U_{h}(𝔤) INVARIANT QUANTIZATION OF COADJOINT ORBITS AND VECTOR BUNDLES OVER THEM

JOSEPH DONIN

DEPT. OF MATH. BAR-ILAN UNIVERSITY MAX-PLANCK-INSTITUT FÜR MATHEMATIK

Abstract. Let M be a coadjoint semisimple orbit of a simple Lie group G. Let U_{h}(𝔤) be a quantum group corresponding to G. We construct a universal family of U_{h}(𝔤) invariant quantizations of the sheaf of functions on M and describe all such quantizations. We also describe all two parameter U_{h}(𝔤) invariant quantizations on M, which can be considered as U_{h}(𝔤) invariant quantizations of the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the sheaves of sections of vector bundles on M as one- and two-sided U_{h}(𝔤) invariant modules over a quantized function sheaf.

1. Introduction

Let G be a simple Lie group with Lie algebra 𝔤, M a semisimple coadjoint orbit of G, i.e., the orbit of G passing through a semisimple element in the coadjoint representation 𝔤^{∗}. Let A be the sheaf of functions on M. It may be the sheaf of smooth, analytic, or algebraic functions. The universal enveloping algebra U(𝔤) acts on the sections of A and the multiplication in A is U(𝔤) invariant. Let U_{h}(𝔤) denote a quantum group that is a deformation of the U(𝔤) as a bialgebra. In the paper [DGS] we considered the following problems.

1) Does there exists a U_{h}(𝔤) invariant deformation quantization of A, i.e., a quantization, A_{h}, having a U_{h}(𝔤) invariant multiplication?

2) Does there exists a two parameter (double) U_{h}(𝔤) invariant quantization, A_{t,h}, such that A_{t,0} is a U(𝔤) invariant quantization of A with Poisson bracket being the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M? Note that A_{t,h} can be considered as a U_{h}(𝔤) invariant quantization of the KKS Poisson bracket on M.

In [DGS], we have classified the Poisson brackets admissible for one and two parameter quantizations. An admissible Poisson bracket for one parameter quantization is the same as a Poisson bracket making M into a Poisson manifold with Poisson action of G, where G is considered to be the Poisson-Lie group with Poisson structure defined by the rmatrix related to U_{h}(𝔤). An admissible Poisson bracket for two parameter quantization, in addition, must be compatible with the KKS bracket on M.

1991 Mathematics Subject Classification. 17B37, 53C35, 81R50.

This research is partially supported by Israel Academy of Sciences grant no. 8007/99-01.

We have shown that all semisimple orbits have admissible Poisson brackets for one parameter quantization and almost all such brackets can be quantized.

In [DGS], we called an orbit having a Poisson bracket admissible for a two parameter quantization a good orbit. We have classified the good semisimple orbits for all simple 𝔤 and shown that in case 𝔤 ≠ sl(n) all the good orbits can be quantized. In the case 𝔤 = sl(n) all semisimple orbits are good but in [DGS] we did not prove the existence of their double quantization.

In this paper we give a complete description of one and two parameter U_{h}(𝔤) invariant quantizations on M. We show that for each semisimple orbit M there exists a universal family of quantizations of A. This family is given by a family of multiplications

m_{f,h} : A ⊗ A → A[[h]], f ∈ X,

where X is the manifold of all admissible Poisson brackets on M. The universality means that any one parameter U_{h}(𝔤) invariant quantization of A is given by the multiplication of the form m_{f(h),h}, where f(h) is a formal path in X, and two different paths give nonequivalent quantizations. As a consequence we obtain that any admissible Poisson bracket on M can be quantized.

There is the analogous description for two parameter quantizations on any good orbit. In particular, we prove that any good orbit (including all semisimple orbits in sl(n)^{∗}) admits a two parameter U_{h}(𝔤) invariant quantization.

Further, we consider the natural polarizations on M with respect to the KKS Poisson bracket. We show that all U_{h}(𝔤) invariant quantizations on M being restricted to functions constant along a polarization have a standard form. In some cases, when M is a coadjoint orbit of a real Lie group G, the polarizations define complex structures on M. In such cases the sheaf of functions constant along polarization specializes to the sheaf of holomorphic (or antiholomorphic) functions on M. So we obtain that in the real case any quantization of smooth functions on M induces a unique quantization of the subsheaves of holomorphic and antiholomorphic functions on M.

In the paper we also consider the quantization of G invariant vector bundles. We identify such a bundle with the sheaf of its smooth sections. Let A_{h} be a U_{h}(𝔤) invariant quantization of the sheaf of smooth functions on M. Under the quantization of a vector bundle V on M with respect to A_{h} we mean the sheaf V [[h]] endowed with a structure of U_{h}(𝔤) invariant left (right, two-sided) A_{h} module. Using a complex polarization, we show that for any U_{h}(𝔤) invariant quantization A_{h} and any G invariant vector bundle V there exists a U_{h}(𝔤) invariant quantization of V as a left A_{h} module. Moreover, we show that there exists a special quantization, A_{h}, such that any vector bundle V admits a quantization as a two-sided module with respect to A_{h}. Note that the papers [Jo], [DG], where the algebra of global holomorphic sections of linear vector bundles on flag varieties was quantized, relate to the problem of quantizing vector bundles.

U_{h}(𝔤) INVARIANT QUANTIZATION 3

The paper is organized as follows. In Section 2 we recall some facts on quantum groups essential for our approach to U_{h}(𝔤) invariant quantization. In particular, we define a quantum group, U_{h}(𝔤), for any classical r-matrix r and show that the problem of constructing U_{h}(𝔤) invariant quantization is equivalent to the problem of constructing U(𝔤) invariant Φ_{h} associative quantization, where Φ_{h} ∈ U(𝔤)^{⊗3}[[h]] defines the Drinfeld associativity constraint (see [Dr2]). Thus, a U(𝔤) invariant Φ_{h} associative quantization defines a U_{h}(𝔤) invariant quantization for all quantum groups associated with different r. In this section we also define φ-brackets, which are infinitesimal parts of U(𝔤) invariant Φ_{h} associative quantizations. We show that a Poisson bracket admissible for U_{h}(𝔤) invariant quantization is the difference of a φ-bracket and the bracket induced on M by the r-matrix associated with U_{h}(𝔤).

In Section 3 we give a classification of φ- and good brackets on semisimple orbits. In particular, we give a description of the variety of those brackets, which is more detailed than in [DGS]. We are needed in this description in the following sections.

In Section 4 we consider Poisson cohomologies of some parameterized complexes, which we use in Sections 5 and 6 for proving the existence of the universal one and two parameter quantizations.

In Sections 7 we consider the natural polarizations on M and prove that any U(𝔤) invariant Φ_{h} associative quantization is trivial when restricted to the sheaf of functions constant along polarization.

Note that up to Section 7 we assume that G is a complex Lie group, M is a complex subvariety in g^{∗}, and A is the sheaf of complex analytic functions on M.

In Section 8 we specify our results to a real Lie group G. We consider complex structures corresponding to polarizations and use them to construct the quantizations of G invariant vector bundles.

Acknowledgments. I am very grateful to P.Bressler and D.Gurevich for useful discussions. I thank Max-Planck-Institut für Mathematik for hospitality and very stimulating working atmosphere.

