Equivariant vector bundles over quantum spheres

We quantize homogeneous vector bundles over an even complex sphere $\mathbb{S}^{2n}$ as one-sided projective modules over its quantized coordinate ring. We realize them in two different ways: as locally finite $\mathbb{C}$-homs between pseudo-parabolic Verma modules and as induced modules of the quantum orthogonal group. Based on this alternative, we study representations of a quantum symmetric pair related to $\mathbb{S}^{2n}_q$ and prove their complete reducibility.


Introduction
In this paper we construct one-sided projective modules over the quantized coordinate ring of the even sphere S 2n , n 1, that are equivariant with respect to the quantized orthogonal group SO(2n + 1). Such modules can be viewed as quantized equivariant vector bundles naturally extending the context of the conventional quantization programme for Poisson manifolds [BFFLS].
Spheres are important manifolds in the classical geometry and topology [H], as well as for applications in other areas [Schw]. There has been a persistent interest to their non-commmutative counterparts [P, VS, FRT, Dab, DL, LM] since the invention of noncommutative geometry [C] and quantum groups [D]. Although in some aspects all quantum spheres admit a uniform description, e.g. as subvarieties of quantum Euclidian planes [FRT], the even-dimensional case has been less covered in the literature. The main reason is a general effect when a pair (group,subgroup) determining a classical homogeneous space may be destroyed in quantization. This is observed for spheres as homogeneous spaces of orthogonal groups, however the odd dimensional case allows for reduced symmetries that survive quantization. Along these lines, S 2n−1 q were approached via the quantum unitary groups in [VS] and S 4n−1 q via the quantum symplectic group in [LPR]. Projective modules over odd quantum spheres were recently studied in [Sheu]. On the contrary, S 2n lacks its quantum stabilizer subgroup but benefits from being a conjugacy class of a connected group, G = SO(2n + 1).
This enables one with the machinery of the category O to construct and study S 2n q via its faithful representation on a highest weight module, [M2]. In the present paper we extend that study to equivariant vector bundles over S 2n .
Semi-simple conjugacy classes of a simple Lie group fall into two classes depending on the type of their stabilizer subgroup. A Levi subalgebra k of a simple Lie algebra g has a basis of simple roots that is a part of the total basis. As a consequence, its universal enveloping algebra U(k) is quantized as a Hopf subalgebra in the total quantum group U q (g). A regular approach to equivariant star product quantization of vector bundles over conjugacy classes with such stabilizer subgroups was developed in [DM]. However, if some simple roots of k are not simple with respect to the fixed polarization of g, it cannot be quantized as a subgroup, which disadvantage makes this case rather challenging. Nevertheless it is known that the trivial vector bundles (algebras of functions) on such classes are quantizable, [M2].
This fact suggests that a uniform quantization scheme should exist for all vector bundles on semi-simple classes of all types (this conjecture is also supported by a local star product on S 2n constructed in [M1], similarly to the Levi case [EEM]). In the present paper we implement this programme for all equivariant vector bundles on S 2n .
The Poisson structure on S 2n under study is restricted from the Semenov-Tian-Shansky bracket on G, which is related to the Reflection Equation [KS]. It makes G a Poisson-Lie manifold over the Poisson group G endowed with the Drinfeld-Sklyanin bracket, with respect to the conjugation action. The quantum sphere is treated as a subvariety of G q and assumes an (conjugation) action of the corresponding quantum group U q (g), g = so(2n + 1), on the quantized polynomial ring C q [S 2n ] (we mean the Poisson group G that gives rise to standard U q (g), [ChP]).
One possible realization of C q [S 2n ] represents it by linear operators on a special highest weight U q (g)-module M, which we call base module [M2]. To construct an equivariant projective C q [S 2n ]-module, one should find an invariant idempotentP in the algebra End(V ) ⊗ C q [S 2n ] for some finite-dimensional U q (g)-module V . Such an idempotent projects V ⊗ M onto a direct summand submodule and therefore can be obtained from an irreducible decomposition of V ⊗ M, privided certain technical conditions are fulfilled. In the case of Levi k, submodules of highest weight in V ⊗ M are parabolically induced from irreducible U q (k)-submodules in V , which in the classical limit turn to the fibers over the initial point.
Inspite of no U q (k) ⊂ U q (g) for non-Levi k, one still may expect that the classical irreducible k-submodules in V parameterize U q (g)-submodules of highest weight in V ⊗ M as in the Levi case.
Thus the problem is to decompose V ⊗ M into a direct sum of highest weight modules, and describe the irreducible summands. The first question is discussed in general in [M3], where a criterion for complete reducibility of V ⊗ M is formulated in terms of a canonical contravariant form on V ⊗M. This form is the product of unique, up to a scalar, contravariant forms on V and M. Then V ⊗ M is completely reducible if and only if the restriction of the canonical form to the span of singular vectors in V ⊗ M is non-degenerate. We apply this criterion to the base module M and all finite-dimensional modules V and prove complete reducibility of V ⊗ M for all q not a root of unity. To answer the second question, we define pseudo-parabolic modules that generalize parabolic modules for Levi subalgebras. We prove that all highest weight submodules in V ⊗ M are pseudo-parabolic, at generic q.
As spheres are symmetric spaces, we take advantage of an alternative presentation of C q [S 2n ] via quantum symmetric pairs. That part of the paper is analogous to [M4] devoted to quantum projective spaces. There is a 1-parameter family of solutions of the Reflection Equation [KS] associated with U q (g). Every solution defines a one-dimensional representation of C q [S 2n ] (a "point" in S 2n q ) and facilitates its realization as a subalgebra in the Hopf dual T to U q (g). At the same time, it defines a left coideal subalgebra B ⊂ U q (g) such that C q [S 2n ] is identified with the subalgebra of B-invariants in T under the action by right translations.
The algebra B is a deformation of U(k ′ ), where k ′ ≃ k is the isotropy Lie algebra of the classical limit of the "quantum point".
We prove that every finite-dimensional U q (g)-module V is completely reducible over B and that simple B-submodules are deformations of irreducible k ′ -submodules. They are constructed via invariant projectors from V ⊗ M onto the corresponding pseudo-parabolic Verma submodules. Finally, we identify projective left C q [S 2n ]-modules with B-invariants in T ⊗ X, where X is a right B-module and T is equipped with the B-action by right translations. It carries a U q (g)-action via the left translations on T . This is a deformation of the classical construction of the equivariant vector bundle associated with the fiber X.
The paper is organized as follows. Section 2 is devoted to certain properties of extremal projectors that we need for construction of singular vectors. We use these results for description of irreducible submodules of V ⊗ M in Section 3, where we also prove its complete reducibility. Therein we describe the pseudo-parabolic subcategory associated with the quantum sphere. Section 4 is devoted to equvariant quantum vector bundles. We present them as projective modules over C q [S 2n ] using an equivalence with the pseudo-parabolic category and compute their equivariant Grothendieck group. Finally, we give a description of vector bundles via invariants of the coideal subalgebra from the quantum symmetric pair.
Appendix A proves that parabolic modules are locally finite over the underlying quantum Levi subgroups. Appendix B contains some useful facts about tensor products of lowest and highest weight vectors.
2 Singular vectors of U q so (5) This is a technical section devoted to certain aspects of representation theory of the quantum group U q so(5) . It is preparatory for the subsequent section, where we define and study the main tool of our approach to quantum vector bundles: generalized parabolic Verma modules.
For a detailed exposition of general quantum groups the reader is referred to [D, ChP]. A specific reminder of the odd orthogonal quantum group of rank n > 0 will be given later in Section 3.1. Here we start with the simplest interesting case of U q so(5) and study singular vectors in its modules with the help of extremal projectors shifted by a "spectral parameter" [Zh, KT]. We present them in a form that is convenient for our applications and establish some useful facts that we have not found in the literature.

