Lie algebra so ( N ), so N is either 2 n or 2 n + 1. Asemisimple conjugacy class O ⊂ G consists of diagonalizable matrices and is determined bythe multi-set of eigenvalues S O = { µ i , µ − i } ni =1 ∪ { } , where { } is present when N is odd.Every eigenvalue µ enters S O with its reciprocal µ − and, in particular, may degenerate to µ = µ − = ±
1. For a class to be non-Levi, both +1 and − S O . Moreover, themultiplicity of − N .In terms of Dynkin diagram, a Levi subgroup is obtained by scraping out a subset ofnodes, while non-Levi isotropy subgroups are obtained from the affine Dynkin diagrams:Levi +1 ❛ × ❛ ❛ . . . ❛ × ❛ ❛ > Non-Levi ❛❛ ❜❜✧✧ ❛ × ❛ . . . ❛ × ❛ > ❛ +1 ∓ g = so (2 n + 1)4 × ❛ ❛ . . . ❛ × ❛ ❛❛ ✧✧❜❜ ± ❛❛ ❜❜✧✧ ❛ × ❛ . . . ❛ × ❛ ❛❛ ✧✧❜❜ ± ∓ g = so (2 n )In other words, a non-Levi subgroup necessarily contains a semisimple orthogonal block ofeven dimension rotating the eigenspace of − N , a semisimple orthogonal blockrotating the eigenspace of +1.With a class O , we associate an integer valued vector n = ( n i ) ℓ +2 i =1 subject to P ℓ +2 i =1 n i = n ,and a complex valued vector µ = ( µ i ) ℓ +2 i =1 . We assume that the coordinates of µ are allinvertible, with µ i = µ ± j for i < j ℓ and µ i = 1 for 1 i ℓ . Finally, we put µ ℓ +1 = − µ ℓ +2 = 1. We reserve the special notation m = n ℓ +1 and p = n ℓ +2 .The initial point o ⊂ O is fixed to the diagonal matrix with the entries µ , . . . , µ | {z } n , . . . , µ ℓ , . . . , µ ℓ | {z } n ℓ , − , . . . , − | {z } m , , . . . , | {z } P , − , . . . , − | {z } m , µ − ℓ , . . . , µ − ℓ | {z } n ℓ , . . . , µ − , . . . , µ − | {z } n , where P = 2 p if N = 2 n and P = 2 p + 1 if N = 2 n + 1. We assume m >
2, also p > N and p > N . The class with p = 1, m = 2 for even N is a ”boundary” casementioned in the introduction, which is not considered here.The stabilizer subgroup of the initial point o ∈ O is the direct product K = GL ( n ) × . . . × GL ( n ℓ ) × SO (2 m ) × SO ( P ) (2.5)and it is determined solely by the vector n . The integer ℓ counts the number of GL -blocks in K of dimension n i , i = 1 , . . . , ℓ , while m and p are the ranks of the orthogonalblocks in K corresponding to the eigenvalues − n = . . . = n ℓ = 0 is formally encoded by ℓ = 0 and referred to as the symmetric case . Then(2.5) reduces to SO (2 m ) × SO ( P ), and the class O ≃ G/K to a symmetric space.Let M K denote the moduli space of conjugacy classes with the fixed isotropy subgroup(2.5), regarded as Poisson spaces. The set of all ℓ + 2-tuples µ as above specified parameter-izes M K although not uniquely. In particular, for even N one can also choose the alternativeparametrization µ ℓ +1 = 1, µ ℓ +2 = −
1, however it is compensated by the Poisson automor-phism A
7→ − A . Therefore, the subset ˆ M K of µ with fixed µ ℓ +1 = − µ ℓ +2 = 1 can beused for parametrization of M K (which is still not one-to-one).The conjugacy class O associated with µ and n is specified by the set of equations( A − µ ) . . . ( A − µ ℓ )( A + 1)( A − A − µ − ℓ ) . . . ( A − µ − ) = 0 , (2.6)Tr( A k ) = ℓ X i =1 n i ( µ ki + µ − ki ) + 2 m ( − k + P, k = 1 , . . . , N, (2.7)5here the matrix multiplication in the first line is understood. This system is polynomial inthe matrix entries A ij and defines an ideal of C [End( C N )] vanishing on O . Theorem 2.1.
The system of polynomial relations (2.6) and (2.7) along with the definingrelations of the group (2.1) generates the defining ideal of the class O ⊂ SO ( N ) .Proof. The proof is similar to the symplectic case worked out in [2], Theorem 2.3. It boilsdown to checking the rank of Jacobian of the system (2.6), (2.7), and (2.1).
The quantum group U ~ ( g ) is a deformation of the universal enveloping algebra U ( g ) alongthe parameter ~ in the class of Hopf algebras, [15]. By definition, it is a topologically free C [[ ~ ]]-algebra. Here and further on, C [[ ~ ]] is the local ring of formal power series in ~ .Let R and R + denote respectively the root system and the set of positive roots of theorthogonal Lie algebra g . Let Π + = ( α , α , . . . , α n ) be the set of simple positive roots.They can be conveniently expressed through an orthonormal basis ( ε i ) ni =1 with respect tothe canonical symmetric inner form ( . , . ) on the linear span of Π + : α i = ε i − ε i +1 , i = 1 , . . . , n − , α n = ε n − + ε n , g = so (2 n ) ,α i = ε i − ε i +1 , i = 1 , . . . , n − , α n = ε n , g = so (2 n + 1) . Given a reductive subalgebra f ⊂ g such that f ⊃ h , we label its root subsystem withsubscript f , as well as the set of positive and simple positive roots: R f , R + f , Π + f . We reservethe notation g k for the orthogonal subalgebra of rank k n corresponding to the positiveroots Π + g k = { α n − k +1 , . . . , α n } .Denote by h the dual vector space to the linear span C Π + . The inner product establishesa linear isomorphism between the C Π + and h . We define h λ ∈ h for every λ ∈ h ∗ = C Π + tobe its image under this isomorphism: µ ( h λ ) = ( λ, µ ) for all h ∈ h .The vector space h generates a commutative subalgebra U ~ ( h ) ⊂ U ~ ( g ) called the Cartansubalgebra. The quantum group U ~ ( g ) is a C [[ ~ ]]-algebra generated by simple root vectors e µ , f µ (Chevalley generators), and h µ ∈ h (Cartan generators), µ ∈ Π + . Both U ~ ( h ) and U ~ ( g ) are completed in ~ -adic topology. The Cartan and Chevalley generators obey thecommutation rule [ h µ , e ν ] = ( µ, ν ) e ν , [ h µ , f ν ] = − ( µ, ν ) f ν , [ e µ , f ν ] = δ µ,ν q h µ − q − h µ q − q − , µ ∈ Π + , q = e ~ . µ , contrary to the usual definition with q µ = e ~ ( µ,µ )2 , µ ∈ Π + , see e.g. [18]. The difference comes from a rescaling of the Chevalleygenerators, which also respects the Serre relations below. With our normalization, thenatural representation of U ~ (cid:0) so ( N ) (cid:1) on C N is determined by the classical matrix assignmenton the generators, which is independent of q .Let a ij = α i ,α j )( α i ,α i ) , i, j = 1 , . . . , n , be the Cartan matrix and put q i := q α i . Define[ z ] q = q z − q − z q − q − for any complex z , and the q -binomial coefficients " nk q = [ n ] q ![ k ] q ![ n − k ] q ! , [0] q ! = 1 , [ n ] q ! = [1] q · [2] q . . . [ n ] q for k, n ∈ N , k n . The positive Chevalley generators satisfy the quantum Serre relations − a ij X k =0 ( − k " − a ij k q i e − a ij − kα i e α j e kα i = 0 . Similar relations hold for the negative Chevalley generators f µ .The comultiplication ∆ and antipode γ are defined on the generators by∆( h µ ) = h µ ⊗ ⊗ h µ , γ ( h µ ) = − h µ , ∆( e µ ) = e µ ⊗ q h µ ⊗ e µ , γ ( e µ ) = − q − h µ e µ , ∆( f µ ) = f µ ⊗ q − h µ + 1 ⊗ f µ , γ ( f µ ) = − f µ q h µ , for all µ ∈ Π + . The counit homomorphism ε : U ~ ( g ) → C [[ ~ ]] annihilates e µ , f µ , h µ . As in[2], our comultiplication is opposite to the comultiplication used in [18].Besides the Cartan subalgebra U ~ ( h ), the quantum group U ~ ( g ) contains the followingHopf subalgebras. The positive and negative Borel subalgebras U ~ ( b ± ) are generated over U ~ ( h ) by, respectively, { e µ } µ ∈ Π + and { f µ } µ ∈ Π + as left (right) regular U ~ ( h )-modules. For anyroot subsystem in R the associated Levi subalgebra U ( l ) is quantized to a Hopf subalgebra U ~ ( l ), along with the parabolic subalgebras U ~ ( p ± ) generated by U ~ ( b ± ) over U ~ ( l ).Let U q ( h ) denote the subalgebra in U ~ ( g ) generated by the exponentials t ± α i = q ± h αi , α i ∈ Π + . By U q ( g ) ⊂ U ~ ( g ) we mean the Hopf subalgebra generated over U q ( h ) by { e µ , f µ } µ ∈ Π + .The other mentioned subalgebras in U ~ ( g ) have their counterparts in U q ( g ) and will bedenoted with the subscript q . Note that all these algebras are considered over C [[ ~ ]] butare not completed in the ~ -adic topology. Also the ~ - and q -versions have different Cartansubalgebras. 