2. Preliminaries

2.1. Quantum groups. We will consider quantum groups in sense of Drinfeld, [Dr2], as deformed universal enveloping algebras. If U(𝔤) is the universal enveloping algebra of a complex Lie algebra 𝔤, then the quantum group (or quantized universal enveloping algebra) corresponding to U(𝔤) is a topological Hopf algebra, U_{h}(𝔤), over ℂ[[h]], isomorphic to U(𝔤)[[h]] as a topological ℂ[[h]] module and such that U_{h}(𝔤)/hU_{h}(𝔤) = U(𝔤) as a Hopf algebra over ℂ. In particular, the deformed comultiplication in U_{h}(𝔤) has the form

Δ_{h} = Δ + hΔ_{1} + o(h),  (2.1)

where Δ is the comultiplication in the universal enveloping algebra U(𝔤). One can prove, [Dr2], that the map Δ_{1} : U(𝔤) → U(𝔤) ⊗ U(𝔤) is such that Δ_{1} − σΔ_{1} = δ (σ is the usual permutation) being restricted to 𝔤 gives a map δ : 𝔤 → ∧^{2}𝔤 which is a 1-cocycle and defines the structure of a Lie coalgebra on 𝔤 (the structure of a Lie algebra on the dual space 𝔤^{∗}). The pair (𝔤, δ) is called a quasiclassical limit of U_{h}(𝔤).

In general, a pair (𝔤, δ), where 𝔤 is a Lie algebra and δ is such a 1-cocycle, is called a Lie bialgebra. It is proven, [EK], that any Lie bialgebra (𝔤, δ) can be quantized, i.e., there exists a quantum group U_{h}(𝔤) such that the pair (𝔤, δ) is its quasiclassical limit.

A Lie bialgebra (𝔤, δ) is said to be a coboundary one if there exists an element r ∈ ∧^{2}𝔤, called the classical r-matrix, such that δ(x) = [r, Δ(x)] for x ∈ 𝔤. Since δ defines a Lie coalgebra structure, r has to satisfy the so-called classical Yang-Baxter equation which can be written in the form

[[r, r]] = φ,  (2.2)

where [[·, ·]] stands for the Schouten bracket and φ ∈ ∧^{3}𝔤 is an invariant element. We denote the coboundary Lie bialgebra by (𝔤, r).

In case 𝔤 is a simple Lie algebra, the most known r-matrix is the Sklyanin-Drinfeld one:

r = ∑_{α} X_{α} ∧ X_{−α},

where the sum runs over all positive roots; the root vectors X_{α} are chosen is such a way that (X_{α}, X_{−α}) = 1 for the Killing form (·, ·). This is the only r-matrix of weight zero, [SS], and its quantization is the Drinfeld-Jimbo quantum group. A classification of all r-matrices for simple Lie algebras was given in [BD].

From results of Drinfeld and of Etingof and Kazhdan one can derive the following

2.1.1. Proposition. Let 𝔤 be a semisimple Lie algebra. Then

a) any Lie bialgebra (𝔤, δ) is a coboundary one;

b) the quantization, U_{h}(𝔤), of any coboundary Lie bialgebra (𝔤, r) exists and is isomorphic to U(𝔤)[[h]] as a topological ℂ[[h]] algebra;

c) the comultiplication in U_{h}(𝔤) has the form

Δ_{h}(x) = F_{h}Δ(x)F_{h}^{−1},  x ∈ U(𝔤),  (2.3)

where F_{h} ∈ U(𝔤)^{⊗2}[[h]] and can be chosen in the form

F_{h} = 1 ⊗ 1 + \frac{h}{2}r + o(h).  (2.4)

Proof. a) follows from the fact that H^{1}(𝔤, ∧^{2}𝔤) = 0. It follows from H^{2}(𝔤, U(𝔤)) = 0 that U(𝔤) does not admit any nontrivial deformations as an algebra, (see [Dr1]), which proves b). From the fact that H^{1}(𝔤, U(𝔤)^{⊗2}) = 0 it follows that any deformation of the algebra morphism Δ : U(𝔤) → U(𝔤) ⊗ U(𝔤) appears as a conjugation of Δ. In particular, the comultiplication in U_{h}(𝔤) looks like (2.3) with some F_{h} such that F_{0} = 1 ⊗ 1. It follows from the coassociativity of Δ_{h} that F_{h} satisfies the equation

(F_{h} ⊗ 1) · (Δ ⊗ id)(F_{h}) = (1 ⊗ F_{h}) · (id ⊗ Δ)(F_{h}) · Φ_{h}  (2.5)

for some invariant element Φ_{h} ∈ U(𝔤)^{⊗3}[[h]].

The element F_{h} satisfying (2.3) and (2.4) can be obtained by a correction of some F_{h} only obeying (2.5), [Dr2]. This procedure also uses a simple cohomological argument, which proves c).

2.1.2. It follows from (2.5) that if F_{h} has the form (2.4), then the coefficient by h in Φ_{h} vanishes. Moreover, the coefficient by h^{2} is the element φ from (2.2), i.e.,

Φ_{h} = 1 ⊗ 1 ⊗ 1 + h^{2}φ + o(h^{2}).  (2.6)

In addition, it follows from (2.5) that Φ_{h} satisfies the pentagon identity

(id^{⊗2} ⊗ Δ)(Φ_{h}) · (Δ ⊗ id^{⊗2})(Φ_{h}) = (1 ⊗ Φ_{h}) · (id ⊗ Δ ⊗ id)(Φ_{h}) · (Φ_{h} ⊗ 1).

2.2. Equivariant deformation quantization. Let G be a simple connected complex Lie group whose Lie algebra is 𝔤. Let G act on a manifold M and A be the sheaf of functions on M. It may be the sheaf of analytic, smooth, or algebraic functions, dependingly of the type of M. Then U(𝔤) acts on sections of A, and the multiplication in A is U(𝔤) invariant.

The deformation quantization of A is a sheaf of associative algebras, A_{h}, which is isomorphic to A[[h]] = A ⊗ ℂ[[h]] (completed tensor product) as a ℂ[[h]]-module, with multiplication in A_{h} having the form m_{h} = ∑_{k=0}^{∞} h^{k}m_{k}, where m_{0} is the usual commutative multiplication in A and m_{k}, k > 0, are bidifferential operators vanishing on constants. The algebra U(𝔤)[[h]] is clearly acts on the ℂ[[h]] module A_{h}.

We will study quantizations of A which are invariant under the U_{h}(𝔤) action, i.e., under the comultiplication Δ_{h}. This means that

bm_{h}(x ⊗ y) = m_{h}Δ_{h}(b)(x ⊗ y) for b ∈ U(𝔤), x, y ∈ A.

A ℂ[[h]] linear map μ_{h} : A_{h} ⊗ A_{h} → A_{h} is called a Φ_{h} associative multiplication if

μ_{h}(Φ_{1}x ⊗ μ_{h}(Φ_{2}y ⊗ Φ_{3}z))) = μ_{h}(μ_{h}(x ⊗ y) ⊗ z) for x, y, z ∈ A,

where Φ_{h} = Φ_{1} ⊗ Φ_{2} ⊗ Φ_{3} (summation implicit).

We say that the Φ_{h} associative multiplication μ_{h} = ∑_{k=0}^{∞} h^{k}μ_{k} gives a Φ_{h} associative quantization of A if μ_{0} = m_{0}, the usual multiplication in A, and μ_{k}, k > 0, are bidifferential operators vanishing on constants.

2.2.1. Proposition. There is a natural one-to-one correspondence between U_{h}(𝔤) invariant and U(𝔤) invariant Φ_{h} associative quantizations of A. Namely, if μ_{h} is a U(𝔤) invariant Φ_{h} associative multiplication in A[[h]], then

m_{h} = μ_{h}F_{h}^{−1}  (2.7)

gives a U_{h}(𝔤) invariant associative multiplication in A[[h]].

Proof. This follows immediately from (2.3) and (2.5). This follows also from the categorical interpretation of Φ_{h} and F_{h}, [Dr2], [DGS].

This proposition shows that given a U(𝔤) invariant Φ_{h} associative quantization of A, we can get the U_{h}(𝔤) invariant quantization of A for any quantum group U_{h}(𝔤) from Proposition 2.1.1 b) by applying F_{h} from (2.4) to the Φ_{h} associative multiplication.