Shifted extremal projector
In what follows, we use the shortcutsq = q −1 , [z] q = q z −q −z q−q −1 , and [x, y] a = xy − ayx for any a ∈ C. The complex parameter q = 0 is assumed to be not a root of unity.
Let e, f, q ±h denote generators of U q sl(2) satisfying To accommodate certain operators of interest, we need to extend U q sl(2) to an algebrâ U q sl(2) . First of all, we extend the Cartan subalgebra to the field C(q h ) of rational functions in q h . Secondly, we include formal series in f k e m of the same weight with coefficients from C(q h ). Such an extension can be done for an arbitrary quantum group, see [KT] for details.
Define a one-parameter family π(s) ∈Û q sl(2) by (2.1) More exactly, it belongs to the zero weight subalgebraÛ 0 q sl(2) ⊂Û q sl(2) . The element π(s) features the following properties, which can be checked by a direct calculation: It follows that π(h+ 1) is the extremal projector, i.e. an idempotent satisfying eπ(h+ 1) = 0, π(h + 1)f = 0, cf. [KT]. Another corollary is that π(s) is well defined on every U q sl(2)module of highest weight, where it returns on a vector v of weight µ such that e m v is the highest vector.
We consider finite-dimensional U q sl(2) -modules whose weights belong to q Z . Then the following fact is important for our exposition.
Proposition 2.1. For all s ∈ C such that q 2s ∈ −q Q , the operator π(s) is invertible on every finite-dimensional module.
Proof. Immediate corollary of (2.3) since µ(h) ∈ Z and q is not a root of unity.
Let σ denote an algebra automorphism of U q sl(2) acting by the assignment f → e, e → f , h → −h. Observe that the operator σ π(s) is well defined on every finite-dimensional module. We are going to relate it to π(s).
To that end, we extend sl(2) to g = sl(3) and use a fact that π(s) is essentially a unique element satisfying a certain identity inÛ q (g). We assume that our sl(2) corresponds to a simple root α. Let β be the other simple root and put γ = α + β.
In the algebraÛ q (g), define The vectors e γ and f γ form a quantum sl(2)-pair in U q (g).
Define a family C(m, s) of rational trigonometric functions of s parameterized by m ∈ Z + (the set of non-negative integers): Note that all terms with k > m vanish, so that the sum is finite.
Proof. Obviously the statement is true for m = 0, as the summation over k > 0 turns zero. It is easy to check that the right-hand side satisfies the identity C(m + 1, s) = 1 − Sinceπ(s) = C(h, −s) modÛ q sl(2) e on every non-negative weight, we have Applying σ toπ(s) and evaluating the result on m < 0, one arrives at π(s) = C(−h, s)
We will also need a "dynamical" version of the vectors f δ and e δ : where p = q 1 2 andp = p −1 . Introduce a one-parameter family π α (s) ∈Û q (g) by It is the image of π(s) under the embedding sl(2) ⊂ g corresponding to the short root α (the parameter q in the preceding subsections is to be replaced with q α = p).