7uantum counterparts e µ , f µ ∈ U ~ ( g ), µ ∈ R + , of higher root vectors are defined througha reduced decomposition of the maximal element of the Weyl group, [18]. Contrary tothe classical case, they depend on such a decomposition. Higher root vectors generate aPoincare-Birkhoff-Witt (PBW) basis in U ~ ( g ) over U ~ ( h ), [18]. This basis establishes a linearisomorphism of the adjoint h -modules U ~ ( g ) and U ( g ) ⊗ C [[ ~ ]]. This isomorphism enables theuse of the same notation for h -submodules in U ( g ) and U ~ ( g ). For instance, by g ⊂ U ~ ( g )we understand the sum of h and the linear span of { f µ , e µ } µ ∈ Π + .The triangular decomposition g = n − l ⊕ l ⊕ n + l gives rise to the triangular factorization U ~ ( g ) = U ~ ( n − l ) U ~ ( l ) U ~ ( n + l ) , (3.8)where U ~ ( n ± l ) are subalgebras in U ~ ( b ± ) generated by the positive or negative root vectorsfrom n ± l , respectively, [24]. This factorization makes U ~ ( g ) a free U ~ ( n − l ) − U ~ ( n + l )-bimodulegenerated by U ~ ( l ). For the special case l = h , we denote g ± = n ± h . Contrary to the classicaluniversal enveloping algebras, U ~ ( n ± l ) are not Hopf subalgebras in U ~ ( g ). ˆ M λ We adopt certain conventions concerning representations of quantum groups, which aresimilar to [2]. We assume that they are free modules over C [[ ~ ]] and their rank will bereferred to as dimension. Finite dimensional U ~ ( g )-modules are deformations of their classicalcounterparts, and we will drop the reference to the deformation parameter in order to simplifynotation. For instance, the natural N -dimensional representation of U ~ ( g ) will be denotedsimply by C N .We shall deal with weight i.e. U ~ ( h )-diagonalizable, modules. If V is an h -invariant sub-space, we mean by [ V ] α the subspace of weight α ∈ h ∗ . We stick to the additive parametriza-tion of weights of U q ( g ) facilitated by the embedding U q ( h ) ⊂ U ~ ( h ). Under this convention,such weights belong to ~ h ∗ [[ ~ ]]. Indeed, for any λ ∈ ~ h ∗ [[ ~ ]] its values on the generators t ± α i are q ± λ ( h αi ) ∈ C [[ ~ ]], so λ is well defined on U q ( h ). By reasons explained below, it is sufficientfor our needs to confine weights to the subspace ~ − h ∗ ⊕ h ∗ ⊂ ~ − h ∗ [[ ~ ]].Let L ⊂ G denote the Levi subgroup L = GL ( n ) × . . . × GL ( n ℓ ) × GL ( m ) × SO ( P ) . It is a maximal Levi subgroup of G among those contained in K , cf. (2.5) (the other oneis obtained by reducing SO ( P ) to GL ( p )). By l we denote the Lie algebra of L . It is areductive subalgebra in g of maximal rank n .8e denote by c l ⊂ h the center of l and realize its dual c ∗ l as a subspace in h ∗ thanksto the canonical inner product. A element λ ∈ C ∗ l = ~ − c ∗ l ⊕ c ∗ l defines a one-dimensionalrepresentation of U q ( l ) denoted by C λ . Its restriction to the Cartan subalgebra acts by theassignment q h α q ( α,λ ) . Since q = e ~ , the pole in λ is compensated, and the representation iscorrectly defined. It extends to U q ( p + ) by nil on n + l ⊂ p + l . Denote by ˆ M λ = U q ( g ) ⊗ U q ( p + ) C λ the parabolic Verma U q ( g )-module induced from C λ , [19]. It plays an intermediate role in ourconstruction: we are interested in a quotient module M λ , which can be defined for certainvalues of λ .Regarded as a U q ( h )-module, ˆ M λ is isomorphic to U q ( n − l ) ⊗ C λ , as follows from (3.8).This implies that ˆ M λ are isomorphic as C [[ ~ []-modules for all λ . Let v λ denote the image of1 ⊗ M λ . It generates ˆ M λ over U q ( g ) and carries the highest weight λ . For any sequenceof Chevalley generators f α k , . . . , f α km we call the product f α k . . . f α km v λ ∈ ˆ M λ Chevalleymonomial or simply monomial.Along with ˆ M λ , we consider the right U q ( g )-module ˆ M ⋆λ = C λ ⊗ U q ( p − ) U q ( g ). Here C λ supports the 1-dimensional representation of U q ( p − ) which extends the U q ( l )-representationby nil on n − l ⊂ p − l . As a U q ( h )-module, it is isomorphic to C λ ⊗ U q ( n + l ) generated v ⋆λ = 1 ⊗ v = f α k . . . f α km v λ we define v ⋆ to be the monomial v ⋆λ e α km . . . e α k ∈ ˆ M ⋆λ .There is a bilinear pairing (Shapovalov form) between ˆ M ⋆λ and ˆ M λ . It is determined by thefollowing requirements: i ) h xu, y i = h x, uy i for all x ∈ ˆ M ⋆λ , y ∈ ˆ M λ , u ∈ U q ( g ), ii ) v ⋆λ isorthogonal to all vectors of weight lower than λ , iii ) it is normalized to h v ⋆λ , v λ i = 1.As in [2], we introduce a subspace of weights that we use for the parametrization of M K ,the moduli space of conjugacy classes with fixed K . Define E i ∈ h ∗ , i = 1 , . . . , ℓ + 2, by E = ε + . . . + ε n , E = ε n +1 + . . . + ε n + n , . . . , E ℓ +2 = ε n − p +1 + . . . + ε n . The vector space c ∗ l is formed by λ = P ℓ +2 i =1 Λ i E i with Λ i ∈ C and Λ ℓ +2 = 0. Put µ k = e k ,for k = 1 , . . . , ℓ + 2. Let c ∗ l ,reg denote the set of all weights λ ∈ c ∗ l such that µ k = ( µ j ) ± for k = j . Denote by c ∗ k ⊂ c ∗ l its subset determined by µ ℓ +1 = − c ∗ k ,reg ⊂ c ∗ k thesubspace of such λ ∈ c ∗ l that µ k = ( µ j ) ± for k, j = 1 , . . . , ℓ + 2, k = j . Obviously c ∗ k ,reg isdense in c ∗ k . Finally, we introduce C ∗ k ,reg ⊂ C ∗ k ⊂ ~ c ∗ l ⊕ c ∗ l by setting C ∗ k = ~ − c ∗ k − P E ℓ +1 and C ∗ k ,reg = ~ − c ∗ k ,reg − P E ℓ +1 . Clearly C ∗ k ,reg is dense in C ∗ k . By construction, all weights from C ∗ k satisfy q α n − p ,λ ) = − q − P .Note that the vector µ = ( µ i ) for λ ∈ c ∗ k ,reg belongs to ˆ M K covering M K , and all pointsin ˆ M K can be obtained this way. 9 .1 Some auxiliary technicalities In this section we introduce some constructions which we use further on. They involvethe quantum subgroup U q (cid:0) gl ( n ) (cid:1) in U ~ ( g ) corresponding to the roots { α i } n − i =1 . Its negativeChevalley generators obey the Serre relations f α i f α i ± − ( q + q − ) f α i f α i ± f α i + f α i ± f α i = 0 , [ f α i , f α j ] = 0 , (4.9)for all feasible i , j , and | i − j | > l = gl (2) ⊕ so ( N − ℓ = 0, m = 2, p = n − >
0, and let ˆ M λ be a parabolic Verma module relative to U q ( l ). Note that f α killsthe generator v λ ∈ ˆ M λ unless α = α . Put κ = p = n − N = 2 n and κ = p + 1 = n − N = 2 n + 1. Introduce the element ω = f α κ . . . f α v λ ∈ ˆ M λ and also ω = v λ for p = n − M λ in the subsequent sections. It is constructed solely out of the gl ( n )-generators and featuresthe following. Lemma 4.1.
Suppose that κ . Then ω is annihilated by f α i , i κ .Proof. Let ∼ denote equality up to a scalar factor. Assuming 3 i < κ , we get f α i ω = f α κ . . . f α i f α i +1 f α i . . . f α v λ ∼ f α κ . . . ( f α i f α i +1 + f α i +1 f α i ) . . . f α v λ , by the Serre relations (4.9). The rightmost dots contain generators with numbers strictly lessthan i . Since they commute with f α i +1 , it can be pushed to the right in the first summand,where it kills v λ . The second summand is equal to f α κ . . . f α i f α i − . . . v λ . Again, using theSerre relation for f α i f α i − we can place at least one factor f α i on the right of f α i − and pushit freely further to the right. This kills the second summand.If i = κ , then we have f α κ ω = f α κ f α κ − . . . f α v λ . Using (4.9), at least one copy of f α κ can be pushed through f α κ − to the right and further on till it kills v λ . Lemma 4.2.
The vector ω is annihilated by e α i , α i ∈ Π + , i = κ .Proof. Obviously, ω is annihilated by e α i if i = 1 and i > κ . Applying e α i with 2 i κ − ω yields f α κ . . . f α i +1 . . . v λ up to a scalar multiplier. Here the dots on the right stand forthe generators with numbers strictly less than i . Since they commute with f α i +1 , the lattercan be pushed to the right, where it kills v λ .10emark that ω is a non-zero vector of weight λ − ε + ε κ +1 . Indeed, one can check thatdim[ ˆ M λ ] λ − ε + ε κ +1 = 1 and all other monomials of this weight turn zero.Further we present another auxiliary construction, which also involves only the gl ( n )-generators. Suppose that 3 n and introduce vectors y k ∈ ˆ M λ , k = 2 , . . . , n − , by y = [ f α , f α ] a f α v λ , y k = f α k . . . f α [ f α , f α ] a f α k . . . f α v λ , k n − , where [ X, Y ] a = XY − aY X . Here and further on we set the parameter a equal to q + q − . Lemma 4.3.