2.3. Poisson brackets associated with equivariant quantizations. A skew-symmetric map f : A^{⊗2} → A we call a bracket if it satisfies the Leibniz rule: f(ab, c) = af(b, c) + f(a, c)b for a, b, c ∈ A. It is easy to see that any bracket is presented by a bivector field on M. Further we will identify brackets and bivector fields on M.

For an element ψ ∈ ∧^{k}𝔤 we denote by ψ_{M} the k-vector field on M which is induced by the action map 𝔤 → Vect(M).

A bracket f is a Poisson one if the Schouten bracket [[f, f]] is equal to zero.

2.3.1. Definition. A G invariant bracket f on M we call a φ-bracket if [[f, f]] = −φ_{M}, where φ ∈ ∧^{3}𝔤 is an invariant element.

2.3.2. Proposition. Let A_{h} be a U(𝔤) invariant Φ_{h} associative quantization with multiplication μ_{h} = m_{0} + hμ_{1} + o(h), where m_{0} is the multiplication in A. Then the map f : A^{⊗2} → A, f(a, b) = μ_{1}(a, b) − μ_{1}(b, a), is a φ-bracket for φ from (2.6).

Proof. A direct computation. Another proof is found in [DGS].

2.3.3. Corollary. Let A_{h} be a U_{h}(𝔤) invariant associative quantization with multiplication m_{h} = m_{0} + hm_{1} + o(h). Then the corresponding Poisson bracket p(a, b) = m_{1}(a, b) − m_{1}(b, a) has the form

p(a, b) = f(a, b) − r_{M}(a, b),  (2.8)

where r is the r-matrix corresponding to U_{h}(𝔤) and f is a φ-bracket with φ = [[r, r]].

Proof. By Proposition 2.2.1 there is a U(𝔤) invariant Φ_{h} associative multiplication μ_{h} such that m_{h} = μ_{h}F_{h}^{−1} with F_{h} as in (2.4). Let f be the φ-bracket corresponding to μ_{h}. Then a direct computation shows that the Poisson bracket of m_{h} is as required.

2.3.4. Remark. For an r-matrix r ∈ ∧^{2}𝔤, denote by r′ and r′′ the left and right invariant bivector fields on G corresponding to r. Then it follows from (2.2) that the bivector field r′ − r′′ defines a Poisson bracket on G which makes G into a Poisson-Lie group. On the other hand, a Poisson brackets on M admitting, in principle, a U_{h}(𝔤) invariant quantization endows M with a structure of (G, r)-manifold. This means that the action G × M → M is a Poisson map. So, Corollary 2.3.3 describes the form of Poisson brackets on M making M into a (G, r)-manifold. One sees, in particular, that the classification of (G, r) Poisson structures on M reduces to the classification of φ-brackets on M.

2.3.5. We will also consider two parameter quantizations on M. A two parameter quantization of A is an algebra A_{t,h} isomorphic to A[[t, h]] as a ℂ[[t, h]] module and having a multiplication of the form

m_{t,h} = m_{0} + tm′_{1} + hm′′_{1} + o(t, h).  (2.9)

With such a quantization one associates two Poisson brackets: the bracket v(a, b) = m′_{1}(a, b) − m′_{1}(b, a) along t, and the bracket p(a, b) = m′′_{1}(a, b) − m′′_{1}(b, a) along h. It is easy to check that p and v are compatible Poisson brackets, i.e., their Schouten bracket [[p, v]] is equal to zero.

2.3.6. Corollary. Let A_{t,h} be a U_{h}(𝔤) invariant associative quantization of the form (2.9). Then the Poisson bracket p(a, b) = m′′_{1}(a, b) − m′′_{1}(b, a) has the form

p(a, b) = f(a, b) − r_{M}(a, b),

where r is the r-matrix corresponding to U_{h}(𝔤) and f is a φ-bracket with φ = [[r, r]]. The Poisson bracket v(a, b) = m′_{1}(a, b) − m′_{1}(b, a) in invariant and compatible with p.

Proof. Similar to Corollary 2.3.3.

2.3.7. In the following, a φ-bracket on M compatible with a nondegenerate Poisson bracket we call a good bracket.

3. Classification of φ- and good brackets on semisimple orbits

3.1. Let G be a complex connected simple Lie group with the Lie algebra 𝔤. Let 𝔩 be a Levi subalgebra of 𝔤, the Levi factor of a parabolic subalgebra. Let L be a Lie subgroup of G with Lie algebra 𝔩. Such a subgroup is called a Levi subgroup. It is known that L is a closed connected subgroup. Denote M = G/L and let o ∈ M be the image of the unity by the natural projection G → M. Then L is the stabilizer of o. It is known, that M may be realized as a semisimple orbit of G in the coadjoint representation 𝔤^{∗}. Conversely, any semisimple orbit in 𝔤^{∗} is a quotient of G by a Levi subgroup.

3.2. Let 𝔥 ⊂ 𝔩 be a Cartan subalgebra of 𝔤 and Ω_{𝔩} ⊂ Ω ⊂ 𝔥^{∗} the sets of roots of 𝔩 and 𝔤 corresponding to 𝔥. Choose root vectors E_{α}, α ∈ Ω, in such a way that

(E_{α}, E_{−α}) = 1  (3.1)

for the Killing form (·, ·) on 𝔤.

3.3. Let Q be a set embedded in a linear space V such that 0 ∉ Q and Q = −Q. We call a subset B ⊂ Q a linear subset if B = Q ∩ V_{B} where V_{B} is the linear subspace in V generated by B. We call a subset B ⊂ Q semilinear if it follows from x, y ∈ B, x + y ∈ Q that x + y ∈ B, and, in addition, B ∩ (−B) = ∅, B ∪ (−B) = Q. For a linear subset B of Q, we denote by Q/B the image of Q without zero by the projection F → F/V_{B}.

3.4. Since 𝔩 is a Levi subalgebra, Ω_{𝔩} is a linear subset in Ω. We put \overline{Ω} = Ω/Ω_{𝔩} and call elements of \overline{Ω} quasiroots. For α ∈ Ω we denote by ᾱ its image in \overline{Ω}. Let Y be a semilinear subset in \overline{Ω}. One can easily shown that there is a subset P ⊂ Y such that any element of Y can be uniquely presented as a linear combination of elements of P with integer coefficients. We call P a set of simple quasiroots corresponding to Y and Y a set of positive quasiroots with respect to P. It is clear that there is a set of simple roots, Π, in Ω such that P = \overline{Π}. Then Y = \overline{Ω}^{+}, where Ω^{+} is the system of positive roots corresponding to Π. For such a Π there is a subset, Γ ⊂ Π, such that 𝔩 coincides with the Lie subalgebra 𝔤_{Γ}, the subalgebra generated by 𝔥 and elements E_{±α}, α ∈ Γ.

3.5. The projection π : G → M induces the map π_{∗} : 𝔤 → T_{o} where T_{o} is the tangent space to M at the point o. Since the ad-action of 𝔩 on 𝔤 is semisimple, there exists an ad(𝔩)-invariant subspace 𝔪 = 𝔪_{𝔩} of 𝔤 complementary to 𝔩, and one can identify T_{o} and 𝔪 by means of π_{∗}. It is easy to see that subspace 𝔪 is uniquely defined and has a basis consisting of the elements E_{γ}, γ ∈ Ω \ Ω_{𝔩}.

3.6. Proposition. The space 𝔪 considered as a 𝔩 representation space decomposes into the direct sum of subrepresentations 𝔪β̄, β̄ ∈ \overline{Ω}, where 𝔪β̄ is generated by all the elements E_{β}, β ∈ Ω, such that the projection of β is equal to β̄. This decomposition have the following properties:

a) all 𝔪β̄ are irreducible;

b) 𝔪_{−β̄} is dual to 𝔪β̄;

c) for β̄_{1}, β̄_{2} ∈ \overline{Ω} such that β̄_{1} + β̄_{2} ∈ \overline{Ω} one has [𝔪β̄_{1}, 𝔪β̄_{2}] = 𝔪_{β̄_{1}+β̄_{2}};

d) for any pair β̄_{1}, β̄_{2} ∈ \overline{Ω} the representation 𝔪β̄_{1} ⊗ 𝔪β̄_{2} is multiplicity free.