On singular vectors in Verma modules
Recall that a weight vector in a U q (g)-module is called singular if it is annihilated by generators of U q (g + ). In this section we derive explicit formulas for singular vectors of weights µ − mδ, m ∈ Z + , in a Verma module of highest weight µ subject to certain conditions.
Proof. The Serre relations imply f α f δ = p 2 f δ f α , cf. Section 3.1. Applying Proposition 2.5 and then formula (2.2)to f αfδ (s)π α (s + 2) we write it as Evaluating these on the highest vector of a generic Verma module and replacing s with s + 1 we prove the lemma.
As a consequence, we obtain the formulâ (2.10) We will also need the commutation relation which readily follows from the defining relations of U q (g).
Proof. It was mentioned that π α (2h α + 1) is the extremal projectorÛ q g (α) , therefore e α Therefore π α (s)f m δ 1 µ is annihilated by e α indeed. Furthermore, notice that e β π α (s + 2m) ∈ U q (g)g + since π α (s + 2m) has zero weight. Using the factorization Using (2.11) for the commutator we rewrite this sum as Pushing f 2 α to the left with the help of (2.10) we get Now pushing π α back to the left with the help of (2.8) we restore the factor f m−1 δ on the right and arrive to Pushing π α (s) to the right with the use of (2.12) and applying to the highest vector we eventually obtain Therefore, the vector in question is annihilated by e β if [s+(µ, β)+m+1] q = 0. Substitution of s = 2(µ, α) − 2m + 1 translates this to the condition on the value of (µ, δ).
In the modules of our interest appearing in the next section, the highest weight µ satisfies . Hence the right-hand side in (2.12) is regular at such weights by Proposition 2.1.
Remark that the assumption on µ of Proposition 2.7 can be written as the condition where ρ is the half-sum of positive roots of so(5). This is a pole of the quantum Shapovalov determinant of the moduleM µ , [DCK].

Generalized parabolic modules
In this section we introduce and study a class of modules over the odd orthogonal quantum group that play a role of "representations" for quantized vector bundles. Some of their properties are derived from corresponding properties of parabolic modules, which are established in Section A. We show that generalized parabolic modules form a nice category that is equivalent to the category of classical finite-dimensional modules over the even orthogonal group, which is the stabilizer of the initial point on the even sphere. In the classical geometry, its modules are fibers over the initial point. They will also parameterize quantized vector bundles in the subsequent section.

The odd orthogonal quantum group
From now on g is the Lie algebra so(2n + 1), n ∈ N, and h ⊂ g its Cartan subalgebra. An ad-invariant form identifies g with g * and h with h * . Let ( . , . ) designate its restriction to h. Denote by R its root system and fix the subset of positive roots R + with basis Π + . We normalize the inner product so that the length of short roots is 1. Then R + contains an By U q (g) we understand the standard orthogonal quantum group [ChP] over the complex field with the set of generators e α , f α , and invertible q hα satisfying The generators e α and f α , α ∈ Π + , obey the Serre relations, Note thatê δ andf δ are deformations of classical root vectors. Although they do not form a quantum sl(2)-pair, they play a role in what follows.
and {f α } α∈Π + , respectively. The Lie subalgebra l = gl(n) ⊃ h with the basis of simple roots is a Levi subalgebra in g. Its universal enveloping algebra is quantized as a quantum subgroup U q (l) ⊂ U q (g). On the contrary, the Lie subalgebra k = so(2n) ⊂ g with the basis of simple roots Π + k = {δ, α 2 , . . . , α n } does not have a natural quantum analog of U(k) ⊂ U(g).
For each α ∈ R + we denote by g (α) the corresponding sl(2)-subalgebra in g. If α is simple, then U q (g (α) ) is a quantum subgroup in U q (g). For compound α, there is a triple of elements (not unique) in U q (g) generating a U q sl(2) -subalgebra (but not a Hopf one), which is a deformation of U(g (α) ). Its generators of weights ±α enter a Poincare-Birkhoff-Witt (PBW) system delivering a basis in U q (g ± ), [ChP]. We denote this subalgebra by U q (g (α) ) assuming its root vectors fixed.
. We also use this definition for classical h-modules (replacing e µ with q µ ) with the obvious definition of weight subspaces.
for all µ ∈ h * . The usual rules for characters hold true: given a short exact sequence and ch(W 3 ) ch(W 2 ). Also, ch(W 1 ⊗ W 2 ) = ch(W 1 )ch(W 2 ) for any W 1 and W 2 .
Recall that finite-dimensional U q (g)-modules are U q (h)-diagonalizable, and the eigenvalues of q hα belong to ±q Z . Throughout the text, we mean by finite-dimensional U q (g)-modules only those whose q hα -eigenvalues belong to q Z and denote their category by Fin q (g). Such modules are in one-to-one correspondence with finite-dimensional U(g)-modules and have the same characters. This weight convention does not apply to finite-dimensional U q (l)-modules and infinite-dimensional U q (g)-modules.
The algebra U q (g) and its modules appearing in this exposition rationally depend on the parameter q which can be specialized to complex numbers except for a certain set of values.
By all q we mean all q that are not a root of unity. Saying almost all we exclude a finite set of q distinct from 1. We say that a property holds for generic q if it is true for almost all q when restricted to every finite-dimensional subspace. That can be formalized by passing to the local ring of rational functions regular at q = 1.

Extremal twist and complete reducibility of tensor products
Let us recall the construction of extremal twist, which is responsible for irreducible decomposition of tensor product V ⊗ Z of two highest weight modules [M3].
Consider an irreducible highest weight U q (g)-module Z as a U q (g − )-module and denote Now let V and Z be a pair of irreducible U q (g)-modules of highest weights ν and ζ, respectively, and highest vectors 1 ν and 1 ζ . Denote by V + Z the kernel of the left ideal I + Z in V and similarly define Z + V . There are linear isomorphisms between V + Z , Z + V , and the subspace (V ⊗ Z) + spanned by singular vectors in V ⊗ Z. To describe this correspondence, where the terms with factors of other weights are suppressed. The isomorphisms are given by the assignmentsδ l (u) = v ν−ζ andδ r (u) = v µ−ν . We denote by δ l and δ r their inverse isomorphisms.
Let ω denote an anti-algebra involution on U q (g) defined as γ −1 • σ. This map preserves the comultiplication on U q (g). Choose a weight basis in V and present a singular vector u This map is independent of the choice of f i in the presentation of u and called extremal twist.
Define a canonical bilinear form on V ⊗ Z as the product of the ω-contravariant forms on V and Z (such forms do exist, are unique up to a scalar multiplier and are non-degenerate).
As ω is a coalgebra map, the canonical form is also contravariant. If ., . is the contravariant form on V , then the form θ V,Z (.), . is the pullback of the canonical form via the isomorphism Similarly one can consider θ V,Z instead of θ Z,V due to the obvious symmetry between V and Z.
Theorem 3.1 ( [M3]). The following are equivalent iv) V ⊗ Z is the sum of submodules of highest weight.
Practically we use ii) and iii) in order to check i); the criterion iv) is included for completeness, As an example, consider the case whith g = sl(2) and Z a Verma module of highest This can be done via presenting This lift can be easily computed for the sl(2)-case. For Z a Verma module, θ V,Z coincides with γ −1 (F 2 )F 1 represented in End(V ). The latter can be expressed through the operators π(s), then (2.3) translates to (3.13).