For all k = 2 , . . . , n − , one has y k = 0 .Proof. For k = 2 we find [ f α , f α ] a f α v λ = f α f α f α v λ − af α f α f α v λ = 0 by the Serrerelation (4.9). For higher k we use induction. Suppose the lemma is proved for some k > y k +1 = f α k +1 f α k . . . f α [ f α , f α ] a f α k +1 f α k . . . f α v λ = ( f α k +1 f α k f α k +1 ) . . . f α [ f α , f α ] a f α k . . . f α v λ . The term in the brackets produces a − ( f α k +1 f α k + f α k f α k +1 ) through (4.9). The first term iszero by the induction assumption. The second term is zero too, because f α k +1 can be pushedto the right till it meets the second copy of f α k . By the Serre relation, one factor f α k +1 canbe pushed through f α k to the right. Then it proceeds freely till it kills v λ . This proves thestatement.Remark that the case N = 5, m = 2, p = 0 is excluded from this construction, and y = 0 then. ˆ M λ for l = gl (2) ⊕ so ( P ) A substantial part of this theory is captured by the special case of symmetric conjugacyclasses. That accounts for the fact that the difference between K and L is confined withinthe orthogonal blocks of K . Because of that, we start from the symmetric case, when thestabilizer k consists of two simple orthogonal blocks of ranks m and p . Furthermore, thegeneral symmetric case can be readily derived from the specialization m = 2 (note that so (4) is the smallest semisimple orthogonal algebra of even dimension). For this reason, westart with k = so (4) ⊕ so ( N − l = gl (2) ⊕ so ( N − U ( k ) is generated over U ( l ) by a pair of root vectors e δ , f δ , where δ = α + 2 n − X i =2 α i + α n − + α n , g = so (2 n ) , δ = α + 2 n X i =2 α i , g = so (2 n + 1) . k = m − ⊕ l ⊕ m + , where m − = ad( l )( f δ ) and m + = ad( l )( e δ ) are abelian Liesubalgebras. The algebra U ( k ) features the triangular decomposition U ( k ) = U ( m − ) × U ( l ) × U ( m + ).In the symmetric case under consideration, the weight λ satisfies the conditions ( α i , λ ) = 0for all i but i = 2. Therefore, ˆ M λ is parameterized by scalar ( α , λ ). Its highest weight vector v λ is annihilated by all e α i and all f α i except for f α . Regarding ˆ M λ as a U ~ ( g − )-moduleconsider its classical limit ˆ M λ / ~ ˆ M λ . It is generated by the root vectors f ε ± ε i , f ε ± ε i , f ε + ε , f ε , f ε ∈ n − l , where i = 3 , . . . , n , and f ε , f ε are present only when N is odd. Therefore, modulo ~ , theweight space [ ˆ M λ ] λ − δ , has the basis of N − f ε ± ε i f ε ∓ ε i v λ , f ε + ε v λ , f ε f ε v λ , where last term counts for odd N . Since ˆ M λ is C [[ ~ ]]-free, dim[ ˆ M λ ] λ − δ = N − v λ − δ ∈ [ ˆ M λ ] λ − δ where λ allows for it. Singularmeans that v λ − δ lies in the kernel of all e α ∈ g + . In order to facilitate the calculations, weneed to choose a suitable basis. Notice that in the classical limit the subspace of weight λ − δ + α = λ − ε (the image e α [ ˆ M λ ] λ − δ for generic λ ) has a basis f ε ± ε i f ε ∓ ε i v λ , f ε f ε v λ , where the last term counts for odd N . Therefore, ker e α | [ M λ ] λ − δ = [ker e α ] λ − δ has dimension n − e α ] λ − δ . First we do it for the lowest dimensions N = 5 , , N = 5 we have only one vector x = [ f α , f α ] a f α v λ , for N = 7 there are two vectors x = [ f α , f α ] a f α ( f α ω ) , x = f α [ f α , f α ] a ( f α ω ) , ω = f α v λ . There are three vectors for N = 8: x = [ f α , f α ] a ( f α f α ω ) , x = f α [ f α , f α ] a f α ω, x = f α [ f α , f α ] a f α ω, ω = f α v λ . N >
8, define n − x i ∈ [ ˆ M λ ] λ − δ by x = [ f α , f α ] a f α < . . . f α n ( f α n ω ) ,x i = f α i > . . . f α [ f α , f α ] a f α i +1 < . . . f α n ( f α n ω ) , i = 3 , . . . , n, for N = 2 n + 1, and by x = [ f α , f α ] a f α < . . . f α n − ( f α n − f α n ω ) ,x i = f α i > . . . f α [ f α , f α ] a f α i +1 < . . . f α n − ( f α n − f α n ω ) , i = 3 , . . . , n − ,x n − = f α n − f α n − > . . . f α [ f α , f α ] a f α n ω,x n = f α n f α n − > . . . f α [ f α , f α ] a f α n − ω, for N = 2 n . The products are ordered with respect to the root numbers as indicated. Theelement ω for all N is defined in the previous section. We emphasize that the generators inthe parenthesis stay within as i varies, while other generators are permuted as specified.The following lemma accounts for the choice of the commutator parameter a . Lemma 4.4.
The vectors x i , i = 2 , . . . , n , belong to ker e α ⊂ ˆ M λ .Proof. Applying e α to x i we get e α x i ∼ . . . [ q h α − q − h α , f α ] a . . . ω = (cid:0) ( q − q − ) − a ( q − q − ) (cid:1) . . . f α . . . ω = 0 . Indeed, observe that h α commutes with everything between the commutator and ω . Further,the weight of ω is λ − ε + ε n − for N = 2 n > λ − ε + ε n for N = 2 n + 1 >
7, and λ − ε for N = 5. This produces the vanishing scalar factor in the brackets.As we already mentioned, the total dimension of [ ˆ M λ ] λ − δ is equal to N −
3. Every vector x i contains the commutator [ f α , f α ] a thus involving two Chevalley monomials. Overall { x i } ni =2 involve 2 n − λ − δ . This is equal to dim[ ˆ M λ ] λ − δ for odd N ,but greater by 1 for even N . However, f α f α n − f α p . . . f α f α n ω ∼ f α f α n f α n − f α p . . . f α ω ∼ f α f α n f α p . . . f α f α n − ω. Therefore, there are effectively 2 n − { x i } ni =2 for N = 2 n , as required.Our search for v λ − δ will be restricted to the subspace [ker e α ] λ − δ , so n − x i annihilated by e α are just enough to form a basis. Next we prove a lemma, which is crucial13or checking the linear independence of x i . Introduce vectors x ′ i ∈ [ ˆ M λ ] λ − δ + α i for i = 2 , . . . , n as follows: x ′ is obtained from x by replacing the commutator [ f α , f α ] a with f α ; to get x ′ i for i >
2, we remove the leftmost copy of f α i from x α i . One can see that e α i x i ∼ x ′ i for i = 2 , . . . , n . Lemma 4.5.
For all i = 2 , . . . , n , x ′ i = 0 .Proof. Observe that dim[ ˆ M λ ] λ − δ + α = 1 and dim[ ˆ M λ ] λ − δ + α i = 2, where i = 3 , . . . , n (for thisverification, one can use the classical PBW basis). Also, notice that dim[ ˆ M λ ] λ − δ + α i + α = 1for such i . Consider the Chevalley monomials x ′′ i of weights λ − δ + α i + α , i = 3 , . . . , n ,obtained from x ′ i by replacing the commutator [ f α , f α ] a with f α . Using Lemma 4.2, onecan easily calculate the matrix elements of the Shapovalov pairing h x ′ ⋆ , x ′ i = h ω ⋆ , ω i , h x ′′ ⋆i , x ′′ i i = q ( α ,λ ) − − q − ( α ,λ )+1 q − q − h ω ⋆ , ω i , i > , and h ω ⋆ , ω i = q ( α ,λ ) − q − ( α ,λ ) q − q − . This calculation proves that x ′ and x ′′ i do not vanish for generic λ and hence for all λ (the U q ( g − )-module ˆ M λ is isomorphic to U q ( g − ) / P α ∈ Π + l U q ( g − ) f α andhence ”independent of λ ”).Further, there are exactly two ways to construct a monomial of weight λ − δ + α i , i =3 , . . . , n , out of x ′′ i : either placing f α before or after the leftmost f α (note that f α is theonly generator which does not commute with f α ). This gives two independent monomialsparticipating in x ′ i , i = 3 , . . . , n . Consequently, x ′ i do not vanish.Note that the vectors x i can be labeled with the simple roots of the subalgebra g n − = g p +1 ⊂ g via the assignment α i x i , i = 2 , . . . , n . The next proposition provides qualitativeinformation about the action of positive Chevalley generators on the system { x i } ⊂ ker e α . Proposition 4.6.
For all α, α i ∈ Π + g p +1 such that ( α, α i ) = 0 the generator e α annihilates x i . If ( α j , α i ) = 0 , then e α j x i ∼ x ′ j .Proof. Suppose first that N is even and put α n − = µ , α n = ν . Denote also x µ = x n − and x ν = x n . Up to a scalar multiplier, e ν x µ is equal to y n − , which is zero due to Lemma 4.3.Further, observe that e µ x i , for i < p , contains the factor f α p f ν f α p producing f α p f ν and f ν f α p via the Serre relation (4.9). In the first term, the generator f ν goes freely to the right andkills v λ . The second term gives rise to the factor f α p ω , which is nil by Lemma 4.1. Due tothe symmetry between the roots µ and ν , this also proves e µ x ν = 0 and e µ x i = 0 for i < p .By Lemma 4.2, e α i kills ω , once 2 i < p . Therefore, such e α i knock out the factor f α i from x µ = f µ f α p . . . f α i +1 f α i . . . [ f α , f α ] . . . ω releasing f α i +1 next to the left. The latter can14e pushed to the right till it meets ω and annihilates it by Lemma 4.1. Hence e α i x µ = e α i x ν =0 for 2 i < p . Similar effect is produced by the action of e α i on x j for 3 i + 1 < j p .If 3 j + 1 < i p , the vector e α i x j contains the factor f α i − f α i +1 . . . ω = . . . f α i − ω, whichis zero due to Lemma 4.1. This completes the proof of the first assertion for even N .Now suppose that N is odd . There is nothing to prove if p = 0, as there is only onevector, x . So we assume p >
0. Let us check that e α i x j = 0 when 3 j + 1 < i n . Then x j has the structure . . . [ f α , f α ] f α j +1 . . . f α i − f α i . . . ( f α n ω ) . Observe that e α i effectively actsonly on the displayed copy of f α i . The other copy is hidden in ω and can be neglected,because e α i kills ω if i = n − , n , by Lemma 4.2. If i = n −
1, then f α n e α i ω = 0 by similararguments. If i = n , then e α i x j still comprises the factor f α n − f α n ω . In all cases, e α i knocksout the leftmost f α i and releases the factor f α i − on the left, which goes freely to the right.It kills ω by Lemma 4.1 if i < n . Still it kills f α n ω if i = n , and the proof is similar toLemma 4.1.If 3 i + 1 < j n , then x j = f α j . . . f α i +1 f α i . . . [ f α , f α ] a . . . f α n ω , and f α i +1 commuteswith everything between f α i and f α n . Further reasoning is similar to the case j + 1 < i , with f α i − replaced by f α i +1 . Therefore, e α i x j = 0 for i + 1 < j . This completes the first part ofthe proposition for odd N .The proof of the second statement becomes quite straightforward on examining the struc-ture of x i . This is left for the reader as an exercise. Corollary 4.7.
The vectors { x i } ni =2 form a basis in [ker e α ] λ − δ .Proof. By Proposition 4.6, the operator E = P ni =2 e α i sends the linear span X = Span { x i } ⊂ [ker e α ] λ − δ to the linear span X ′ = Span { x ′ i } . By Lemma 4.5, all x ′ i = 0, have differentweights and hence independent. We shall see in the next section (cf. Proposition 5.2) thatker E | X = { } for generic λ . Hence { x i } ni =2 are independent for generic λ . Thanks to the U q ( g − )-module isomorphism ˆ M λ ≃ U q ( g − ) / P α ∈ Π + l U q ( g − ) f α , they are independent for all λ .This proves the statement, since dim[ker e α ] λ − δ = n − { x i } ni =2 affects uniqueness of v λ − δ ∈ Span { x i } ni =2 , for special λ , but not its existence. M λ In this section we construct the highest weight U q ( g )-module M λ that supports quantizationof the class G/K . We define it as a quotient of ˆ M λ by a proper submodule generated by a15ingular vector of certain weight. First we do it for symmetric G/K and afterwards extendthe solution for general K . Consider the simplest symmetric case m = 2, p = n −
2. In other words, assume k = so (4) ⊕ so ( P ), and l = gl (2) ⊕ so ( P ).Define a vector v λ − δ ∈ [ ˆ M λ ] λ − δ by v λ − δ = c x + . . . c n − x n − + c n − x n − + c n x n , with the scalar coefficients c i set to be c i = ( − i q n − i − + ( − i q − ( n − i − ) , i n, for N = 2 n + 1 and c i = ( − q ) n − − i + ( − q ) − ( n − − i ) , i n − , c n − = c n = 1 , for N = 2 n . The n − c i satisfy the recurrent system of n − c i − + ac i + c i +1 = c n − + c n = 0 , N = 2 n + 1 ,c i − + ac i + c i +1 = c n − + ac n − + c n − + c n = c n − + ac n − = c n − + ac n = 0 , N = 2 n, (5.10)where i varies from 3 to n − n − c i ) ni =2 up to a common multiplier. Lemma 5.1.