Proof. Statements a), b), and c) are proven in [DGS], Remark 3.1. Statement d) follows from the fact that the weight subspaces of all 𝔪β̄ have dimension one (see N.Bourbaki, Groupes et algèbres de Lie, Chap. 8, §9, Ex. 14).

3.7. Restricting to the point o ∈ M defines the natural one-to-one correspondence between G invariant tensor fields on M and 𝔩 invariant tensors over 𝔪.

Since 𝔩 contains a Cartan subalgebra, 𝔥, each 𝔩 invariant tensor over 𝔪 is of weight zero with respect to 𝔥. It follows that there are no invariant vectors in 𝔪. Hence, there are no invariant vector fields on M.

3.8. Proposition. A bivector v ∈ ∧^{2}𝔪 is 𝔩 invariant if and only if it has the form v = \frac{1}{2} ∑ c(ᾱ)E_{α} ∧E_{−α} where the sum runs over α ∈ Ω\Ω_{𝔩} (we suppose c(−ᾱ) = −c(ᾱ)).

Proof. Follows from (3.1) and Proposition 3.6 (see also [DGS], Proposition 3.2).

3.9. Denote by [[v, w]] ∈ ∧^{k+l−1}𝔪 the Schouten bracket of polyvector fields v ∈ ∧^{k}𝔪 and w ∈ ∧^{l}𝔪 defined by the formula

[[X_{1} ∧ ⋯ ∧ X_{k}, Y_{1} ∧ ⋯ ∧ Y_{l}]] = ∑(−1)^{i+j}[X_{i}, Y_{j}]_{𝔪} ∧ X_{1} ∧ ⋯ X̂_{i} ⋯ Ŷ_{j} ⋯ ∧ Y_{l},

where [·, ·]_{𝔪} is the composition of the Lie bracket in 𝔤 and the projection 𝔤 → 𝔪. The defined Schouten bracket is compatible with the Schouten bracket on M under identifying 𝔩 invariant polyvectors over 𝔪 and G invariant polyvector fields on M.

3.10. It is obvious that any 𝔩 invariant bivector is θ anti-invariant for the Cartan automorphism θ, θ(E_{α}) = −E_{−α}, of 𝔤. Hence, if v, w ∈ ∧^{2}𝔪 are 𝔩 invariant, then [[v, w]] is θ invariant, i.e., is of the form [[v, w]] = ∑ e(α, β)E_{α+β} ∧ E_{−α} ∧ E_{−β} where e(α, β) = −e(−α, −β). Hence, in order to calculate [[v, w]] for such v and w it is sufficient to calculate coefficients e(α, β) for positive α and β by any choice of the system of positive roots.

3.11. Lemma. Let v = ∑ c(α)E_{α} ∧ E_{−α}, w = ∑ d(α)E_{α} ∧ E_{−α} be elements from 𝔤 ∧ 𝔤. Choose a system of positive roots. Then for any positive roots α, β, (α + β) the coefficient by the term E_{α+β} ∧ E_{−α} ∧ E_{−β} in [[v, w]] is equal to

N_{α,β}(d(α)(c(β) − c(α + β)) + d(β)(c(α) − c(α + β)) − d(α + β)(c(α) + c(β))), (3.2)

where the number N_{α,β} is defined by relation [E_{α}, E_{β}] = N_{α,β}E_{α+β}.

Proof. Direct computation, see [KRR].

3.12. Let φ ∈ ∧^{3}𝔤 be an invariant element. Since 𝔤 is simple, φ is defined uniquely up to a factor. Denote by φ_{M} the invariant three-vector field on M induced by φ with the help of the action map 𝔤 → Vect(M). It is easy to check that φ_{M} is θ invariant and up to a factor has the form

φ_{M} = \frac{1}{3} ∑_{α,β,α+β∈Ω\Ω_{𝔩}} N_{α,β}E_{α+β} ∧ E_{−α} ∧ E_{−β}.  (3.3)

3.13. From Lemma 3.11 it follows that the Schouten bracket of bivector v = \frac{1}{2} ∑ c(ᾱ)E_{α}∧ E_{−α} with itself is equal to K^{2}φ_{M} for a complex number K, if and only if the following equations hold

c(ᾱ + β̄)(c(ᾱ) + c(β̄)) = c(ᾱ)c(β̄) + K^{2}  (3.4)

for all the pairs of quasiroots ᾱ, β̄ such that ᾱ + β̄ is a quasiroot.

Let X_{K^{2}} be the algebraic variety consisting of the points \{c(ᾱ), ᾱ ∈ \overline{Ω}\} satisfying (3.4) (we always assume c(ᾱ) = −c(−ᾱ)). So, for a given φ the variety X = X_{K^{2}=−1} is the variety of all φ-brackets. It is clear that all the varieties X_{K^{2}}, K ≠ 0, are isomorphic to X.

3.14. Let {c(ᾱ)} be a solution of (3.4) for a number K, i.e., {c(ᾱ)} ∈ X_{K^{2}}. It is easy to derive the following properties.

(*) If c(ᾱ) + c(β̄) = 0 then necessarily c(ᾱ) = ±K, c(β̄) = ∓K.

(**) If c(ᾱ) = ±K and c(β̄) ≠ ±K, then c(ᾱ + β̄) = ±K and c(ᾱ − β̄) = ±K.

(***) If c(ᾱ) = ±K and c(β̄) = ±K, then c(ᾱ + β̄) = ±K.

3.15. Formally, all the solutions of (3.4) for a fixed K can be obtained in the following way. Choose a system of positive quasiroots, \overline{Ω}^{+}. Denote by \overline{Π} the corresponding set of simple quasiroots. Given c(ᾱ) and c(β̄), we find from (3.4) that

c(ᾱ + β̄) = \frac{c(ᾱ)c(β̄) + K^{2}}{c(ᾱ) + c(β̄)} .  (3.5)

Assume, ᾱ, β̄, γ̄ are positive quasiroots such that ᾱ+β̄, β̄+γ̄, ᾱ+β̄+γ̄ are also quasiroots. Then the number c(ᾱ + β̄ + γ̄) can be calculated formally (ignoring possible division by zero) in two ways, using (3.5) for the pair c(ᾱ), c(β̄ + γ̄) on the right hand side and also for the pair c(ᾱ + β̄), c(γ̄). But it is easy to check that these two ways give the same value of c(ᾱ + β̄ + γ̄). In this sense the system of equations corresponding to (3.5) for all pairs is consistent. So, taking arbitrary values c(ᾱ) for simple quasiroots ᾱ one can try to find c(ᾱ) for all ᾱ ∈ \overline{Ω}^{+} recursively. We say that a solution, \{c(ᾱ)\}, of (3.4) can be obtained recursively if in the course of the recursive procedure started with the values c(ᾱ) for simple quasiroots ᾱ the denominators in (3.5) will be not equal to zero.

3.16. Proposition. For K ≠ 0 the following holds.

a) Any solution of (3.4) can be obtained recursively by choosing a respective system of positive quasiroots.

b) The variety X_{K^{2}} is without singularities, connected, and of dimension k, where k is equal to the number of simple quasiroots.

Proof. For proving a) we have to show that for any solution {c(ᾱ)} ∈ X_{K^{2}} one can choose a system of positive quasiroots in such a way that the denominators appearing in (3.5) by the recursive procedure are not equal to zero. It follows from (**) that the set Ψ consisting of ᾱ such that c(ᾱ) ≠ ±K is a linear subset of \overline{Ω}. Moreover, the function c(ᾱ) is constant on the cosets of \overline{Ω}/Ψ. Let Y is the set of cosets on which this function has the value K. It follows from (***) that Y is a semilinear subset of \overline{Ω}/Ψ. Let \overline{Ω}^{+} be a semilinear subset of \overline{Ω} projecting on Y. Then it follows from (*) that for ᾱ, β̄ ∈ \overline{Ω}^{+} c(ᾱ) + c(β̄) ≠ 0, which proves a).