Description of base module and its generalized extremal spaces
The base module M for quantum S 2n has a PBW basis that makes it isomorphic to the vector space of polynomials in n variables. To describe this basis, we need root vectors of weights −ε i for i = 1, . . . , n. Define e ε 1 = e α 1 and f ε 1 = f α 1 and proceed recurrently for i > 1 as Fix λ ∈ h * by the condition q 2(λ,ε i ) = −q −1 for all i = 1, . . . , n and (α i , λ) = 0 for i > 1. We call it base weight. The Verma moduleM λ of highest weight λ with the canonical generator 1 λ has singular vectors f α i 1 λ with i > 1 andf δ 1 λ . Define M as the quotient ofM λ by the sum of submodules generated by these vectors. As proved in [M1], it has a basis of weight vectors f m 1 ε 1 . . . f mn εn 1 λ where m i take all possible values in Z + (we use the same notation for the image of 1 λ in M). In particular, ch Lemma 3.2. The quantum group U q (g) acts on M by where the suppressed numerical factors are non-zero once q is not a root of unity.
Proof. By weight arguments the action has the specified form up to numerical factors. That they are not zero can be proved as follows. First of all, they are given explicitly for the f α iaction in Lemma 3.2 in [M1]. One can check that as operators in End(M). Using these relations, one can easily prove that the factors arising from the e α i -action are non-zero too.
Remark that irreducibility of M readily follows from these formulas since all weights in M are multiplicity free, and M clearly has no singular vectors. Alternatively, that can be proved via the contravariant form on M, which is found to be non-degenerate in [M1].
Denote by ρ ∈ h * the half-sum of positive roots and put α ∨ = 2α (α,α) for all α ∈ h * . Fix a finite-dimensional U q (g)-module V of highest weight ν and introduce ℓ i = (ν +ρ, α ∨ i )−1 ∈ Z + for i = 1, . . . , n. It is known that V is a quotient of the Verma moduleM ν by the submodule [Jan]. Set I + V = σ(I − V ). Then singular vectors in V ⊗ M are parameterized by M + V = ker I + V ⊂ M, as explained above. where k i > ℓ i for some i = 1, . . . , n.
Proof. Observe that ω( which proves the right equality. The left equality follows from Lemma 3.2. Since ω(I + V )M is orthogonal to M + V and the form is non-degenerate due to irreducibility of M, the proposition is proved.
Singular vectors in V ⊗ M can be alternatively parameterized by the subspace V + M ⊂ V , which is the joint kernel of {e α i } n i=2 andê δ . The weight subspaces in V + Z have dimension 1 and carry weights ν − n i=1 m i ε i , with 0 m i ℓ i . In the classical limit, V + M is spanned by highest vectors of irreducible k-modules, according to the Gelfand-Zeitlin reduction.
Consider, for example, the case of g = so(5) and the U q (g)-module V of highest weight ν = 3ν 1 + 2ν 2 , where ν i are fundamental weights, (ν i , α ∨ j ) = δ ij , i, j = 1, 2. The subspace M + V is spanned by f m 1 ε 1 f m 2 ε 2 1 λ with m 1 3 and m 2 2. Its reciprocal space V + M is spanned by vectors of weights ν − m 1 ε 1 − m 2 ε 2 of multiplicity 1, which are depicted on the weight lattice of V by the fat circles: An interesting problem is evaluation of the quantum reduction algebra of the pair (g, k) or, more exactly, of the left ideal U q (g)I + M . Its positive part is generated by a pair of commuting elements z ε 1 = e ε 1 [h α 2 ] q − qe ε 2 f α 2 and z ε 2 = e ε 2 . It allows to restore V + M from the vector of lowest weight in V + M . Negative generators, which are unknown to us, would deliver V + M out of the highest vector of V .
The subspace V + M is spanned by "singular vectors" of a "subalgebra" U q (k) ⊂ U q (g), which is also unknown (and whose existence is under question). We believe that such a subalgebra does exist, maybe in an appropriate extension of U q (g). One indication in favor of this conjecture is the presence of a coideal subalgebra, which is a quantization of the stabilizer of a different point in S 2n , cf. Section 4.3.

