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Introduction
In this paper, we incorporate an example of homogeneous space with non-Levi stabilizer into a uniform quantization scheme for closed conjugacy classes of simple algebraic groups. Originally, this approach was developed in 2003 for Levi classes and utilized the presence of quantum isotropy subgroup in the total quantum group, [1,2]. The key distinction of non-Levi classes is the absence of a natural candidate for such a subgroup because its root basis cannot be made a part of the total root basis. Still the coordinate ring of the class can be quantized by an operator realization on certain modules [3]. Such a quantization is formulated in terms of generators and relations and is not apparently local. On the other hand, a dynamical twist constructed from the Shapovalov form yields a local version of the star product on Levi classes [1,2] (see Dedicated [4,5] for coadjoint orbits with the Kirillov bracket). It is natural to extend that approach to all closed conjugacy classes. Such a possibility for S 4 was pointed out without proof in [6]. Here we give a solution for all even-dimensional spheres.
Sphere is a relatively simple curved space endowed with a rich structure that has numerous applications. An interest to its quantum version started to grow with the invention of quantum groups [7], and S 2 q was the first quantum G-space [8] after Manin's C 2 q [9]. A review of various constructions of the q-sphere in small dimensions and some references to its applications can be found in [10].
An even sphere admits several independent although isomorphic equivariant quantizations: a subvariety of the quantum Euclidean plane [11], an induced representation of a quantum symmetric pair (cf. [12]), and a subalgebra of linear operators on a highest weight module of the orthogonal quantum group [13]. Each particular reincarnation has its pluses that help tackling hard issues arising in other approaches. For instance, the operator realization of C q [S 2n ] allows to study representations of the coideal subalgebra in the corresponding symmetric pair [12]. All realizations of C q [S 2n ] known to date appeal to generators and relations. At the same time, a local formulation may be of interest for some applications, like Fedosov's star product approach to the index theorem [14]. The present work fills that gap. Note that, like in the Levi case [1], this problem can be placed in a more general context of quantum vector bundles addressed in [12]. This is also a part of the Gelfand-Zetlin reduction for orthogonal quantum groups, that is open by now. It turns out that local quantization of the function algebra on S 2n (the trivial bundle) can be done with elementary means and deserves a special consideration.
The original approach to the star product on Levi classes was as follows. Let k ⊂ g be the isotropy Levi subalgebra of a point t and p ± ⊂ g its parabolic extensions. One associates with t a certain weight λ ∈ h * and a pair of modules M λ , N λ of, respectively, highest and lowest weights λ and − λ. There is a (essentially unique) U q (g)-invariant form M λ ⊗ N λ → C, which is non-degenerate if and only if the modules are irreducible. In that case, there exists the inverse form C → N λ ⊗ M λ and its lift assigning 1 → F ∈ U q (p + ) ⊗ U q (p − ) (a completed tensor product). The element F gives rise to a "bidifferential" operator via the left coregular action on the Hopf dual A = U * q (g). With this operator, the multiplication in A is twisted to a non-associative operation invariant under the right coregular action of U q (g). The key observation is that the new multiplication becomes associative when restricted to the subspace A k of U q (k)-invariants in A. As a (right) U q (g)-module, A k has the same structure as the U (g)-module C[G/K ], where K ⊂ G is the centralizer subgroup of the point t. Hence, A k is a flat deformation of C[G/K ]. It is known that the initial star product on A is local [15]; therefore, the resulting multiplication is local as well.
In the non-Levi case, one can go along those lines and define A k as the joint kernel of certain operators that deform generators of k. Then the new product will be associative on A k as in the Levi case [16]. However, those operators do not close up to a deformation of U q (k), so one cannot be sure that A k has the proper size. (Observe that kernel can decrease under deformation.) Therefore, the problem is to check the size of A k . We do it for S 2n via a harmonic analysis relative to quantized SO(2n + 1).
Note that odd-dimensional spheres belong to the second connected component of the orthogonal group O(2n), and the current methods are not directly applicable.
The paper consists of five sections: after Introduction, we recall quantization of C[S 2n ] via operator realization on a highest weight module M λ in Sect. 2. In Sect. 3 we construct a system of vectors that spans M λ . We prove it to be a basis in Sect. 4 by computing the Shapovalov form on M λ . This way, we show that M λ is irreducible and the form is invertible. In the final section we show that for finite U q (g)-module V q , the dimension of V k q is equal to dim V k of the classical k-invariants. We do it via realization of V q with dim V k q > 0 in the coordinate ring of the quantum Euclidean plane C 2n+1 q .