Let \overline{Π} = \{ᾱ_{i}, i = 1, ..., k\} be the set of simple quasiroots corresponding to \overline{Ω}^{+}. Let c(ᾱ_{i}) = c_{i}. It is clear that starting the recursive procedure with c(ᾱ_{i}) = c′_{i} for c′_{i} arbitrary but close enough to c_{i}, the denominators in (3.5) remain not equal to zero. This proves that any point of X_{K^{2}} is non-singular and X_{K^{2}} has dimension k.

Let us prove that X_{K^{2}} is connected. Fix a set of positive quasiroots, \overline{Ω}^{+}, and the corresponding set of simple quasiroots, \overline{Π} = \{ᾱ_{i}, i = 1, ..., k\}. We say that a k-tuple of complex numbers (c_{1}, ..., c_{k}) is admissible, if starting with c(ᾱ_{i}) = c_{i} one obtains a solution of (3.4) by the recursive procedure. It is clear that the admissible tuples form a subset, A, of ℂ^{k} complement to an algebraic subset of lesser dimension, therefore A is connected. On the other hand, the set of points {c(ᾱ)} ∈ X_{K^{2}} such that c(ᾱ_{i}) form an admissible k-tuple is obviously dense in X_{K^{2}}. This proves the connectness of X_{K^{2}}.

3.17. Let \{c(ᾱ)\} be a solution of (3.4) and Ψ ∈ \overline{Ω} the linear subset of \overline{Ω} such that c(ᾱ) ≠ ±K for ᾱ ∈ Ψ. Then, using the formula for coth(x + y), similar to (3.5), one can see that there exists a linear form λ : Ψ → ℂ such that

λ(ᾱ) ∉ \frac{2πi}{K}ℤ ᾱ ∈ Ψ  (3.6)

and

c(ᾱ) = K coth (\frac{K}{2} λ(ᾱ))  for any ᾱ ∈ Ψ.  (3.7)

As K → 0, (3.6) makes into

λ(ᾱ) ≠ 0 ᾱ ∈ Ψ  (3.8)

and (3.7) tends to

c(ᾱ) = \frac{1}{λ(ᾱ)}  for any ᾱ ∈ Ψ.  (3.9)

So, we come to

3.18. Proposition. a) For K ≠ 0, any solution of (3.4) is determined by: choosing a linear subset, Ψ, in \overline{Ω}, a semilinear subset, B, in \overline{Ω}/Ψ, and a linear form, λ : Ψ → ℂ, satisfying (3.6). The respective solution {c(ᾱ)} is the following: for ᾱ ∈ Ψ c(ᾱ) is defined by (3.7); for ᾱ ∉ Ψ, c(ᾱ) = K if the projection of ᾱ in \overline{Ω}/Ψ belongs to B, c(ᾱ) = −K if the projection belongs to −B.

b) For K = 0, any solution of (3.4) is defined by: choosing a linear subset, Ψ, in \overline{Ω} and a linear form, λ : Ψ → ℂ, satisfying (3.8). The solution {c(ᾱ)} is the following: for ᾱ ∈ Ψ c(ᾱ) is defined by (3.9); for ᾱ ∉ Ψ c(ᾱ) = 0.

3.19. Remark. As mentioned in [Lu], for 𝔩 = 𝔥 the solutions described in Proposition 3.18 relate to solutions of classical dynamical Yang-Baxter equations, [EV].

3.20. Note that by K = 0 the solutions of (3.4) define Poisson brackets on M, so Proposition 3.18 b) describes all the Poisson brackets on M. We see that nondegenerate Poisson brackets on M are in one-to-one correspondence with the linear forms λ : \overline{Ω} → ℂ such that λ(ᾱ) ≠ 0 for all ᾱ ∈ \overline{Ω} and have the form

\frac{1}{2} ∑_{ᾱ∈\overline{Ω}} \frac{1}{λ(ᾱ)}E_{α} ∧ E_{−α}.  (3.10)

This is exactly the KKS bracket on the orbit in 𝔤^{∗} passing through the linear form on 𝔤 being the trivial extension of λ.

3.21. Denote by X_{0} the variety of nondegenerate Poisson brackets on M. Since X_{0} coincides with all solutions of (3.4), {c(ᾱ)}, such that Πᾱc(ᾱ) ≠ 0, it is clear that X_{0} is an affine connected algebraic variety without singularities.

3.22. Fix a Poisson bracket s of the form (3.10). Let us describe the invariant brackets f = ∑ c(ᾱ)E_{α} ∧ E_{−α} satisfying the conditions

[[f, f]] = K^{2}φ_{M},  (3.11)
[[f, s]] = 0.  (3.12)

with some K ≠ 0.

A direct computation shows that conditions (3.11) and (3.12) are equivalent to the system of equations for the coefficients c(ᾱ) of f, [DGS],

c(β̄)λ(β̄) = c(ᾱ)λ(ᾱ) ± Kλ(ᾱ + β̄)  (3.13)
c(ᾱ + β̄)λ(ᾱ + β̄) = c(ᾱ)λ(ᾱ) ± Kλ(β̄)  (3.14)

with the same sign before K, for all the pairs of quasiroots ᾱ, β̄ such that ᾱ + β̄ is a quasiroot.

3.23. Definition. Let M be an orbit in 𝔤^{∗} (not necessarily semisimple). The invariant bracket f on M is said to be good if f satisfies conditions (3.11) and (3.12) for s the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We call M a good orbit, if there exists a good bracket on it.

3.24. Proposition.

a) For 𝔤 of type A_{n} all semisimple orbits are good.

b) For all other 𝔤, the orbit M is good if and only if 𝔩 = 𝔤_{Γ} where Γ ⊂ Π for a system Π of simple root for 𝔤 and the set Π \ Γ consists of one or two roots which appear in the representation of the maximal root with coefficient 1.

c) For a given K ≠ 0 the good brackets f on a good orbit form a one-dimensional variety: all such brackets have the form

±f_{0} + ts,

where t ∈ ℂ and f_{0} is a fixed bracket satisfying (3.11), (3.12).

Proof. The proof reduces to solving the system of equations defined by (3.11) and (3.12), see [DGS].

3.25. So, if the set Π \ Γ consists of one root, M is exactly a hermitian symmetric space. As follows from the classification of simple Lie algebras, the case when the set Π \ Γ consists of two roots appears (besides A_{n}) for 𝔤 of types D_{n} and E_{6}.

3.26. Proposition 3.24 shows that the property for M to be a good orbit depends only on the pair (𝔤, 𝔩) but not on the realization of M as an orbits. The pair (𝔤, 𝔩) corresponding to a good orbit we call a good pair.

3.27. Remark. It is clear that if f satisfies (3.11) and (3.12) then ±f + ts also satisfies the same conditions (with the same K) for all numbers t. Proposition 3.24 c) shows that, conversely, all good brackets on a good semisimple orbit are contained in these families ±f + ts, t ∈ ℂ.

Denote by Y the variety of good brackets f on M satisfying (3.11) and (3.12) for a fixed K ≠ 0 and some nondegenerate Poisson bracket s. From the above it follows that there is a projection, Y → X_{0}, f ↦ s, where s is a Poisson bracket such that [[f, s]] = 0. The fiber over s ∈ X_{0} consists of two components, {±f_{0} + ts}, t ∈ ℂ, isomorphic to ℂ. These components correspond to choosing the sign in (3.13), (3.14).