Pseudo-parabolic Verma modules
Suppose that k ⊂ g is a reductive subalgebra of maximal rank. Suppose that its polarization is induced by the polarization of g, so there are inclusions R ± k ⊂ R ± g of their root subsystems. Take a weight ξ ∈ h * that is integral dominant with respect to k and let X be the irreducible finite-dimensional k-module X of highest weight ξ. For each α ∈ Π + k the sum of ξ and the base weight λ satisfies the condition q 2(ξ+λ+ρ,α)−mα(α,α) = 1, (3.14) where m α = (ξ, α ∨ )+1 a positive integer. Then there is a submoduleM ξ+λ−mαα in the Verma moduleM ξ+λ , [DCK]. Let M X,λ denote the quotient ofM ξ+λ by the sum ofM ξ+λ−mαα over all α ∈ Π + k . If k were a Levi subalgebra in g, that is, if Π + k ⊂ Π + g , then M X,λ would be a parabolic Verma module. That justifies the name of pseudo-parabolic Verma module for M X,λ when Π + k ⊂ Π + g . For non-Levi k, M X,λ is factored through a parabolic module MX λ relative to the maximal Levi subalgebra l among those sitting inside k. The finite-dimensional U q (l)-moduleX λ = X ⊗ C λ is of the same highest weight ξ + λ. As a U q (l)-module, MX λ is isomorphic to U q (n − ) ⊗X λ , where U q (n − ) ⊂ U q (g) is a subalgebra invariant under the adjoint action of U q (l). It is a deformation of U(n − ), where n − = g − ⊖ l − , see Section A.3 for details.
In the case of concern, l = gl(n), and M X,λ is a quotient of MX λ by the submodule generated by the image of π α 1 (2h α + 1)f m δ 1 ξ+λ with m = m δ . By Proposition 2.7, it is singular unless it is zero.
Thus M X,λ is a quotient of MX λ by a proper submodule. Since MX λ is locally finite over U q (l) by Proposition A.3, so is M X,λ . Proof. In the classical limit q → 1, the vectorf mδ (s) goes over to f m δ , and the U q (g − )-module M X,λ turns into the induced module U(g − ) ⊗ U (k − ) X. Therefore ch(M X,λ ) ch(M)ch(X) for generic q, since deformation does not increase quotients.

Pseudo-parabolic category O(S 2n )
In this section we prove that generalized parabolic modules of highest weight are irreducible, and tensor products of the base module with finite-dimensional modules are completely reducible. We end up showing that the corresponding subcategory in the category O is equivalent to the category of finite-dimensional k-modules.
Fix a finite-dimensional U q (g)-module V . The subspace V + M ⊂ V is the kernel of the left ideal I + M ⊂ U q (g + ) generated by e α i , i > 1, and byê δ (s), with s = 2(λ, α 1 ) − 1 (note that e δ (s) ∝ê δ mod U q (g + )e α 2 for such s). In the classical limit, V + M is the kernel of k + .
Denote by M ′ X i ,λ the image of the module M X i ,λ in V ⊗ M, then ch(M ′ X i ,λ ) ch(M X i ,λ ). We aim to prove that V ⊗ M is a direct sum of M ′ X i ,λ and essentially M ′ X i ,λ = M X i ,λ . The proof will be done by induction on the rank of g. For each k = 1, . . . , n, we identify the Lie subalgebra g (k) = so(2k + 1) ⊂ g whose root basis is {α 1 , . . . , α k }. The base module M contains base modules for S 2k , which we denote by M (k) . Obviously, M (k) ⊂ M (k+1) for all k < n.
Theorem 3.7. Let V be a finite-dimensional U q (g)-module. Then

V ⊗ M is completely reducible for all q and splits into a direct sum
3. M X,λ ≃ M ′ X,λ for generic q.
Note that irreducibility of M ′ X i ,λ is equivalent to complete reducibility of V ⊗ M, by Theorem 3.1, iii). Below we give a full proof only for generic q. For all q, complete reducibility of V ⊗M follows from a direct calculation of θ M,V that employs a relation of θ M,V with extremal projectors, in generalization of (3.13). That is based on a more advanced development of [M3] and done in [M5].
Proof. Consider the case of n = 1. Then g ≃ sl(2), k = h, and M is a Verma module. Let ν be the highest weight of V . The extremal twist acts on a vector f m ε 1 1 ν ∈ V of weight ξ = ν − mε 1 as in (3.13), where α = ε 1 . It does not vanish once q 2(λ,ε 1 ) = −q −1 , therefore This proves the theorem for n = 1 and all q.
Suppose that the theorem is proved for n 1. An irreducible U q (g (n+1) )-module V is parameterized by the highest weight ν = l 1 ε 1 + . . . + l n+1 ε n+1 with 0 l i l i+1 . Observe through the Gelfand-Zeitlin reduction that all irreducible U q (g (n) )-modules can be obtained by considering the special case of l n = l n+1 . This corresponds to the zero n + 1-th coordinate in the expansion of ν over the fundamental weights. Then V + M is contained in the U q (g (n) )submodule V (n) generated by the highest vector of V , and the corresponding singular vector lies in V (n) ⊗ M (n) . By the assumption, it is completely reducible over U q (g (n) ), therefore θ V (n) ,M (n) is bijective. Since the extremal twist is independent of the choice of lift for singular vectors, θ V (n) ,M (n) = θ V,M , and thus θ V,M is bijective. Therefore V ⊗M is completely reducible for special V (and for all q). Furthermore, Corollary 3.5 implies These lines can be obtained from three special modules of highest weights ℓ(ε 1 + ε 2 ) with ℓ = 3, 5, 7. They are marked on the diagram with large circles. Their weight subspaces are in U q (g (1) )-submodules V (1) of dimensions 4, 6, 8, therefore we can restrict to U q (g (1) ) when calculating singular vectors.
Conjecture 1. For any irreducible k-module X, the module M X,λ is isomorphic to M ′ X,λ for all q.
Let us comment that its proof is equivalent to computing ch(M X,λ ). In the light of Theorem 3.7, 3), the inequality of Corollary 3.5 becomes equality at generic q and can be rectified as ch(M X,λ ) ch(M)ch(X) for all q.
Recall that the category O is formed by finitely generated U q (h)-diagonalizable U q (g)modules that are locally finite over U q (g + ). Define pseudo-parabolic category O(S 2n q ) as the full subcategory in O whose objects are submodules in V ⊗ M, where V ∈ Fin q (g). Theorem 3.7 can be reworded as Proposition 3.8. O(S 2n q ) is a semi-simple Abelian category equivalent to Fin(k). It is a Fin q (g)-module category generated by M.