Up to a scalar factor, v λ − δ ∈ ˆ M λ is a unique vector of weight λ − δ annihilatedby e α , α ∈ Π + − { α } .Proof. First of all, v λ − δ is annihilated by e α , due to Lemma 4.4. Further proof is based onProposition 4.6, stating that e α i v λ − δ = E i x ′ i , for some scalars E i , i = 2 , . . . , n . This yields asystem of n − E i = 0, which is written down in (5.10) for each parity of N . Thecoefficients ( c i ) ni =2 are determined uniquely, up to a common factor.Recall that a vector in a U q ( g )-module is called singular if it is annihilated by g + . In amodule with highest weight, singular vectors generate proper submodules. Proposition 5.2.
Suppose that λ satisfies the condition q α ,λ ) = − q − P . Then the vector v λ − δ ∈ ˆ M λ is singular. Up to a scalar factor, it is a unique singular vectorof weight λ − δ and it exists only if λ satisfies the above condition. roof. By Corollary 4.7, v λ − δ = 0. In view of Lemma 5.1, we only need to satisfy thecondition e α v λ − δ = 0. From Corollary 4.7, we get e α v λ − δ = E x ′ for some scalar E .Evaluating E x ′ and equating it to zero we get conditions on λ for v λ − δ to be singular.If N = 5, we find q ( α ,λ ) (1 − q ) = q − ( α ,λ ) (1 − q − ), which immediately gives required q α ,λ ) = − q − . For for N = 8 we obtain q ( α ,λ ) q c + q ( α ,λ ) qc + q ( α ,λ ) qc = q − ( α ,λ ) q − c + q − ( α ,λ ) q − c + q − ( α ,λ ) q − c . For N = 2 n > N = 2 n + 1 > q ( α ,λ ) q c + q ( α ,λ ) qc = q − ( α ,λ ) q − c + q − ( α ,λ ) q − c . Plugging the expressions for c , c , c in these equations we find that v λ − δ is singular only if λ satisfies the hypothesis. Its uniqueness follows from Lemma 5.1. M λ for general k In this section we abandon the simplifying ansatz ℓ = 0, m = 2 and allow for generalisotropy subgroup K , as in (2.5). The Lie algebra k of the subgroup K and the maximalLevi subalgebra l read k = gl ( n ) ⊕ . . . ⊕ gl ( n ℓ ) ⊕ so (2 m ) ⊕ so ( P ) , (5.11) l = gl ( n ) ⊕ . . . ⊕ gl ( n ℓ ) ⊕ gl ( m ) ⊕ so ( P ) . (5.12)Consider the subalgebra g ′ = g p +2 ⊂ g with the simple positive roots ( α n − p − , . . . , α n ).Under this embedding, the root α goes over to α n − p , and the root δ reads δ = α n − p − + 2 n − X i = n − p α i + α n − + α n , g = so (2 n ) , δ = α n − p − + 2 n X i = n − p α i , g = so (2 n + 1) . Assuming λ ∈ C ∗ k , let ˆ M λ be the parabolic Verma module over U q ( g ). Regarded as a g ′ -weightby restriction, λ satisfies the assumptions of Proposition 5.2. Therefore, there is a singularvector v λ − δ in the U q ( g ′ )-submodule of ˆ M λ generated by v λ . Proposition 5.3.
Suppose that λ ∈ C ∗ k . Then v λ − δ ∈ ˆ M λ is a unique, up to a scalarmultiplier, singular vector of weight λ − δ .Proof. The vector v λ − δ is annihilated by e α for α ∈ Π + g ′ , by to Proposition 5.2. Furthermore, v λ − δ is constructed out of f β ∈ Π + g ′ , which commute with e α for α ∈ Π + − Π + g ′ . Consequently,such e α annihilate v λ − δ too. Therefore v λ − δ is singular in ˆ M λ . It is unique up to a factor asit is so for U q ( g ′ ). 17 efinition 5.4. Assuming λ ∈ C ∗ k , we denote by ˆ M λ − δ ⊂ ˆ M λ the submodule generated by v λ − δ and we denote by M λ the quotient module ˆ M λ / ˆ M λ − δ . The module M λ is the key object of our approach to quantization and of our further study.Next we prove that M λ is free as a C [[ ~ ]]-module. There is a PBW basis in U ~ ( g − )generated by ordered quantum root vectors f µ , µ ∈ R + , which are defined through an actionof the quantum Weyl group on the generators, [18]. This basis establishes a natural U ~ ( h )-linear isomorphism of U ~ ( g − ) and U ( g − ) ⊗ C [[ ~ ]]. It is argued in [2] that, over the ring ofscalars C [[ ~ ]], one can arbitrarily change the ordering of f µ and arbitrarily deform f µ within[ U ~ ( g − )] µ . Reordered deformed root vectors still generate a PBW-like basis. To apply thisargument to the orthogonal case, we must find an appropriate element f δ ∈ [ U ~ ( g − )] − δ . Lemma 5.5.
There exists a deformation f δ ∈ U ~ ( g − ) = U q ( g − ) of the classical root vectorof root − δ such that v λ − δ = f δ v λ .Proof. Put U − ~ = U ~ ( g − ), U − = U ( g − ), and define φ ∈ U − ~ /U − ~ l − from the presentation v λ − δ = φv λ . Note that φ is independent of λ . By Lemma 5.1, φ ∈ [ U − ~ /U − ~ l − ] − δ is unique,up to a scalar multiplier, solution of the system [ e α , φ ] = 0 mod U − ~ l , α ∈ Π + l . Modulo ~ , the classical root vector f ∈ [ g ] − δ solves this system. Indeed, it commutes with allsuch e α as δ − α is not a root once α ∈ Π + l . Therefore, upon a proper normalization, theprojection of f to U − /U − l − coincides with the zero fiber of φ . Regarding f as an element of U − ~ = U − ~ /U − ~ l − ⊕ U − ~ l − under the natural linear isomorphism U − ~ ≃ U − ⊗ C [[ ~ ]], we defineits deformation f δ by changing the U − ~ /U − ~ l − -component to φ . The U − ~ l − -component of f can be replaced with its arbitrary deformation within [ U − ~ l − ] − δ .Note that, in the symplectic case [2], the element f δ participates in construction of v λ − δ .In this exposition, we have introduced v λ − δ in a simpler way, at the price of Lemma 5.5. Proposition 5.6.
The module M λ is free over C [[ ~ ]] .Proof. The quantum ”root vector” f δ can be included in a PBW-like basis as discussed in[2]. The rest of the proof is similar to the proof of Proposition 6.2 therein.We are going to prove that the quantization of the conjugacy class G/K can be realizedby linear operators on M λ . To this end, we study the module structure of the tensor product C N ⊗ M λ in the following section. Again we start with the symmetric case ℓ = 0. Fortechnical reasons we process separately the cases of even and odd N .18 The U q ( g ) -module C N ⊗ M λ in the symmetric case In this section, we study the U q ( g )-module C N ⊗ M λ . To a large extent, the difference betweenLevi and non-Levi classes is concentrated in the ”symmetric part” of the stabilizer, so weconsider this case first, as we did in the preceding sections. We assume that the isotropysubalgebra k consists of two orthogonal blocks of rank m and p , k = so (2 m ) ⊕ so ( P ), where P = 2 p for the D -series and P = 2 p + 1 for the B -series.When restricted to the Levi subalgebra l = gl ( m ) ⊕ so ( P ), the natural g -representation C N splits into three irreducible sub-representations, C N = C m ⊕ C P ⊕ C m . The submodule C P carries the natural representation of so ( P ) ⊂ l while the two copies of C m are the naturaland conatural submodules of gl ( m ) ⊂ l . This reduction extends to the pair of the quantumgroups U q ( l ) ⊂ U q ( g ).The natural and conatural gl ( m )-submodules C m glue up to the natural module of theblock so (2 m ) ⊂ k leading to the irreducible decomposition C N = C m ⊕ C P over k . We cannotwrite a quantum version of this reduction because we do not know a natural candidate forthe subalgebra U q ( k ) ⊂ U q ( g ). Yet we bypass this obstacle.We fix the standard basis { w i } Ni =1 ⊂ C N of columns with the only non-zero entry in the i -th position. The highest weights of the irreducible U q ( l )-submodules in C N are ε , ε m +1 , − ε m ,and the corresponding weight vectors are w , w m +1 , w N +1 − m . For generic λ ∈ C ∗ l , the tensorproduct C N ⊗ ˆ M λ splits into the direct sum of three U q ( g )-modules C N ⊗ ˆ M λ = ˆ M ⊕ ˆ M ⊕ ˆ M ,of highest weights ν = λ + ε , ν = λ + ε m +1 , and ν = λ − ε m , respectively. We shall prove,for almost all λ ∈ C ∗ k , the direct decomposition C N ⊗ M λ = M ⊕ M , where M i are theimages of ˆ M i under the projection C N ⊗ ˆ M λ → C N ⊗ M λ . This results in a degree reductionof the minimal polynomial for the quantum coordinate matrix (a similar effect is producedon the classical coordinate matrix by the transition from G/L to G/K ). This is the key stepof our strategy .Let u ν i , i = 1 , ,
3, denote the canonical generators of ˆ M i ⊂ ˆ M λ . It can be shown that u ν ∼ w ⊗ v λ − δ and vanishes in C N ⊗ M λ , λ ∈ C ∗ k . The vector u ν = w ⊗ v λ carries thehighest weight λ + ε in C N ⊗ ˆ M λ and generates the submodule ˆ M . The singular vector ofweight λ + ε m +1 that generates ˆ M reads u ν = q ( α,λ ) − q − ( α,λ ) q − q − w m +1 ⊗ v λ + ( − q ) − w m ⊗ f α m v λ + . . . + ( − q ) − m w ⊗ f α . . . f α m v λ . It is calculated in [2] for the symplectic case and still valid for orthogonal g , because itinvolves only the generators of gl ( n ) ⊂ g . Note that vector u ν is also singular in C N ⊗ M λ ,19s it is not nil there. The following fact is established in [2]. Lemma 6.1.