Operator realization of C q [S 2n ]
Throughout the paper, g stands for the Lie algebra sp(2n + 1). We are looking for quantization of the polynomial ring C[S 2n ] that is invariant under an action of the quantized universal enveloping algebra U q (g). We regard S 2n as a conjugacy class of the Poisson group G = SO(2n + 1) equipped with the Drinfeld-Sklyanin bracket corresponding to the standard solution r ∈ g⊗g of the classical Yang-Baxter equation [7]. The group G supports the Semenov-Tian-Shansky bivector field making it a Poisson G-space with respect to conjugation. Here r − and r + are, respectively, the skew-symmetric and invariant symmetric parts of r , and the superscripts designate the vector fields where ξ ∈ g and f is a smooth function on G. This bivector field (2.1) is tangent to every conjugacy class of G. In particular, the sphere S 2n becomes a homogeneous Poisson manifold over the Poisson group G [17]. Quantization of C[G] along (2.1) gives rise to the reflection equation dual of U q (g) [18]. Accordingly, the algebra C q [S 2n ] can be presented as its quotient. Here we recall that construction.
Let h ⊂ g denote the Cartan subalgebra equipped with the inner product restricted from an ad-invariant form on g. We endow the dual space h * with the inverse form (., .) and normalize it so that short roots have length 1. For any μ ∈ h * we denote by h μ ∈ h the vector such that ν(h μ ) = (ν, μ) for all ν ∈ h * .
The root system R contains an orthonormal basis = {ε i } n i=1 ⊂ h * of short roots. We choose the basis of simple positive roots as We define the subalgebra l gl(n) ⊂ g of maximal rank with the root basis l = {α i } n i=2 . Throughout the paper we assume that q ∈ C is not a root of unity and use the notation is a C-algebra generated by q ±h α , e ±α , α ∈ , such that q h α e ±β q −h α = q ±(α,β) e ±β and [e α , e −β ] = δ α,β [h α ] q for all α, β ∈ . The generators e ±α satisfy the q-Serre relations The subset k = {δ, α 1 , . . . , α n } ⊂ R + forms a root basis for a subalgebra k ⊂ g isomorphic to so(2n). Although e ±δ are deformations of classical root vectors, they do not generate an sl(2)-subalgebra in U q (g), so we have no natural subalgebra U q (k) in U q (g). Still e ±δ play a role in what follows.
By U q (h) ⊂ U q (g) we denote the subalgebra generated by {q ±h α } α∈ . We use the notation g ± ⊂ g for the Lie subalgebras generated by {e ±α } α∈ . They generate , α ∈ g , and by zero on the generators on nonzero weight. Extend them to representations of U q (p ± ) by zero on e ±α for all α ∈ g . Then set Denote by 1 λ ∈ M λ and 1 * λ ∈ N λ their highest/lowest weight generators. Due to the special choice of λ, the vectors e −δ 1 λ ∈M λ and e δ 1 * λ ∈N λ are killed by e α and, respectively, by e −α for all α ∈ . They generate submodulesM λ−δ ⊂ M λ and The module M λ supports quantization of C[S 2n ] in the following sense. The sphere S 2n is isomorphic to a subvariety in G of orthogonal matrices with eigenvalues ±1, where 1 is multiplicity-free. It is a conjugacy class with a unique point of intersection with the maximal torus relative to h. The isotropy subalgebra of this point is k. Quantization of C[G] along the Poisson bracket (2.1) can be realized as a subalgebra [13] for details.

Spanning M λ
In this section we introduce a set of vectors in M λ which is proved to be a basis in the subsequent section. Here we demonstrate that it spans M λ . Put f α = e −α for all simple roots and define The elements f ε i can be included in the set of composite root vectors generating a Poincare-Birkhoff-Witt basis in U q (g − ), [19]. By deformation arguments, the set of ..,m n ∈Z + is a basis in M λ extended over the local ring C[[h]], whereh = log q, see [13] and references therein. We will prove that B is a C-basis once q is not a root of unity.
Let k − m denote the subspace C f δ +Span{ f α 2 , . . . , f α m } ⊂ U q (g − ) assuming m 2.  [6], which completes the proof for m = 2. Suppose that m = 3. All calculations below are done modulo U q (g)k − m . Denote a = [2] q , then where the left equality means f δ ∈ U q (g)k − 3 and the last two equalities are obtained from it and from [ f δ , f α 1 ] = 0 (a Serre relation). Furthermore, the Serre relations along with (3.2) yield Multiply the first equality by a and subtract from the second: This completes the case m = 3.
m , for all m i > 3, as required.

Corollary 3.2 The set B spans M λ . The action of U q (g − ) on M λ is given by
. . . f m n ε n 1 λ , i > 1. Proof The first summand vanishes since f ε i −ε 2 commutes with f ε i+1 , by Lemma A. 1

. The internal commutator in the second summand is
i+1 as well. Now we can complete the proof. The linear span CB is invariant under the obvious action of f α 1 = f ε 1 . For i > 0, we push f α i+1 to the right in the product For any form . . , f ε n )1 λ by Lemma 3.1. This yields the action of f α i+1 on CB and proves its U q (g − )-invariance. Since CB 1 λ , it coincides with M λ . The elements f ε i , e ε i are known to generate U q (sl (2))-subalgebras in U q (g) with the commutation relation [e ε i ,

Invariant bilinear form M λ ⊗ N λ → C
Fix the comultiplication on U q (g) as in [19]: Proof The δ-symbols are due to orthogonality of weight subspaces M λ [μ] and N λ [ν] unless μ = − ν. Now we prove factorization of the matrix coefficients on setting  There is also an ω-contravariant form on M λ defined by x1 λ ⊗ y1 λ → ω(x)y , for all x, y ∈ U q (g). It is called the Shapovalov form and related to the invariant form in the obvious way.
Proof (1) Corollary 3.2 with (4.3) proves the completeness of B and independence. All weight subspaces in M λ have dimension 1, and the form is non-degenerate; hence, the basis B is orthogonal with respect to the Shapovalov form.