Proof. The category O(S 2n
q ) is additive semi-simple, hence it is Abelian. By construction, it is a module category over Fin q (g).
The functor O(S 2n q ) → Fin(k) is constructed as follows. Given a module M 1 from O(S 2n q ) defineX λ to be its finite-dimensional U q (l)-submodule generated by the span of singular vectors in M 1 . LetX 0 be the classical limit ofX λ ⊗ C −λ equipped with the corresponding action of U(l). Proceed to the k-module parabolically induced fromX 0 and then to its finite-dimensional quotient X by the maximal submodule that has zero intersection withX 0 .
These steps, apart from maybe the last one, respect morphisms. But morphisms of parabolic k-modules induced from the l-modules descent to morphisms of their finite-dimensional quotients. Now let M 2 be another module from O(S 2n q ) and defineY λ ,Y 0 , and Y accordingly. We have where + designates span of singular vectors. These imply the isomorphisms The middle isomorphism is due to [EK], while the other two are consequences of semisimplicity.
Finally, every module from Fin(k) is isomorphic to one constructed this way because it can be realized as a submodule of a module from Fin(g).
In the next section we consider two other Fin q (g)-module categories: equivariant projective modules over C q [S 2n ] and representations of a coideal subalgebra in U q (g). These categories will be shown equivalent to O(S 2n q ).
Since the two coproducts are conjugated via an R-matrix, this modification does not affect the conclusions of the previous sections.
In the deformation context, we pass to the C[[ ]]-extension of U q (g) completed in theadic topology, which we denote by U (g). Note that the C[[ ]]-extension of the base module does not support the action of U (g). Nevertheless, the U (g)-action on End(M) is well defined.

Given a C-algebra A we call a C[[ ]]-algebra A quantization of A if it is a free C[[ ]]module and A / A is isomorphic to A as a C-algebra. The quantization is called equivariant
if A is a U (g)-module algebra such that the U (g)-action is a deformation of a U(g)-action -extension of a U q (g)-algebra A q , then A q is also called (equivariant) quantization of A. This should not cause a confusion as the ring of scalars is always clear from the context.
Suppose that Γ is a projective A-module with an U(g)-action such that the multiplication A ⊗ Γ → Γ is equivariant. Suppose that A is an equivariant quantization of A. We call a projective left A -module Γ quantization of A if Γ is also a U (g)-module such that the multiplication A ⊗ Γ → Γ is equivariant and the action of U (g) is a deformation of the action of U(g). Equally we consider the version of right A -modules Γ . Our convention about indexing with q instead of applies to A q -modules as well.
By the Serre-Swan theorem [S, Sw], finitely generated projective A-modules are sections of vector bundles in the classical algebraic geometry. Such modules over A can be regarded as quantized vector bundles.

Projective modules over
By a classical equivariant vector bundle over S 2n with fiber X we understand the (left or right) finitely generated projective C [S 2n ]-module Γ(S 2n , X) of its global sections. It can be realized as the subspace of k-invariants in C[G] ⊗ X, where G is either SO(2n + 1) or its simply connected covering if X is a spin representation of k.
The quantum polynomial algebra C q [S 2n ] is represented as a subalgebra A ⊂ End(M), cf. [M2]. In accordance with the above definitions, quantization of an equivariant vector bundle on S 2n is a U q (g)-equivariant deformation of Γ(S 2n , X) in the class of right C q [S 2n ]-modules.
idempotents can be constructed via decomposition of V ⊗ M due to the following fact.
One can prove it using the fact that A exhausts all of the locally finite part of the U q (g)module End(M), which can be checked with the help of Corollary B.2. This is the answer to a Kostant's problem for quantum groups, [KST].
For each weight vector v ∈ V + M there is a projectorP ∈ End(V ) ⊗ A to a particular copy of M ′ X,λ generated by the singular vector δ l (v) ∈ V ⊗ M.
Proof. We can assume that X is irreducible of highest weight ξ. The U q (g)-moduleP (V ⊗A) is isomorphic to the locally finite part of Hom C (M, M X,λ ). Let us prove that it has a similar U q (g)-module structure as the U(g)-module Γ(S 2n , X).
First of all, observe that the preceding considerations can be conducted in the opposite category O of modules whose weights are bounded from below, leading to similar conclusions with regard to semi-simplicity etc. Denote by N X,λ the module of lowest weight −ξ − λ that is opposite to M X,λ . The annihilator of its lowest vector is obtained from I − M X,λ by the involution σ. Let N ′ X,λ ≃ (M ′ X,λ ) * stand for its irreducible image in V * ⊗ N. Since M ′ X,λ and M are irreducible, the vector space Hom W, Hom C (M, M ′ X,λ ) is isomorphic to Hom N ′ X,λ ⊗ M, W * for any finite-dimensional module W . Using Corollary B.2, one can show that Hom N ′ X,λ ⊗ M, W * ⊂ Hom N X,λ ⊗ M, W * as vector subspaces in W * , and the latter is isomorphic to Hom k (X * , W * ) ≃ Hom k (W, X). Include X in an irreducible decomposition V = ⊕ i X i over k and write Since the rightmost term is isomorphic to ⊕ i Hom k (W, X i ), we get Hom (W, Hom C (M ′ X i ,λ ) ≃ Hom k (W, X i ) for all X i by dimensional reasons.
Denote by Proj(A, g) the additive category of equivariant finitely generated projective right A-modules. The morphisms in Proj(A, g) commute with the left action of U q (g) and the right action of A. It is a module category over Fin q (g), since tensor product with a module from Fin q (g) preserves direct sums.
, where ξ is the highest weight of a k-submodule X ⊂ V in the classical limit.
In conclusion of this subsection, we compute the equivariant group K 0 of S 2n q . Let R(g) denote the representation ring generated by isomorphism classes of finite-dimensional gmodules. It is known to be isomorphic to Z(Λ) W , where Λ is the integral weight lattice, Z(Λ) the group ring, and W the Weyl group [FH]. The isomorphism is implemented by the character ch. The ring R(k) has an additional structure of (left) R(g)-module via the homomorphism R(g) → R(k) induced by the restriction functor g ↓ k.
The representation ring of finite-dimensional U q (g)-modules of quasiclassical type is isomorphic to the classical representation ring R(g). Let K 0 (A, g) denote the Grothendieck group of the category Proj(A, g).
Theorem 4.5. The group K 0 (A, g) is a left R(g)-module isomorphic to R(k).
Proof. The isomorphism of Abelian groups readily follows from the equivalence of categories Proj(A, g) ∼ O (S 2n q ) ∼ Fin(k), established by Propositions 3.8 and 4.3. The isomorphism of R(g)-modules is a consequence of the isomorphisms Hom k (X, V ) ≃ Hom Uq(g) (M X,λ , V ⊗ M) for all k-modules X and all U q (g)-modules V , by Theorem 3.7, 1).