3.28. Remark. It is shown in [Do1] that in case A_{n}, i.e., when 𝔤 = sl(n), all the coadjoint orbits (not necessarily semisimple) are good. Moreover, there exists a unique quadratic φ-bracket on sl(n)^{∗} which can be restricted to all orbits to give good brackets on them. This quadratic bracket on sl(n)^{∗} can be quantized, [Do2].

4. Poisson complexes

4.1. Let C^{k} = (Λ^{k}𝔪)^{𝔩} be the space of 𝔩 invariant k-vectors on 𝔪. This space is identified with the space of G invariant k-vector fields on M. Denote by C^{k} the sheaf of holomorphic functions on X_{K^{2}} with values in C^{k}. We form the complex (C^{•}, δ_{f}) where δ_{f} is the differential given by the Schouten bracket with a bivector f ∈ X_{K^{2}},

δ_{f} : u ↦ [[f, u]] for u ∈ C^{•}.

The condition δ_{f}^{2} = 0 follows from the Jacobi identity for the Schouten bracket together with the fact that [[f_{x}, f_{x}]] = K^{2}φ_{M}.

We also consider the complex of sheaves (C^{•}, δ) on X_{K^{2}}. The operator δ is defined as

δ(u)(f) = [[f, u(f)]] = δ_{f}(u(f)),  (4.1)

where u is a section of C^{•} and f ∈ X_{K^{2}}.

We denote by H^{k}(M, δ_{f}) and H^{k}(C^{•}, δ) the cohomologies of (C^{•}, δ_{f}) and (C^{•}, δ) , respectively, whereas the usual de Rham cohomologies are denoted by H^{k}(M).

4.2. Proposition. a) For any nondegenerate Poisson brackets f ∈ X_{0} and, if K ≠ 0, for almost all f ∈ X_{K^{2}} (except an algebraic subset of lesser dimension) one has

H^{k}(M, δ_{f}) = H^{k}(M)  (4.2)

for all k. In particular, H^{k}(M, δ_{f}) = 0 for odd k.

b) Let K ≠ 0. Then H^{2}(M, δ_{f}) = H^{2}(M) for all f ∈ X_{K^{2}}.

Proof. The proof of a) follows [DGS]. First, let v be a nondegenerate Poisson bracket on M, in particular, v ∈ X_{0}. Then the complex of polyvector fields on M, Θ^{•}, with the differential δ_{v} is well defined. Denote by Ω^{•} the de Rham complex on M. Since none of the coefficients c(ᾱ) of v are zero, v is a nondegenerate bivector field, and therefore it defines an A-linear isomorphism ṽ : Ω^{1} → Θ^{1}, ω ↦ v(ω, ·), which can be extended up to the isomorphism ṽ : Ω^{k} → Θ^{k} of k-forms onto k-vector fields for all k. Using Jacobi identity for v and invariance of v, one can show that ṽ gives a G invariant isomorphism of these complexes, so their cohomologies are the same.

Since 𝔤 is simple, the subcomplex of 𝔤 invariants, (Ω^{•})^{𝔤}, splits off as a subcomplex of Ω^{•}. In addition, 𝔤 acts trivially on cohomologies, since for any g ∈ G the map M → M, x ↦ gx, is homotopic to the identity map, (G is a connected Lie group corresponding to 𝔤). It follows that cohomologies of complexes (Ω^{•})^{𝔤} and Ω^{•} coincide.

But ṽ gives an isomorphism of complexes (Ω^{•})^{𝔩} and (Θ^{•})^{𝔤} = ((Λ^{•}𝔪)^{𝔩}, δ_{v}). So, cohomologies of the latter complex coincide with de Rham cohomologies, which proves a) for v being Poisson brackets.

Denote by X the variety of points \{c(ᾱ), ᾱ ∈ \overline{Ω}; K\} satisfying (3.4). Consider the family of complexes ((Λ^{•}𝔪)^{𝔩}, δ_{v}), v ∈ X. It is clear that δ_{v} depends algebraicly on v. It follows from the uppersemicontinuity of dim H^{k}(M, δ_{v}) and the fact that H^{k}(M) = 0 for odd k, [Bo], that H^{k}(M, δ_{v}) = 0 for odd k and almost all v ∈ X. Using the uppersemicontinuity again and the fact that the number ∑_{k}(−1)^{k} dim H^{k}(M, δ_{v}) is the same for all v ∈ X, we conclude that dim H^{k}(M, δ_{v}) = dim H^{k}(M) for even k and almost all v ∈ X. We have that there is a number K_{0} ≠ 0 and v ∈ X_{K_{0}} such that (4.2) holds for δ_{v}. Since the cohomologies do not change when v replaces by cv with any number c ≠ 0, we conclude that for any K ≠ 0 there is f satisfying (4.2). Since by Proposition 3.16 b) X_{K^{2}} is connected, it follows that (4.2) holds for almost all f ∈ X_{K^{2}}.

Let us prove b). Since (Λ^{1}𝔪)^{𝔩} = 0, H^{2}(M, δ_{f}) coincides with Ker(δ_{f} : C^{2} → C^{3}). According to Proposition 3.16, let us choose a system of positive quasiroots \overline{Ω}^{+} such that f has the form f = ∑_{ᾱ∈\overline{Ω}^{+}} c(ᾱ)E_{α} ∧ E_{−α}, and for ᾱ, β̄, ᾱ + β̄ ∈ \overline{Ω}^{+} c(ᾱ) + c(β̄) ≠ 0. By Lemma 3.11, any v ∈ C^{2} such that [[f, v]] = 0 has the form v = ∑_{ᾱ∈\overline{Ω}^{+}} d(ᾱ)E_{α} ∧ E_{−α} where d(ᾱ) obey the equation

d(α)(c(β) − c(α + β)) + d(β)(c(α) − c(α + β)) − d(α + β)(c(α) + c(β)) = 0. (4.3)

Starting with arbitrary values d(ᾱ) for simple quasiroots we find d(ᾱ) for all ᾱ ∈ \overline{Ω}^{+} recursively using the formula following from (4.3)

d(ᾱ + β̄) = \frac{d(α)(c(β) − c(α + β)) + d(β)(c(α) − c(α + β))}{c(α) + c(β)}.  (4.4)

Indeed, denominators of (4.4) are not equal to zero by choosing of \overline{Ω}^{+}. Moreover, assume ᾱ, β̄, γ̄ are positive quasiroots such that ᾱ+β̄, β̄+γ̄, ᾱ+β̄+γ̄ are also quasiroots. Then the number d(ᾱ + β̄+ γ̄) can be calculated in two ways, using (4.4) for the pair d(ᾱ), d(β̄+ γ̄) on the right hand side and also for the pair d(ᾱ + β̄), d(γ̄). But it is easy to check that these two ways give the same value of d(ᾱ + β̄ + γ̄).

So we see that dim H^{2}(M, δ_{f}) = {the number of simple quasiroots} for all f ∈ X_{K^{2}}.

4.3. Proposition. Let K ≠ 0. Then

H^{3}(ΓC^{•}, δ) = 0,

where ΓC^{k} is the space of global sections of C^{k} over X_{K^{2}}.

Proof. Since X_{K^{2}} is a Stein manifold, it is enough to prove that H^{3}(C^{•}, δ) = 0. Note that according to Proposition 4.2 b) Ker(δ : C^{2} → C^{3}) = H^{2}(C^{•}, δ) is a subbundle (direct subsheaf) of C^{2}. Therefore, Im(δ) is a subbundle of C^{3}. On the other hand, Ker(δ : C^{3} → C^{4}) being a subsheaf of C^{3} is a torsion free sheaf. It follows that H^{3}(C^{•}, δ) is a torsion free sheaf. According to Proposition 4.2 a) the support of H^{3}(C^{•}, δ) is an algebraic subset of X_{K^{2}} of lesser dimention. Hence H^{3}(ΓC^{•}, δ) = 0.