The singular vector u ν is equal to q − m q ( α,λ )+ m − q − ( α,λ ) − m q − q − w m +1 ⊗ v λ modulo ˆ M . To proceed with the analysis of module structure of C N ⊗ ˆ M λ , we have to develop a specialdiagram technique, which takes the rest of this section. The action of a positive (negative)Chevalley generator on the standard basis { w i } Ni =1 features the following property: the line C w i is either annihilated or mapped onto the line C w k for some k . It is convenient to depictsuch an action graphically. Further we consider negative generators, since positive can beobtained by reversing the arrows.Up to an invertible scalar multiplier, the action of the family { f α } α ∈ Π + ⊂ U q (cid:0) so (2 n + 1) (cid:1) on the standard basis { w i } n +1 i =1 in C n +1 is encoded in the following scheme: w n +1 ❜ ✛ ❜ . . . ✛ w n ❜ ✛ w n +2 ✛ ❜ w n +1 ✛ ❜ w n ✛ ✛ ❜ w . . . ✛ ❜ w f α f α f α n − f α n f α n f α n − f α f α This diagram has simple linear structure without branching and cycles. The diagram for g = so (2 n ) is more complicated: w n ❜ ❜ ✛ ✛ . . . ✛ w n − ❜ w n +2 ✛ ❜ w n +1 ✙ ✙ ❜ w n ✛ ❜ w n − . . . ✛ ✛ ❜ w ✛ ❜ w f α f α f α n − f α n − f α n f α n − f α n − f α n f α f α The arrows designate the action up to a non-zero scalar, which is equal to − f α produces −
1, while the second gives +1. In thematrix language, the non-zero entries above and below the skew diagonal are +1 and − { f α } and { e α } determine therepresentation of U q ( g ) on C N .Further we adopt the following convention. If a vector v is proportional to a vector u with a scalar coefficient c = 0, i.e. v = cu , we write v ≃ u and say that v is equivalent to u .If the difference v − cu belongs to a vector space W , we write v ≃ u mod W .To study the action of { f α } α ∈ Π + on C N ⊗ ˆ M λ , we develop our diagram language further.First of all, we transpose the above diagrams of the natural representation to columns, so thearrows become vertical and oriented downward. Suppose ( v i ) li =1 ∈ ˆ M λ is a finite sequenceof vectors. We associate a horizontal graph with nodes ( v i ) li =1 and arrows designating the20ction of { f α } α ∈ Π + on ˆ M λ , in a similar fashion as vertical but with the following difference:it involves not all possible arrows v k ← v i but only those of our interest. We still assumethat the chosen arrows are isomorphisms of lines spanned by v i . This implies that, up to anon-zero scalar factor, the nodes are determined by the subset of maximal nodes (having noinward arrows) and by the set of arrows. In our case, there will be only one maximal vector v = v λ , hence the other nodes are determined by arrows. This implies that the horizontalgraph is connected.Let Arr( w k ) denote the set of negative Chevalley generators whose arrows are directedfrom w k . Similarly, Arr( v i ) denote the set of generators whose arrows are directed from v i .For instance, Arr( w k ) consists of only one element for k < n + 1 and is empty for k = 2 n + 1,in the series B . We say that arrow f has length k if it sends node i to the node i + k . Allvertical arrows but f α n for even N have length 1.We construct a ”tensor product” of vertical and horizontal graphs to be a diagram withthe nodes w k ⊗ v i , k = 1 , . . . , N , i = 1 , . . . , l . The factor w k marks the rows from top tobottom, while v i marks the columns from right to left. The vertical and horizontal arrowsdesignate the action of the designated Chevalley generators on the tensor factors , up to ascalar multiplier. Under these assumptions, such diagrams provide information about theaction of { f α } α ∈ Π + not just on the tensor factors but on the entire tensor product C N ⊗ ˆ M λ ,in the following sense: Proposition 6.2.
Suppose a horizontal arrow designates the action of a Chevalley generator f α on v i . If f α Arr( w k ) , then f α ( w k ⊗ v i ) ≃ w k ⊗ ( f α v i ) , otherwise f α ( w k ⊗ v i ) ≃ w k ⊗ f α v i modulo C f α w k ⊗ v i .Proof. This statement follows from the definition of diagram and quasi-primitivity of theChevalley generators (cf. the comultiplication in Section 3). In particular, f α Arr( w k ) ifand only if f α w k = 0, hence the first alternative. The second alternative is also an immediateconsequence of quasi-primitivity of f α . Corollary 6.3.
Suppose that f α ∈ Arr( v i ) − Arr( w k ) for some i and k . Suppose that for k j k the nodes w j ⊗ v i lie in a submodule ˆ M ⊂ C N ⊗ ˆ M λ and all arrows from Arr( w j ) have length . Then Span { w j ⊗ f α v i } kj = k ⊂ ˆ M .Proof. We have f α ( w k ⊗ v i ) ≃ w k ⊗ f α v i , as f α w k = 0. Therefore, Span { f α ( w j ⊗ v i ) } kj = k =Span { w j ⊗ f α v i } kj = k modulo Span { w j ⊗ v i } kj = k ⊂ ˆ M . Now the proof is immediate.Suppose there are intervals of vertical nodes ( w k ), k ∈ I v = [ k , k ], and horizontal nodes( v i ), i ∈ I h = [ i , i ], such that: all vertical arrows directed from w k , k ∈ I ′ v = [ k , k −
1] are21f length 1; for each i ∈ I ′ h = [ i , i − there is a horizontal arrow of length 1. In particular,Arr( w k ) consists of one element for all k ∈ I ′ v . Let us denote the subset of these selectedhorizontal arrows by A h . Consider the subgraph with nodes ( w k ⊗ v i ), ( k, i ) ∈ I v × I h , thevertical arrows from Arr( w k ), k ∈ I ′ v , and the horizontal arrows from the selected subset A h .We call such a subgraph simple rectangle . In particular, the entire diagram may be simple.Its horizontal and vertical ”tensor factor” subgraphs are topologically simply connected,having no cycle or branching.The diagrams of interest will be specified in the next section. Here we establish a generalfact, which will be used in what follows. Lemma 6.4.
Suppose that a diagram D contains an equilateral rectangular triangle T (lev-eled by top and right edges) belonging to a simple rectangle in D . Suppose that the right edgeof T belongs to a submodule ˆ M ⊂ C N ⊗ ˆ M λ . Then the entire triangle T belongs to ˆ M .Proof. Without loss of generality we may assume that D is simple and T sits in the north-east corner of D , i.e. w ⊗ v ∈ ˆ M is its maximal node. Suppose that its edge contains t nodes. We do induction on column’s number i , which is illustrated below (on the left). f i +1 i ❅❅❅❅❅❅❅❅❅ ❄✛✛ f i +1 i ❅❅❅❅❅❅ ❄✛✛ By the hypothesis, the column { w k ⊗ v } tk =1 lies in ˆ M . Suppose that the statement is provedfor 1 i < t . Let f ∈ U q ( g ) be the operator assigned to the horizontal arrow v i +1 ← v i .Each node w k ⊗ v i , k = 1 , . . . , t − i, is sent by f to w k ⊗ v i +1 possibly modulo C w k +1 ⊗ v i ,which belongs to ˆ M by the induction assumption. Therefore, the i + 1-st column of T doesbelong to ˆ M .Remark that the triangle can be replaced with a trapezoid obtained by cutting off theleft end of T with vertical line, as shown on the right.The sequence ( v i ) is assumed to be finite and contain the unique minimal node (nooutward arrows). This node is in the focus of our interest. By construction, it will carry theweight λ − δ , and the whole diagram yield a path (paths) to it from the maximal node v λ .This way, we associate a diagram with every non-vanishing Chevalley monomial in [ ˆ M λ ] λ − δ participating in the singular vector v λ − δ . 22 .1 Series B , symmetric case k = so (4) ⊕ so (2 n − Suppose first that m = 2. Later on we drop this restriction. Given a permutation s of1 , . . . , n , we define Chevalley monomials v is ∈ ˆ M λ , i = 1 , . . . , n , through the graph ❜ . . . ✛ v ns ❜ ✛ v n +1 s ✛ ❜ v ns ✛ . . . ✛ ❜ ✛ ❜ v s = v λ f α s (1) f α s ( n − f α s ( n ) f α n f α f α In particular, the minimal node of the graph is v ns = f α s (1) . . . f α s ( n ) f α n . . . f α v λ ∈ [ ˆ M λ ] λ − δ .For s = id we omit the subscript s and denote v is simply by v i .With a permutation s of 1 , . . . , n such that v ns = 0 we associate a diagram D s , asexplained in the preceding section. Essential are the nodes { w k ⊗ v i } with i + k n + 1,so we display only this triangular part of D s : D sw ⊗ v ns w ⊗ v n +1 s w ⊗ v n w ⊗ v ✛ . . . ✛ ✛ ✛ . . . ✛❄ ❄ ❄ ... ... ... ❵❵❵ ❄ ❄ ❄✛ ✛ . . . ✛ w n ⊗ v n +1 s w n ⊗ v n w n ⊗ v ❄ ❄ w n +1 ⊗ v n w n +1 ⊗ v ✛ . . . ✛ ❄ ... ❄ w n ⊗ v f αs (1) f αs ( n − f αs ( n ) f αn f α f α f αn f αn f αn − f α ❵❵❵ Note that the first n columns (counting from the right) in all D s are the same.Denote by D ′ id ⊂ D id the sub-graph above the principal diagonal, i.e. consisting of nodes { w k ⊗ v j } such that k + j n . Given s = id let i be the highest of 1 , . . . , n displaced by s ,i.e. s ( i ) = i . Denote by D ′ s ⊂ D s the trapezoid of nodes { w k ⊗ v js } obeying k + j n + 1, k i . D ′ s , s = id w ⊗ v ns ✛ . . . ✛ w ⊗ v n − i +1 s ✛ w ⊗ v n − is ✛ . . . ✛ w ⊗ v ns ✛ . . . ✛ w ⊗ v λ ❄ ... ❄❄ ... ❄❄ ... ❄❄ ... ❄ ❵❵❵ w i ⊗ v n − is w i ⊗ v n − i +1 s ✛ ✛ . . . ✛ w i ⊗ v ns ✛ . . . ✛ w i ⊗ v λ f αs (1) f αs ( i ) f αi +1 f αn f αn f α f α f αi − Let ˆ M denote the U q ( g )-submodule ˆ M + ˆ M ⊂ C N ⊗ ˆ M λ .23 emma 6.5. Suppose that q − p = − . Then D ′ s lies in ˆ M .Proof. First suppose that s = id. The statement is trivial for the node u ν = w ⊗ v λ generating ˆ M . By Lemma 6.1, u ν ≃ w ⊗ v λ mod ˆ M , under the assumption q − p = − w ⊗ v λ belongs to ˆ M too. Further observe that the rightmost column of D ′ id liesin ˆ M . For this part of D ′ id , the vertical arrows actually depict the action on the whole tensorsquare, as they kill the factor v λ . Notice that D s is simple. One is left to apply Lemma 6.4to the triangle T = D ′ id .Now we consider the case s = id and let i be the highest integer displaced by s . Observethat the right rectangular part of D ′ id up to column 2 n − i is the same as in D id andbelongs to D ′ id . Hence it lies in ˆ M , as already proved. Since s ( i ) = i , the horizontal arrow f α s ( i ) ∈ Arr( v n − is ) is distinct from vertical f α i constituting Arr( w i ) (suppressed in the graph).By Corollary 6.3, column 2 n − i + 1 of D ′ id belongs to ˆ M . The remaining part of D ′ s is atriangle bounded on the right with column 2 n − i + 1. By Lemma 6.4, it belongs to ˆ M .Now we are ready to prove the following Proposition 6.6.