Coideal stabilizer subalgebra
As for projective spaces in [M4], we give an alternative realization of quantum vector bundles over spheres in terms of quantum symmetric pairs. Here we pass to the left A-module version of vector bundles.
Let T denote the Hopf dual of U q (g) that is a quantization of the function algebra on the spin group covering SO(2n + 1). It contains the quantum function algebra of the orthogonal group, which is generated by matrix coefficients T ij , i, j = 1, . . . , 2n + 1, of the natural representation in C 2n+1 . The matrix T is invertible with (T −1 ) ij = γ(T ij ), where γ stands for the antipode of T . There are two commuting left and right translation actions of U q (g) on T expressed through the Hopf paring and the comultiplication in T by h ⊲ a = a (1) (h, a (2) ), a ⊳ h = (a (1) , h)a (2) , a ∈ T , h ∈ U q (g).
They are compatible with multiplication on T .
Let R be a universal R-matrix of U q (g). Fix a representation of U q (g) such that the image R ∈ End(C 2n+1 ) ⊗ End(C 2n+1 ) of R is proportional to the orthogonal R-matrix from [FRT]. The element R 21 R commutes with the coproduct ∆(x) for all x ∈ U q (g). Its image Q in End (C 2n+1 ) ⊗ End(M) is a matrix whose entries generate A. The matrix with c ∈ C\{0}, solves the Reflection Equation where the indices mark the tensor factors. It also satisfies other equations of the quantum sphere, cf. [M2], and defines a one-dimensional representation χ : A → C, Q ij → A ij . The assignment Q → T −1 AT extends to an equivariant embedding A ⊂ T , where T is regarded as a U q (g)-module under the left translation action. The character χ factors to a composition of this embedding and the counit ǫ. The entries of the matrix R 21 A 1 R 12 ∈ End(C 2n+1 ) ⊗ U q (g) generate a left coideal subalgebra B ⊂ U q (g), such that a ⊳ b = ǫ(b)a for all b ∈ B and a ∈ A.
It is a deformation of U(k ′ ), where k ′ ≃ so(2n).
Proposition 4.6. 1. Every finite-dimensional right U q (g)-module V is completely reducible over B.

Every irreducible
The proof is similar to [M4]. Note that Proposition 4.6 holds for almost q for each V .
Proposition 4.6 also can be given a category theoretical interpretation. Denote by Fin q (k ′ ) the additive category of B-modules whose objects are submodules in some V from Fin q (g) and morphisms are B-invariant maps. Since it is semi-simple, it is Abelian and it is also a module category over Fin q (g). Given an equivariant projective A-module Γ, the assignment Γ → Γ ⊗ A C via the character χ is a functor Proj(A, g) → Fin q (k ′ ). This is an equivalence of module categories.

Vector bundles via symmetric pairs
The realization of quantum vector bundles as linear maps between pseudo-parabolic Verma modules has no classical analog. In this section we follow a different approach presenting an associated vector bundle by B-invariants in the tensor product of T and a B-module. This construction is quasi-classical: in the limit q → 1 we recover the standard construction of equivariant vector bundles as induced modules. It is convenient to pass to left A-modules, which corresponds to the right coset picture.
It is known that every finite-dimensional right U q (g)-module V is a left T -comodule.
We use a Sweedler-like notation for the left coaction We define a left U q (g)-action on V by h ◮ v = v ⊳ γ(h), h ∈ U q (g), v ∈ V , and consider A ⊗ V as a left U q (g)-module. The tensor product T ⊗ V is also a left U q (g)-module with respect to the left translations on T and the trivial action on V .
LetP ∈ End(V ) ⊗ A be a U q (g)-invariant idempotent and P = (id ⊗ χ)(P ) its B- on V ) are isomorphic as left A-modules and U q (g)-modules, [M4].
Theorem 4.7. The A-module (T ⊗ X) B is a quantization of the vector bundle Γ(S 2n , X).
By considering vector (integer-spin) modules, one can restrict consideration to T the function algebra on the quantum orthogonal group.