Star product on S 2n
Denote by A q the RTT dual of U q (g) with multiplication • and the Hopf paring (., .). It is equipped with the two-sided action (here f (1) ⊗ f (2) = ( f ) in the Sweedler notation) making it a U q (g)-bimodule algebra. The multiplication • is known to be local [15]. We define a new operation by It is obviously equivariant with respect to the right coregular action of U q (g). However, is not associative on the entire A q . As a two-sided U q (g)-module, A q is isomorphic to ⊕ V V * ⊗ V , where the summation is over all equivalence classes of irreducible finite-dimensional representations of U q (g). This is a q-version of the Peter-Weyl decomposition.
For every U q (g)-module V we define V k ⊂ V to be the intersection of the space V l of U q (l)-invariants with the joint kernel of the operators e δ and f δ . For q = 1, this definition gives the subspace of U (k)-invariants.

Proposition 5.1 A k
q is an associative U q (g)-algebra with respect to .
Proof Identify M * λ with N * * λ and the locally finite part of M λ ⊗ M * λ with the locally finite part End • C (M λ ) of End C (M λ ) regarded as a U q (g)-module. For every completely reducible module V , there is a uniqueφ ∈ Hom(V * , End • C (V )) for each φ ∈ Hom(M λ ⊗ N λ , V ), due to the natural isomorphism of the hom-sets.
LetM λ andN λ denote the Verma modules, i.e., induced from the U q (b ± )-modules C ∓λ . Every homomorphism M λ ⊗ N λ → V amounts to a homomorphismM λ ⊗N λ → V vanishing on α∈ kM λ−α ⊗N λ + α∈ kM λ ⊗N λ−α . Therefore, φ corresponds to a unique zero weight element (φ) ∈ ∩ α∈± k ker e α = V k . Given also ψ ∈ q (with respect to the -action). Then f g is the image of f g ∈ (A q ⊗ A q ) k under the multiplication •: A q ⊗ A q → A q , which is again in A k q since • is -equivariant. Associativity of follows from associativity of • and •.
Proof We only need to make sure that A k It is done in Proposition 6.2 below.
Note that, though A k q goes over to C[G] k at q = 1, the fact A k q C[G] k ⊗ C(q) as a C(q)-vector space needs a proof because ker e δ and ker f δ may decrease under deformation. That is done in the next section.

Quantum Euclidian plane
To complete the proof of Theorem 5.2, it is sufficient to check dim V k q = dim V k for all finite-dimensional modules V that appear in C[S 2n ]. They all can be realized in the polynomial ring of the Euclidian plane C 2n+1 [20]. So we are going to look at its quantum version.
Choose a basis {x i } n i=−n ⊂ C N , N = 2n + 1, and define a representation of U q (g) on C N by the assignment . . , n. Then x i carry weights ε i subject to ε i = − ε −i . The quantum Euclidian plane C q [C N ] is an associative algebra generated by {x i } n i=−n with relations They are equivalent to the relations presented in [11].
The representation on C N extends to an action on C q [C N ] making it a U q (g)-module algebra. Let θ denote the involutive algebra and anti-coalgebra linear automorphism of U q (g) determined by the assignment e α → − f α , q h α → q −h α . Define also an anti-algebra linear involution on C q [C N ] by ι(x i ) = (x −i ). They are compatible with the action in the sense that ι(u x) = θ(u) ι(x) for all u ∈ U q (g), x ∈ C q [C N ]. Proof Put c k = q −k −1 q −1 −1 for k ∈ Z + . Since f α 2 x k 0 = 0, the equality f δ x k 0 = 0 follows from Finally, e δ x k The q-version of the quadratic invariant is C q = 1 1+q denote the vector space of polynomials of degree m and P m q the irreducible submodule of harmonic polynomials of degree m. Then Let P m and H m denote their classical counterparts.

Proposition 6.2
For any finite-dimensional U q (g)-module V q , dim V k q is equal to dim V k of the classical k-invariants.
Proof It is sufficient to show that dim(P m q ) k = dim(P m ) k . In the classical limit, the trivial k-submodule in H m−2l is multiplicity-free, so its dimension in P m is l=0 and has the same dimension. Since all U q (l)-invariants are killed by e δ , f δ by Lemma 6.1, this proves the statement.
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A
For reader's convenience, we prove an algebraic identity that is useful for the study of the subalgebras U q (g ± ) ⊂ U q (g).  This yields (1 + q −2 )[[y, z] q , [x, y]q ] = 0, which proves the first formula. Using the "Jacobi identity" with X = x, Y = y, Z = [y, z] q , a =q, b = 1, and c =q, we get