4.4. Let M be a good orbit, s the KKS Poisson bracket on M, and f_{0} a good bracket on M. Hence, [[f_{0}, f_{0}]] = K^{2}φ_{M} for some K ≠ 0 and [[f_{0}, s]] = 0. Consider the family of brackets

ϕ = f_{h,t} = hf_{0} + ts, h, t ∈ ℂ.

One has [[f_{h,t}, f_{h,t}]] = h^{2}K^{2}φ_{M}, therefore, according to Proposition 3.24 c) this family contains all the good brackets on M for all K. We will consider f_{h,t} as a linear map of ℂ^{2} with the coordinates (h, t) to the space (Λ^{2}𝔪)^{𝔩}.

Denote by C^{k}(m) the space of homogeneous maps ℂ^{2} → (Λ^{k}𝔪)^{𝔩} of degree m. Thus, f_{h,t} ∈ C^{2}(1). Define the differential δ : C^{•}(m) → C^{•}(m + 1) as (δ(a))(h, t) = [[f_{h,t}, a(h, t)]].

4.5. Proposition. a) For all (h, t) ∈ ℂ^{2} \ 0

H^{2}(M, δ_{f_{h,t}}) = H^{2}(M).

b) Let m ≥ 0 and b ∈ C^{3}(m + 1) such that δb = 0. Then there exists a ∈ C^{2}(m) such that δa = b.

Proof. Let us prove a). If h ≠ 0 then f_{h,t} ∈ X_{hK} with hK ≠ 0. Hence H^{2}(M, δ_{f_{h,t}}) = H^{2}(M) by Proposition 4.2 b). If h = 0 then t ≠ 0 and f_{0,t} is a nondegenerate Poisson bracket on M, therefore, H^{2}(M, δ_{f_{0,t}}) = H^{2}(M) by Proposition 4.2 a).

Let us prove b). Denote by L(m) a linear holomorphic vector bundle of degree m over the Riemann sphere 𝕊. The space of global sections of L(m) may be naturally identified with the space of homogeneous polynomials of two variables of degree m. Taking as the variables h and t, one can consider the space C^{k}(m) as the space of global sections of a vector bundle, E^{k}(m), that is a direct sum of dim(Λ^{k}𝔪)^{𝔩} copies of L(m). It is obvious that f_{h,t} defines a map of sheaves, δ_{m}^{k} : E^{k}(m) → E^{k+1}(m + 1). Denote H^{k}(m) = Ker(δ_{m}^{k})/ Im(δ_{m−1}^{k−1}).

It follows from a) that δ_{m}^{2} : E^{2}(m) → E^{3}(m + 1) is a map of bundles, i.e., its image is a direct subsheaf of E^{3}(m+1). Hence, H^{3}(m+1) is a sheaf over 𝕊 without torsion. But over a neighborhood of the point of 𝕊 with homogeneous coordinates (0, t) H^{3}(m + 1) = 0. This follows from the fact that f_{0,t} is a nondegenerate Poisson bracket and, therefore, H^{3}(M, δ_{f_{0,t}}) = 0. Since 𝕊 is connected, H^{3}(m + 1) = 0 over 𝕊. So, Ker δ_{m+1}^{3} = Im δ_{m}^{2}.

Consider the exact sequence of sheaves over 𝕊:

0 ⟶ H^{2}(m) ⟶ E^{2}(m) ⟶ Im δ_{m}^{2} ⟶ 0.

To complete the proof of b) one needs to show that any global section of Im δ_{m}^{2} can be lifted to a global section of E^{2}(m). But this follows from the fact that H^{1}(𝕊, H^{2}(m)) = 0. To prove the last fact we observe that H^{2}(m) is a subbundle of E^{2}(m), therefore, is a direct sum of a number of copies of L(m). Since m ≥ 0, H^{1}(𝕊, L(m)) = 0. This implies that H^{1}(𝕊, H^{2}(m)) = 0, too.

5. The G invariant Φ_{h} associative quantization in one parameter

5.1. Denote by (HC_{M}^{•}, ∂) the Hochschild complex on M, where each space HC_{M}^{k} consists of holomorphic k-differential operators on M. Let X be a complex analytic manifold. The map ψ : X → HC_{M}^{k} is called to be holomorphic if for any open subsets U ⊂ X and V ⊂ M and any holomorphic functions a(x, y)_{1}, ..., a(x, y)_{k} on U × V ψ(a_{1}, ..., a_{k}) is also a holomorphic function on U × V. We will denote the map ψ by ψ_{x}, x ∈ X, and call ψ_{x} a holomorphic family of k-differential operators on M.

Denote by HC_{M}^{k}(X) the space of all holomorphic maps X → HC_{M}^{k} and by (HC_{M}^{•}(X), ∂) the corresponding complex. It is clear that (HC_{M}^{•}(X), ∂) is naturally identified with the subcomplex of (HC_{X×M}^{•}, ∂) consisting of polydifferential operators along M. Denote by Λ^{k}T_{M}(X) the space of analytic maps of X to the space of polyvector fields on M.

5.2. Proposition. Let X be a Stein manifold. Then

H^{k}(HC_{M}^{•}(X), ∂) = Λ^{k}T_{M}(X).

Proof. The proof can be proceed in the similar way as for (HC_{M}^{•}, ∂), using that M and X are Stein manifolds, [HKR].

5.3. Proposition. Let 𝔤 be a simple Lie algebra, M a semisimple orbit in 𝔤^{∗}. Let X = X_{K^{2}=−1} be the manifold of invariant brackets f satisfying [[f, f]] = −φ_{M}. Then there exists a holomorphic family of multiplications μ_{f,h} on A of the form

μ_{f,h}(a, b) = ab + (h/2)f(a, b) + ∑_{n≥2} h^{n}μ_{f,n}(a, b), f ∈ X,  (5.1)

that is U(𝔤) invariant and Φ_{h} associative.

Proof. The proof is essentially follows to [DGS], Proposition 5.1, but here we construct the multiplication for all f simultaneously using parameterized Poisson cohomologies from the previous section.

To begin, consider the multiplication μ^{(1)}(a, b) = ab + (h/2)f(a, b). The corresponding obstruction cocycle is given by

obs_{2} = \frac{1}{h^{2}} (μ^{(1)}(μ^{(1)} ⊗ id) − μ^{(1)}(id ⊗ μ^{(1)})Φ_{h})

considered modulo terms of order h. No \frac{1}{h} terms appear because f is a biderivation and, therefore, a Hochschild cocycle. The fact that the presence of Φ_{h} does not interfere with the cocyle condition and that this equation defines a Hochschild 3-cocycle was proven in [DS1] (see the proof of Proposition 4.1 there). By Proposition 5.2 the differential Hochschild cohomology of A in dimension p is the space of holomorphic families of p-polyvector fields on M parameterized by X. Since 𝔤 is reductive, the subspace of 𝔤 invariants splits off as a subcomplex and has cohomology given by (Λ^{p}𝔪)^{𝔩}(X). The complete antisymmetrization of a p-tensor projects the space of invariant differential p-cocycles onto the subspace (Λ^{p}𝔪)^{𝔩}(X) representing the cohomology. The equation [[f, f]] + φ_{M} = 0 implies that the obstruction cocycle is a coboundary, and we can find a 2-cochain μ_{f,2}, so that μ^{(2)} = μ^{(1)} + h^{2}μ_{f,2} satisfies

μ^{(2)}(μ^{(2)} ⊗ id) − μ^{(2)}(id ⊗ μ^{(2)})Φ_{h} = 0 mod h^{3}.

Assume we have defined the deformation μ^{(n)} to order h^{n} such that Φ_{h} associativity holds modulo h^{n+1}, then we define the (n + 1)-st obstruction cocycle by

obs_{n+1} = \frac{1}{h^{n+1}} (μ^{(n)}(μ^{(n)} ⊗ id) − μ^{(n)}(id ⊗ μ^{(n)})Φ_{h}) mod h.