A Parabolic Verma modules
It is known that a parabolic Verma module over a classical universal enveloping algebra U(g) of a simple Lie algebra g relative to a Levi subalgebra l ⊂ g is locally finite over U(l), cf. [Hum]. The goal of this auxiliary section is to prove an analogous statement for quantum groups. We are not aware if this topic is covered in the literature and include it for completeness.
We start with a few general definitions. Let A be an associative algebra with unit. Fix an element a ∈ A and define its quasi-normalizer QNorm(a) ⊂ A as the subset of elements b ∈ A such that a m+k b = b k a k for some m ∈ Z + , b k ∈ A and all positive k ∈ Z + . Clearly it is a unital subalgebra in A.
Here is a way of checking if b ∈ QNorm(a). Consider a sequence of complex numbers Proof. One can prove by induction that a n b = n k=0 c n,k D k a (b)a n−k for all positive integer n, where c n,k are some complex coefficients. It follows that D l a (b) = 0 for l > m, and a n b = ( m k=0 c n,k D k a (b)a m−k )a n−m , as required.
We call an element a ∈ A quasi-normal if QNorm(a) = A. Their set is stable under the group of automorphism of A. It is straightforward that invertible elements of A are quasi-normal, as well as elements of the center of A. We aim to prove that Lusztig root vectors in U q (g) are quasi-normal.
Recall that a total order in R + is called normal if every sum of positive roots is between the summands. Fix such an order and set α i ∈ R + , i = 1, . . . , N = #R + . Then there exists a system of "root vectors" {e i } N i=1 of weights α i such that the monomials {e m 1 1 . . . e m N N } m i ∈Z + deliver a PBW basis in U q (g + ). Similarly there are negative root vectors f i ∈ U q (g − ) giving rise to a PBW basis of ordered monomials in U q (g − ).
Proposition A.2. The Lusztig root vectors are quasi-normal.
Proof. It is sufficient to prove that generators of U q (g) belong to QNorm(e i ) for each e i . First suppose that e i = e α for a simple root α. Then clearly U q (h) ⊂ QNorm(e α ). Furthermore, e n α f β = (δ αβ f β e α +[n] q [h α +n−1] q )e n−1 α , so U q (g − ) ⊂ QNorm(e α ). Finally, the Serre relations can be written as D m eα (e β ) = 0 for certain integer m and a sequence (q i ) m i=1 depending on α and β. Then by Proposition A.1, U q (g + ) ⊂ QNorm(e α ), and therefore e α is quasi-normal. Now observe that compound root vectors are obtained from the Chevalley generators via automorphisms of U q (g), [ChP]. This completes the proof.
Let l be a Levi subalgebra in g with a basis of simple roots Π + l ⊂ Π + g . One can choose a normal order so that R + \R + l < R + l . Then Span{f m 1 1 . . . f m l l } m i ∈Z + , where l = #(R + \R + l ), is a subalgebra, cf. [KT1], Proposition 3.3 (see also [Ke]). It is a quantization of the U(n − ), where n ± are the nil-radicals of the parabolic Lie algebras p ± = l + g ± . Similarly Span{e m l l . . . e m 1 1 } m i ∈Z + is a subalgebra, U q (n + ). The PBW factorization then yields U q (g) = U q (n − )U q (p + ) with U q (p + ) = U q (l)U q (n + ). Now we use an alternative construction of parabolic Verma modules via parabolic induction. GivenX ∈ Fin q (l) (relax the quasi-classical weight convention) make it a U q (p + )module by setting it trivial on U q (n + ). Then MX = U q (g)⊗ U (p + )X is said to be parabolically induced fromX. In the special case of irreducibleX, this construction recovers the parabolic Verma module as the quotient of the ordinary Verma module whose highest weight is that ofX. The PBW factorization of U q (g − ) implies that MX is a free U q (n − )-module generated byX.
Recall that an A-module V is called locally finite if for each v ∈ V the cyclic submodule Av ⊂ V is finite-dimensional. An element a ∈ A is called locally nilpotent if a n v = 0 for all v ∈ V and some positive n ∈ Z + depending on v.
Proposition A.3. The parabolic module MX is locally finite over U q (l).
Proof. It is sufficient to prove that MX is locally finite over U q (l − ) since it is clearly so over U q (h) and U q (l + ). Indeed, every f α with α ∈ R + l is locally nilpotent onX and therefore on MX ≃ U q (n − )X since it is quasi-normal. Therefore all but a finite number of PBWmonomials in U q (l − ) vanish on every v ∈ MX . This implies the statement.
It follows that MX is completely reducible over U q (l) and is an infinite direct sum of irreducible finite-dimensional modules. As another consequence, U q (n − ) is a locally finite adjoint U q (l)-module because it is isomorphic to the parabolic Verma module induced from the trivial representation of U q (p + ).

B Tensor product of modules of highest and lowest weight
For reader's convenience, we prove some facts that we use in the proof of Proposition 3.6.
Recall that a lower Verma module is freely generated over U q (g − ) by its lowest weight vector.
Lemma B.1. LetM µ be a Verma module of highest weight µ andN ν a Verma module of lowest weight µ. Then the tensor productM µ ⊗N ν is isomorphic to the U q (g)-module Ind g h (C µ+ν ) induced from the U q (h)-module C µ+ν .
Proof. Let us prove that the homomorphism Ind g h (C µ+ν ) →M µ ⊗N ν determined by the assignment d : x → ∆(x)(1 µ ⊗ 1 ν ), x ∈ U q (g), is an isomorphism. Introduce a Z-grading on U q (g ± ) by assigning degree 1 to the Chevalley generators and denote by U k ± their homogeneous component of degree k. This makes the vector spaces Ind g h (C µ+ν ) = U q (g − )U q (g + )1 µ+ν andM µ ⊗N ν graded by the semi-group Z 2 + . The map d is an isomorphism on the components (0, j) for all j ∈ Z + . Since f α , α ∈ Π + α , are quasi-primitive, the map d sends the U i − U j + 1 µ+ν onto the U i − 1 µ ⊗ U j + 1 ν modulo l<i,k j U l − 1 µ ⊗ U k + 1 ν . Induction on i proves that d is surjective and therefore injective on l i,k j U l − U k + 1 µ+ν for all j, as the graded components are finite-dimensional.
It follows from Lemma B.1 that Hom(M ⊗N , V ) ≃ V [µ + ν] for any V . Suppose {f i } ⊂ U q (g − ) are such that f i 1 µ are singular vectors inM µ and let M be the quotient ofM by the sum of submodules they generate. Similarly let {e j } ⊂ U q (g + ) be such that e i 1 ν are killed by all Chevalley generators f α and let N denote the corresponding quotient ofN ν .