Suppose that q m − p − = − . Then the tensor product C N ⊗ M λ splitsinto the direct sum M ⊕ M .Proof. We will use an operator
Q ∈
End( C N ⊗ ˆ M λ ) defined by (7.13). Here we need to knowthat Q is U ~ ( g )-invariant and turns scalar on ˆ M and ˆ M with the eigenvalues, respectively, µ = − q − p − and µ = q − m , cf. (7.16). All eigenvalues of Q are calculated in [3] andpresented in (7.15). Since µ = µ , the modules M and M have zero intersection, and theirsum is direct.We prove the statement if we show that C N ⊗ v λ ⊂ M = M ⊕ M . We consider thecase m = 2 first. Then w n is the highest weight vector of the l -submodule in C N of weight − ε . Our strategy is to show that the vertical scheme representing the U q ( b − )-action on C N yields the action on C N ⊗ v λ modulo M . Starting from w ⊗ v λ ∈ M we obtain all w i ⊗ v λ mod M by applying the Chevalley generators. The hardest part of the job is the transitionfrom w n − ⊗ v λ to w n ⊗ v λ .By Lemma 6.5, the diagonal of D id right over the principal diagonal lies in ˆ M . Notice thatthe vertical and horizontal arrows applied to all nodes in this diagonal coincide. Therefore, upto a non-zero scalar factor, the elements on the principal diagonal are all equivalent moduloˆ M . For instance, apply f α to w ⊗ v n − ∈ ˆ M and get q w ⊗ v n − + w ⊗ v n ∈ ˆ M , hence w ⊗ v n = − q w ⊗ v n − mod ˆ M . Moving further down the principal diagonal, we find24 ⊗ v n ≃ w n ⊗ v λ mod ˆ M . Now notice that x = v n − av ns , where s is the transposition(1 , x i are linear combinations of v ns for certain s = id. Since v ns ∈ ˆ M once s = id, by Lemma 6.5, w ⊗ v λ − δ ≃ w n ⊗ v λ modulo ˆ M . Hence w n ⊗ v λ ∈ M .Applying f α to w n ⊗ v λ we get w n +1 ⊗ v λ ∈ M , up to a scalar factor.We have proved the inclusion C N ⊗ v λ ⊂ M under the assumption m = 2. Now we dropthis restriction. First of all, w ⊗ v λ ∈ M and w i ⊗ v λ = f α i − ( w i − ⊗ v λ ) ∈ M for i = 2 , . . . , m .The transition from w m ⊗ v λ to w m +1 ⊗ v λ is facilitated by Lemma 6.1 and is similar to thecase m = 2. Namely, w m +1 ⊗ v λ ≃ u ν ∈ M ⊂ M modulo M under the assumption q m − p − = −
1. This is reducing the proof to the case m = 2. Recall g ′ = g p +2 ⊂ g definedin Section 5.2. Acting by { f α } α , α ∈ Π + g ′ , on w m − ⊗ v λ we proceed as in the m = 2-caseand check that w i ⊗ v λ ∈ M , i = m + 2 , . . . , N − m + 2. Applying l − to w N +2 − m ⊗ v λ ∈ M we get w i ⊗ v λ ∈ M for i = N − m + 3 , . . . , N , as required. The inclusion C N ⊗ M λ ⊂ M isproved. D , symmetric case k = so (4) ⊕ so (2 n − We have to consider two types of diagrams for g = so (2 n ) associated with Chevalley mono-mials constituting the vectors x i , i = 2 , . . . , n − x n − and x n , on the other.Let us start with the first type. Given a permutation s of (1 , . . . , n −
2) we define vectors v is ∈ ˆ M λ , i = 1 , . . . , n −
1, through the graph v n − s ❜ ❜ ✛ ✛ . . . ✛ v n − s ❜ v n +1 s ✛ ❜ v ns ✙ ✙ ❜ v n − s ✛ ❜ v n − s . . . ✛ ✛ ❜ v s ✛ ❜ v s = v λ f α s (1) f α s (2) f α s ( n − f α n − f α n f α n − f α n − f α n f α f α The minimal element of this sequence is v n − s = f α s (1) . . . f α s ( n − f α n . . . f α v λ . The first n + 1nodes are independent of s . As for odd N , we drop the subscript s from v is for s = id.With every permutation s such that v n − s = 0 we associate the diagram D s . As before,we restrict consideration to the triangular part of it, retaining only w k ⊗ v is with k + i n .25 ✙ ✙✙ ✙ w ⊗ v n − s ✛ . . . ✛ w ⊗ v n +1 ✛ w ⊗ v n w ⊗ v n − ✛ w ⊗ v n − ✛ . . . ✛ w ⊗ v · · · · · · ❄ ... ❄❄ ... ❄❄ ... ❄❄ ... ❄❄ ... ❄ w n − ⊗ v n +1 ✛ w n − ⊗ v n w n − ⊗ v n − ✛ w n − ⊗ v n − ✛ . . . ✛✛ . . . ✛✛ . . . ✛✛ . . . ✛ w n ⊗ v n w n ⊗ v n − ✛ w n ⊗ v n − w n +1 ⊗ v n − ✛ w n +1 ⊗ v n − w n +2 ⊗ v n − w n − ⊗ v ❄❄❄❄ w n ⊗ v w n +1 ⊗ v ❄❄ w n +2 ⊗ v ☛☛☛☛☛ ❄ ... ❄ w n − ⊗ v f α f αn − f αn − f αn f αn f αn − f αn − f α f αn f αn f αs (1) f αs ( n − f αn − f αn − f αn − f α D s Only the part of D s which lies to the left of column n + 1 depends on s . We have emphasizedthis by omitting the subscript s in the right part.The diagrams D s account for Chevalley monomials v n − s participating in { x i } n − i =2 . Thevectors x n − and x n bring about different diagrams. Due to the symmetry between x n − and x n we will consider only x n . Define the set { v i } n − i =1 ⊂ M λ as follows. The first n − v = v λ , v i = f α i v i − , i = 2 , . . . , n −
1. The remaining n vectors are set to be v n − k = f α k v n + k − , k = 1 , . . . , n. The arrows v i − ← v i are uniquely determined by the set of nodes { v i } . Given a permutation s of 1 , . . . , n − { v is } n − i =1 by v is = v i , i = 1 , . . . , n − v n − ks = f α s ( k ) v n + k − s , k = 1 , . . . , n. In fact, these vectors are zero for most s . Of all s we only need the transposition (1 , x n , which comprises two Chevalley monomials, due to the factor [ f α , f α ] a in it. D n id w ⊗ v n − ✛ . . . ✛ w ⊗ v n +1 ✛ w ⊗ v n − ✛ w ⊗ v n − ✛ . . . ✛ w ⊗ v · · · ❄ ... ❄❄ ... ❄❄ ... ❄❄ ... ❄✛ w n − ⊗ v n +1 ✛ w n − ⊗ v n − ✛ w n − ⊗ v n − ✛ . . . ✛ w n − ⊗ v f α f αn − f αn f α f α f αn − f αn − f α f α n − , f α n ∈ Arr( w n − ), which are directed from the bottom line. Note thatthe rightmost square of ( n − × ( n −
1) nodes is the same in D ns for all s . It is also asub-graph in D id .Denote by D ′ id ⊂ D id the sub-graph above the principal diagonal, i.e. { w k ⊗ v j } suchthat k + j n −
1. For s = id, let i ∈ [1 , n −
2] the maximal integer displaced by s . Wedenote by D ′ s ⊂ D s the trapezoid rested on line i , i.e. the set of nodes { w k ⊗ v j } such that k + j n − k i . Lemma 6.7.
Suppose that q − p = − . Then D ′ s and D ns ′ lie in ˆ M .Proof. The situation is slightly different from the settings of Lemma 6.4 (odd N ), as thediagrams D s are not simple. Applying similar arguments as in the proof of Lemma 6.5 wecheck that the trapezoid in D ′ id on the right of column w i ⊗ v n − inclusive lies in ˆ M . Thegenerator f α n − sends the nodes of this column one step to the left modulo maybe one stepdown. Since the node w n ⊗ v n − is sent strictly leftward, column n − D ′ id is mappedonto column n −
1, modulo its column n −
2, which is proved to be in ˆ M . Therefore,column n − D ′ id lies in ˆ M . The bottom node of column n of D ′ id is w n − ⊗ v n . Modulo w n +1 ⊗ v n − ∈ ˆ M , it is the f α n -image of w n − ⊗ v n − ∈ ˆ M . Hence w n − ⊗ v n ∈ ˆ M . Thenodes higher in this column are also obtained from column n − f α n , which now actsstrictly leftward. Therefore, the right part of D ′ id lies in ˆ M up to column n . The remainingpart of D ′ id to the left of column n inclusive is a triangle in a simple rectangle (just ignorethe leftmost f α n ) and falls into Lemma 6.4.Now suppose that s = id and and let i be the highest integer displaced by s . Contrary to s = id, this case is pretty similar to Proposition 6.6 for odd N . Notice that right rectangularpart of D ′ s up to column 2 n − i − D ′ id ⊂ D id and lies in ˆ M as argued.Since f α s ( i ) = f α i , column 2 n − i of D ′ id lies in ˆ M , by Corollary 6.3. The remaining part of D ′ id is the triangle bounded by column 2 n − i on the right. It belongs to ˆ M by Lemma 6.4.The proof for D ns ′ for s = id , (1 ,
2) is similar to D ′ s with s = id. The key observation isthat right rectangular part up to column n − D ′ id and hence lie in ˆ M .Further arguments are based on Lemma 6.3 applied to column n − D ′ s .Now we are ready to prove the main result of this section. Proposition 6.8.
Suppose that q m − p = − . Then the tensor product C N ⊗ M λ splits tothe direct sum M ⊕ M . roof. Similar argument as in the proof of Proposition 6.6 tells us that M ∩ M = { } .Indeed, the eigenvalues of Q on ˆ M and ˆ M are µ = − q p , µ = q − m , cf. (7.16). Theyare distinct by the hypothesis, hence the sum M + M is direct. As for g = so (2 n + 1), weneed to show that M = M ⊕ M exhausts all of C N ⊗ M λ , and it is sufficient to prove theinclusion C N ⊗ v λ ⊂ M .First we consider the case m = 2. Then w n − is the highest weight vector of the l -submodule C m ⊂ C N of highest weight − ε . As before, we intend to reach w n − ⊗ v λ from w ⊗ v λ through all w i ⊗ v λ in between staying within M . Then we get w n ⊗ v λ ∈ M byapplying f α to w n − ⊗ v λ .By Lemma 6.7, the diagonal of D id over the principal diagonal lies in ˆ M , as it belongsto D ′ id . The vertical and horizontal arrows applied to this diagonal coincide. The same istrue regarding the nodes w n − ⊗ v n − and w n ⊗ v n − . Therefore, up to a non-zero scalarfactor, the elements in the principal diagonal are all equivalent modulo ˆ M . In particular, w ⊗ v n − ≃ w n − ⊗ v λ mod ˆ M . Now notice that x = v n − − av n − s , where s is thetransposition 1 ↔
2. Observe that all other x i , i < n − v n − s for certain s = id. Since v n − s ∈ ˆ M for such s by Lemma 6.7, all x i with i < n − M . The vector x n is a combination of two monomials associated with D ns , s = id, s = (1 , w ⊗ x n ∈ ˆ M . Due to thesymmetry between x n − and x n , we conclude that w ⊗ x n − ∈ ˆ M too.Since the singular vector v λ − δ is a linear combination of x i , i = 2 , . . . , n , the vector w n − ⊗ v λ is equivalent to w ⊗ v λ − δ modulo ˆ M . Hence w n − ⊗ v λ ∈ M as required.Now we lift the restriction m = 2. This is done similarly to the g = so (2 n + 1)-case.Applying the f α , . . . , f α m − ∈ l − we get w i ⊗ v λ ∈ M for i = 1 , . . . , m . Lemma 6.1 facilitatestransition to from w m ⊗ v λ ∈ M to w m +1 ⊗ v λ ∈ M , under the assumption q m − p = − U q ( g ′ ), g ′ = g p +2 , and reduce consideration tothe case m = 2. This gives w i ⊗ v λ ∈ M , i = m + 2 , . . . N − m + 2. Finally, applying thegenerators f α , . . . , f α m − ∈ l − , we descend from w N − m +2 ⊗ v λ ∈ M to w N ⊗ v λ ∈ M . Thiscompletes the proof. Remark 6.9.
The assumption q m − P = − q if one considers the quantum group U q ( g ) over the complex field with q ∈ C , or over the field C ( q ) of rational functions of q . This condition is fulfilled for anopen set including q = 1 and therefore over the formal series in ~ with q = e ~ . Observe that q m − P = − Q coincide, cf. (7.16). This is accountable byLemma 6.1, because for such q we get the inclusion ˆ M ⊂ ˆ M .28 .3 The module C N ⊗ M λ , general k The symmetric case worked out in detail in the preceding sections will serve as an illustrationto the case of general k considered below. However, our strategy will be slightly different, inorder to save the effort of calculating singular vectors in C N ⊗ M λ . We pay a price for thatby getting a weaker result about the structure of C N ⊗ M λ . Namely, instead of direct sumdecomposition of C N ⊗ M λ we construct a filtration by highest weight modules. Still it issufficient for our purposes, as all we need to know is the spectrum of the quantum coordinatematrix Q , cf. (7.13). Under certain conditions, it can be extracted from the graded moduleassociated with filtration as well as from direct sum decomposition.We have the irreducible decomposition C N = C n ⊕ . . . ⊕ C n ℓ ⊕ C m ⊕ C P ⊕ C m ⊕ C n ℓ ⊕ . . . ⊕ C n of the natural g -representation C N into l -blocks. The submodules C n i carry the naturaland conatural representations the block gl ( n i ) ⊂ l , i = 1 , . . . , ℓ + 1, with n ℓ +1 = m . Thesubmodule C P supports the natural representation of the block so ( P ) ⊂ l . We enumeratethese submodules from left to right as W i , i = 1 , . . . , ℓ +3. This decomposition is compatiblewith the standard basis { w i } , and the basis element with the lowest number falling into theblock is its highest weight vector. Let ν i , i = 1 , . . . , ℓ + 3, be the highest weights ofthe irreducible blocks and let w ν i ∈ W i denote their highest weight vectors. As we said,they form a subset of the standard basis. Explicitly, the highest weights of the blocks are ν i = ε n + ... + n i − +1 for i = 1 , . . . , ℓ + 2 and ν ℓ +4 − i = − ε n + ... + n i for i = 1 , . . . , ℓ + 1.For generic λ this decomposition gives rise to the decomposition C N ⊗ ˆ M λ = ⊕ ℓ +3 i =1 ˆ M i where each ˆ M i is the parabolic Verma module induced from W i ⊗ C λ . Let M i denote itsimage under the projection to C N ⊗ M λ .Transition to the isotropy subalgebra k ⊃ l merges two copies of C m up into a singleirreducible k -submodule. As a result, M ℓ +3 should disappear from C N ⊗ M λ . We saw thiseffect for ℓ = 0 and we expect it for general k . However, constructing the direct sumdecomposition of C N ⊗ M λ along the same lines requires the knowledge of singular vectorsfor all ˆ M i . Instead, we work with a filtration, which construction is much easier. We do noteven check that each graded component, apart from the ℓ + 3-d, survives in the projection C N ⊗ ˆ M λ → C N ⊗ M λ . We just need to make sure that the elementary divisor correspondingto the quotient V ℓ +3 /V ℓ +2 drops from the minimal polynomial of Q .For all j = 1 , . . . , ℓ + 3 we denote by ˆ V j the submodule in C N ⊗ ˆ M λ generated by { w ν i ⊗ v λ } i =1 ,...,j . Let V j denote its image in C N ⊗ M λ . We have the obvious inclusions29 V j − ⊂ ˆ V j , V j − ⊂ V j . It is convenient to set ˆ V and V to { } .Propositions 6.6 and 6.8, which are formulated for the symmetric case, can be restated ina milder setting as C N ⊗ M λ ≃ V ⊕ V /V . The equality C N ⊗ M λ = V = V remains true ifwe relax the assumption on λ . This assumption facilitates the equivalence u ν ≃ w m +1 ⊗ v λ mod V , so the proof remains essentially the same if u ν is replaced with w m +1 ⊗ v λ . Herewe establish a generalization of this fact. Proposition 6.10.
The submodules { } = V ⊂ V ⊂ . . . ⊂ V ℓ +3 form an ascendingfiltration of C N ⊗ M λ . For each k = 1 , . . . , ℓ + 3 , the graded component V k /V k − is either { } or generated by (the image of ) w ν k ⊗ v λ , which is the highest weight vector in V k /V k − .In particular, V ℓ +2 = V ℓ +3 .Proof. Our strategy is similar to the proof of Propositions 6.6 and 6.8. We mean to showthat ⊕ ki =1 W i ⊗ v λ ⊂ V k , which in particular imples C N ⊗ v λ ⊂ V ℓ +3 for k = 2 ℓ + 3. Then e α ( w ν k ⊗ v λ ) = 0 mod V k − , i.e. w ν k ⊗ v λ is a singular vector in V k − /V k if not zero. Since V k − /V k is generated by w ν k ⊗ v λ , it is then the highest weight vector. This will imply C N ⊗ v λ ⊂ V ℓ +3 and V ℓ +3 = M λ .Thus, we wish to prove that W k ⊗ v λ ⊂ V k . This is true for k = 0 if we set W = { } .Suppose we have done this for some k >
0. By construction, w ν k +1 ⊗ v λ ∈ V k +1 . Consecutivelyapplying the Chevalley generators from the block of l − which does not vanish on W k +1 weconclude that W k +1 ⊗ v λ ⊂ V k +1 . Induction on k proves W k ⊗ v λ ⊂ V k for all k .Finally, the equality V l +2 = V l +3 follows from the inclusion W l +3 ⊗ v λ ⊂ V l +2 , and thisboils down to the symmetric case. Indeed, let g ′ = g p +2 ⊂ g be as defined in Section 5.2.Let ˆ M ′ λ ⊂ ˆ M λ be its parabolic Verma submodule generated by v λ and let V ′ ℓ +2 be the U q ( g ′ )-submodule generated by w µ ℓ +1 ⊗ v λ , w µ ℓ +2 ⊗ v λ . As we discussed in the symmetric case, W ℓ +3 ⊗ v λ ⊂ V ′ ℓ +2 ⊂ V ℓ +2 . Hence V l +3 = V l +2 , and the proof is complete. Similarly to classical conjugacy classes, their quantum counterparts are described througha matrix A of non-commutative ”coordinate functions” or its image Q ∈
End( C N ) ⊗ U q ( g ),which should be regarded as ”restriction” of A to the ”quantum group G q ”. In this sectionwe study algebraic properties of Q .The operator Q is defined through the universal R-matrix R , which is an invertible30lement of (completed) tensor square of U ~ ( g ): Q = ( π ⊗ id)( R R ) ∈ End( C N ) ⊗ U q ( g ) . (7.13)Here π is the representation homomorphism U ~ ( g ) → End( C N ). The matrix Q commuteswith ( π ⊗ id) ◦ ∆( u ) for all u ∈ U q ( g ) producing an invariant operator on C N ⊗ V for every U q ( g )-module V .Let ρ denote the half-sum of all positive roots ρ = P α ∈ R + α . In the orthogonal basisof weights { ε i } , it reads ρ = n X i =1 ρ i ε i , ρ i = ρ − ( i − , ρ = ( n − for g = so (2 n + 1) ,n − g = so (2 n ) . Regarded as an operator on C N ⊗ ˆ M λ , the element Q satisfies a polynomial equation withthe roots q λ + ρ,ν i ) − ρ,ε )+( ν i ,ν i ) − = ( q λ,ν i )+2( ρ,ν i − ε ) for p > ,q − n for p = 0 , i = ℓ + 2 , g = so (2 n + 1) . . where ν i , i = 1 , . . . , ℓ + 3 are the highest weights of the irreducible l -submodules in C N ,[3], Theorem 4.2. The bottom line corresponds to zero ν i , which is present only for odd N if p = 0.Assuming λ ∈ C ∗ l ,reg , put Λ i = ( λ, ε n + ... + n i − +1 ) = ( λ, ε n + ... + n i ) for i = 1 , . . . , ℓ + 2 (recallthat n ℓ +1 = m and n ℓ +2 = p , by our convention). The weight λ depends on the parameters(Λ i ), with Λ ℓ +2 = 0. Define the vector µ by µ i = q i − n + ... + n i − ) , i = 1 , . . . , ℓ + 2 . (7.14)The eigenvalues of Q on End( C N ⊗ ˆ M λ ) are expressed through µ by µ i , µ − i q − ρ +2( n i − = µ − i q − N +2( n i +1) , i = 1 , . . . , ℓ + 1 , µ ℓ +2 . (7.15)As was mentioned the operator Q on C N ⊗ ˆ M λ satisfies a polynomial equation of degree2 ℓ + 3. Formula (7.15) implies that, at generic point λ ∈ C ∗ l ,reg , the roots of the polynomialare pairwise distinct for almost all q . Hence Q is semisimple for almost all q at generic λ .In particular, the eigenvalues µ ℓ +1 , µ ℓ +2 , and µ ℓ +3 read µ ℓ +1 = − q − p , µ ℓ +2 = q − m , µ ℓ +3 = − q − n +2 , N = 2 n,µ ℓ +1 = − q − p − , µ ℓ +2 = q − m , µ ℓ +3 = − q − n +1 , N = 2 n + 1 . (7.16)31ote that µ ℓ +1 may be equal to µ ℓ +3 only for m = 1, which case is excluded from ourconsideration. In other words, the minimal polynomial of Q remains semisimple on C N ⊗ ˆ M λ for almost all q upon specialization of λ to generic point of C ∗ k ,reg . Therefore Q is semisimpleon C N ⊗ ˆ M λ and hence on C N ⊗ M λ for an open set in C ∗ k ,reg , for almost all q .The vector µ belongs to ˆ M K modulo ~ for λ ∈ C ∗ k ,reg . Recall that ˆ M K parameterizesthe moduli space M K of conjugacy classes with given K . Proposition 7.1.
For λ ∈ C ∗ k ,reg the operator Q satisfies a polynomial equation of degree ℓ + 2 on C N ⊗ M λ with the roots µ i , µ − i q − N +2( n i +1) , i = 1 , . . . , ℓ, µ ℓ +1 , µ ℓ +2 . (7.17) Proof.
For generic q , the operator Q ∈
End( C ⊗ ˆ M λ ) is semisimple, and the roots (7.15) arepairwise distinct. Therefore, the projection of Q to End( C ⊗ M λ ) is semisimple for almostall q and satisfies the same polynomial equation. Since the graded components V k /V k − arehighest weight modules, the projection of Q is scalar on each V k /V k − , which is one of theeigenvalues of Q . By Proposition 6.10, the eigenvalue µ ℓ +3 drops from the spectrum of Q on C ⊗ M λ , hence the simple divisor Q − µ ℓ +3 is invertible and can be canceled from thepolynomial.The matrix Q produces the center of U q ( g ) via the q-trace construction. For any invariantmatrix X ∈ End( C N ) ⊗ A with the entries in a U q ( g )-module A , one can define an invariantelement Tr q ( X ) := Tr (cid:0) q h ρ X (cid:1) ∈ A . (7.18)Recall that h ρ is an element from h such that α ( h ρ ) = ( α, ρ ) for all α ∈ h ∗ . The q -trace,when applied X = Q k , k ∈ Z + , gives a series of central elements of U q ( g ). We will use theshortcut notation τ k = Tr q ( Q k ).A module M of highest weight λ defines a one dimensional representation χ λ of the centerof U q ( g ), which assigns a scalar to each τ ℓ , [3], formula (24): χ λ ( τ k ) = X ν q k ( λ + ρ,ν ) − k ( ρ,ε )+ k ( ν,ν ) − k Y α ∈ R + q ( λ + ν + ρ,α ) − q − ( λ + ν + ρ,α ) q ( λ + ρ,α ) − q − ( λ + ρ,α ) . (7.19)The summation is taken over weights ν of the module C N . The term k ( ν, ν ) − k survivesfor ν = 0, which is the case only for odd N . Restriction of λ to C ∗ k ,reg makes the righthand side a function of µ defined in (7.14). We denote this function by ϑ k n ,q ( µ ), where32 = ( n , . . . , n ℓ , m, p ) is the integer valued vector of multiplicities. In the limit ~ → ϑ k n ,q ( µ ) goes over into the right hand side of (2.7), where µ i = lim h → q λ,ν i ) , i = 1 , . . . , ℓ . By quantization of a commutative C -algebra A we understand a C [[ ~ ]]-algebra A ~ , whichis free as a C [[ ~ ]]-module and A ~ / ~ A ~ ≃ A as a C -algebra. Quantization is called U ~ ( g )-equivariant if A and A ~ are, respectively U ( g )- and U ~ ( g )-module algebras and the U ~ ( g )-action on A ~ is a deformation of the U ( g )-action on A . Below we describe the quantizationof C [ G ] along the Poisson bracket (2.4).Recall from [20] that the image of the universal R-matrix of the quantum group U ~ ( g ) inthe defining representation is equal, up to a scalar factor, to R = N X i,j =1 q δ ij − δ ij ′ e ii ⊗ e jj + ( q − q − ) N X i,j =1 i>j ( e ij ⊗ e ji − q ρ i − ρ j e ij ⊗ e i ′ j ′ ) . The coefficients ρ i are defined as ρ n +1 = 0, ρ i = − ρ i ′ = ( ρ, ε i ) = n + − i for N = 2 n + 1and ρ i = − ρ i ′ = ( ρ, ε i ) = n − i for N = 2 n , where i runs over 1 , . . . , n .Denote by S the U ~ ( g )-invariant operator P R ∈ End( C N ) ⊗ End( C N ), where P is theordinary flip of C N ⊗ C N . This matrix has three invariant projectors to its eigenspaces, amongwhich there is a one-dimensional projector κ to the trivial U ~ ( g )-submodule, proportional to P Ni,j =1 q ρ i − ρ j e i ′ j ⊗ e ij ′ . Denote by C ~ [ G ] the associative algebra generated by the entries of a matrix A =( A ij ) Ni,j =1 ∈ End( C N ) ⊗ C ~ [ G ] modulo the relations S A S A = A S A S , A S A κ = q − N +1 κ = κA S A . (8.20)These relations are understood in End( C N ) ⊗ End( C N ) ⊗ C ~ [ G ], and the indices distinguishthe two copies of End( C N ), in the usual way.The algebra C ~ [ G ] is a quantization of C [ G ] along the Poisson bracket (2.4). It carriesa U ~ ( g )-action, which is a deformation of the conjugation action of U ( g ) on C [ G ]. Thisaction is determined by the requirements that A commutes with ( π ⊗ id) ◦ ∆ U ~ ( g ) in thetensor product End( C N ) ⊗ C ~ [ G ] ⋊ U ~ ( g ), where π : U ~ ( g ) → End( C N ) is the representationhomomorphism. It is important that C ~ [ G ] can be realized as a U ~ ( g )-invariant subalgebra in33 q ( g ), with respect to the adjoint action. The embedding is implemented via the assignmentEnd( C N ) ⊗ C ~ [ G ] ∋ A
7→ Q ∈
End( C N ) ⊗ U q ( g ) . The following properties of C ~ [ G ] will be of importance. Denote by I ~ ( G ) ⊂ C ~ [ G ] thesubalgebra of U ~ ( g )-invariants, which also coincides with the center of C ~ [ G ]. For N = 2 n +1it is generated by the q-traces Tr q ( A l ), l = 1 , . . . , N . Not all traces are independent, butthat is immaterial for this consideration. Traces of A l are not enough for N = 2 n , and oneshould add one more invariant τ − in order to get entire I ~ ( G ). On a module of highestweight λ , this invariant returns χ λ ( τ − ) = Q ni =1 ( q λ + ρ,ε i ) − q − λ + ρ,ε i ) ), see Proposition 7.4,[3]. However, it vanishes on modules with highest weight λ ∈ C ∗ l , so we take no care of it. Theorem 8.1.
Suppose that λ = C ∗ k ,reg is admissible, and let µ be as in (7.14). The quotientof C ~ [ G ] by the ideal of relations ℓ Y i =1 ( Q − µ i ) × ( Q − µ ℓ +1 )( Q − µ ℓ +2 ) × ℓ Y i =1 ( Q − µ − i q − N +2( n i +1) ) = 0 , (8.21)Tr q ( Q k ) = ϑ k n ,q ( µ ) (8.22) is an equivariant quantization of the class lim ~ → µ = µ ∈ ˆ M K . It is the image of C ~ [ G ] in the algebra of endomorphisms of the U q ( g ) -module M λ .Proof. The proof is similar to [2], Theorem 10.1. and [3], Theorem 8.2., and we give its sketchhere. It is based on equivariant homomorphism from C ~ [ G ] to End( M λ ) ⊂ End( M λ ) ⊗ C (( ~ )),where the extension of M λ by Laurent series in ~ is taken to enable the U ~ ( g )-action. While M λ is only a U q ( g )- but not a U ~ ( g )-module, End( M λ ) is U ~ ( g )-invariant as well as the imageof C ~ [ G ] in it, see [2] for details. This homomorphism factors through a homomorphismΨ : C ~ [ G ] /J λ → End( M λ ), where J λ is the ideal generated by the kernel of the centralcharacter χ λ ; it is defined by the relations (8.22). The U ~ ( g )-algebra C ~ [ G ] /J λ is a directsum of isotypical components of finite rank over C [[ ~ ]], which follows from [21], Theorem 5.4.Therefore, the image of Ψ is free over C [[ ~ ]]. It can be shown that the algebra C ~ [ G ] /J λ is freeover C [[ ~ ]], hence (ker Ψ) = ker Ψ / ~ (ker Ψ) is isomorphically embedded in C ~ [ G ] /J λ mod ~ .The kernel ker Ψ contains the ideal J generated by the entries of the matrix polynomial inthe left-hand-side of (8.21). In the classical limit, J goes over to the defining ideal of theconjugacy class, by Theorem 2.1. The latter is a maximal proper invariant ideal and henceequal to (ker Ψ) . Therefore, the embedding J ⊂ ker Ψ is an isomorphism by the Nakayamalemma. 34heorem 8.1 describes quantization in terms of the matrix Q , which is the image of thematrix A . To obtain the description in terms of A , one should replace Q with A in (8.21)and (8.22) and add the relations (8.20).The quantization we have constructed is equivariant with respect to the standard orDrinfeld-Jimbo quantum group U ~ ( g ). Other quantum groups are obtained from standard U ~ ( g ) by twist, [22]. Formulas (8.21) and (8.22) are valid for any quantum group U ~ ( g ) uponthe following modifications. The matrix Q is expressed through the universal R-matrixas usual. The q-trace should be redefined as Tr q ( X ) = q ρ,ε ) Tr (cid:16) π (cid:0) γ − ( R ) R (cid:1) X (cid:17) = q N − Tr (cid:16) π (cid:0) γ − ( R ) R (cid:1) X (cid:17) . This can be verified along the lines of [23]. Acknowledgements . The author is extremely grateful to the Max-Planck Institute forMathematics in Bonn, where this work has been done, for warm hospitality and excellentresearch atmosphere. This study is also supported in part by the RFBR grant 09-01-00504.
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