In [DS1] (Proposition 4.1) it is shown that the usual proof that the obstruction cochain satisfies the cocycle condition carries through to the Φ_{h} associative case. The coboundary of obs_{n+1} appears as the h^{n+1} coefficient of the signed sum of the compositions of μ^{(n+1)} with obs_{n+1}. The fact that Φ_{h} = 1 mod h^{2} together with the pentagon identity implies that the sum vanishes identically, and thus all coefficients vanish, including the coboundary in question. Let obs′_{n+1} ∈ (Λ^{3}𝔪)^{𝔩}(X) be the projection of obs_{n+1} on the totally skew symmetric part, which represents the cohomology class of the obstruction cocycle. The coefficient of h^{n+2} in the same signed sum, when projected on the skew symmetric part, is [[f, obs′_{n+1}]] which is the coboundary of obs′_{n+1} in the complex ((Λ^{•}𝔪)^{𝔩}(X), δ = [[f, .]]). Thus obs′_{n+1} is a δ cocycle. By Proposition 4.3, this complex has zero 3-cohomology. Now we modify μ^{(n+1)} by adding a term h^{n}μ_{f,n} with μ_{f,n} ∈ (Λ^{2}𝔪)^{𝔩} and consider the (n + 1)-st obstruction cocycle for μ^{′(n+1)} = μ^{(n+1)} + h^{n}μ_{f,n}. Since the term we added at degree h^{n} is a Hochschild cocyle, we do not introduce a h^{n} term in the calculation of μ^{(n)}(μ^{(n)} ⊗ id) − μ^{(n)}(id ⊗ μ^{(n)})Φ_{h} and the totally skew symmetric projection h^{n+1} term has been modified by [[f, μ_{f,n}]]. By choosing μ_{f,n} appropriately, we can make the (n+1)-st obstruction cocycle represent the zero cohomology class, and we are able to continue the recursive construction of the desired deformation.

5.4. Let X be as in Proposition 5.3. Let (o, ℂ[[h]]) be the formal manifold which is the formal neighborhood of 0 ∈ ℂ. We call a morphism π : o → X a formal path in X. A formal path π may be given by a formal series in h,

f(h) = f_{0} + hf_{1} + h^{2}f_{2} + ⋯, f_{k} ∈ (Λ^{2}𝔪)^{𝔩}

satisfying [[f(h), f(h)]] = −φ_{M}. It is clear that f_{0} ∈ X; we call it the origin of the path f(h). The element f_{1} belongs to the tangent space to X at the point f_{0}, which consists of elements v such that [[f_{0}, v]] = 0.

It is clear that if we put f = f(h) in the multiplication μ_{f,h}, we obtain a U(𝔤) invariant Φ_{h} associative multiplication which depends only on h and has the form

μ_{f(h),h}(a, b) = ab + (h/2)f_{0}(a, b) + ⋯

with φ_{M}-bracket f_{0}. In particular, we obtain

5.5. Corollary. Any φ_{M}-bracket on a semisimple orbit in 𝔤^{∗} can be quantized.

Proof. Indeed, let f_{0} be a φ_{M}-bracket. The multiplication corresponding to any path f(h) = f_{0} + hf_{1} + ⋯ with origin in f_{0} is as required.

5.6. Proposition. The multiplication μ_{f,h} of the form (5.1) has the following universal properties:

a) For any U(𝔤) invariant Φ_{h} associative multiplication m_{h}, there exists a formal path in X, f(h), such that m_{h} is equivalent to μ_{f(h),h}.

b) Multiplications corresponding to different paths are not equivalent.

We will need the following

5.6.1. Lemma. Let m_{h}(a, b) = ab + hf_{0}(a, b) + h^{2}m_{2}(a, b) + ⋯ be a multiplication and f(h) a path such that the multiplication μ_{f(h),h} coincides with m_{h} modulo h^{n+1}. Then there exist a path f′(h) = f(h) + h^{n}p_{1} + ⋯ and a differential operator D such that the multiplication m′_{h} = (1 + h^{n+1}D) ∘ m_{h},

m′_{h}(a, b) = (1 + h^{n+1}D)^{−1}m_{h}((1 + h^{n+1}D)a, (1 + h^{n+1}D)b),

coincides with μ_{f′(h),h} modulo h^{n+2}.

Proof. We have

μ_{f(h),h} = ab + hf_{0} + h^{2}m_{2} + ⋯ + h^{n}m_{n} + h^{n+1}μ_{n+1} + ⋯,

where m_{k}, k = 2, 3, ..., are terms appearing in the expansion of m_{h}. It is easy to check that μ_{n+1} − m_{n+1} is a Hochschild cocycle, because both μ_{n+1} and m_{n+1} resolve the same obstruction obs_{n+1},

obs_{n+1}(a, b, c) = ∑_{i,j≥1^{i+j=n}} (m_{i}(m_{j}(a, b), c) − m_{i}(a, m_{j}(b, c))) ,

depending only on m_{k}, k ≤ n. Hence, one has μ_{n+1} = m_{n+1} + ∂D + p_{1}, where D is a Hochschild 1-chain, i.e., a differential operator, and p_{1} is a bivector field. Applying 1 + h^{n+1}D to m_{h} we obtain

(1 + h^{n+1}D) ∘ m_{h} = μ_{f(h),h} − h^{n+1}p_{1} mod h^{n+2}.

Observe now that p_{1} is a δ_{f_{0}} cocycle, i.e., [[f_{0}, p_{1}]] = 0. This follows from the fact that (1 + h^{n+1}D) ∘ m_{h} is a Φ_{h} associative multiplication. Indeed, if [[f_{0}, p_{1}]] is not equal to zero, its contribution to obs_{n+2} is not a Hochschild coboundary. So, p_{1} is a tangent vector to X at f_{0}. Since, by Proposition 3.16 b), X is without singularities, there exists a formal path in X of the form p(h) = f_{0} +h^{n}p_{1} +· · ·. Let f′(h) be the path that in local coordinates on a neighborhood of f_{0} in X is the sum f(h)+p(h). It is clear that f′(h) = f(h)+h^{n}p_{1}+· · ·. Putting in μ_{f,h} f′(h) instead f(h) does not change the coefficients by h^{k}, k ≤ n, in μ_{f(h),h} and changes the (n + 1)-st coefficient adding p_{1} to it. So we have

m′_{h} = (1 + h^{n+1}D) ∘ m_{h} = μ_{f′(h),h} mod h^{n+2},

as required

5.6.2. Proof of Proposition 5.6. Let us prove a). Let f_{0} be the φ_{M}-bracket corresponding to the multiplication m_{h}. By Corollary 5.5 one can assume that m_{h} coincides modulo h^{2} with μ_{f(h),h} for the trivial path f(h) = f_{0}. Using Lemma 5.6.1 we find D_{2} and p_{1} such that the multiplication m_{h}^{(2)} = (1 + h^{2}D_{2}) ∘ m_{h} is equal modulo h^{3} to the multiplication corresponding to a path f_{1}(h) = f_{0} + hp_{1} + · · ·. Now we may apply the lemma to m_{h}^{(2)}, and so on. On the n-st step we obtain m_{h}^{(n)} = (1+h^{n}D_{n})...(1+h^{2}D_{2})∘m_{h} that corresponds modulo h^{n+1} to a path f_{n−1}(h) = f_{0} + hp_{1} + ⋯ + h^{n−1}p_{n−1} + · · ·. Let D = lim(1 + h^{n}D_{n})...(1 + h^{2}D_{2}), f(h) = lim f_{n−1}(h) in h-adic topology. It is clear that such limits exist. We obtain that D ∘ m_{h} = μ_{f(h),h}, which proves a).

The proof of b) using (Λ^{1}𝔪)^{𝔩} = 0 is left to the reader.

5.7. Remark. Let X_{0} be the variety of all nondegenerate Poisson brackets on M (see (3.21)). As in the proof of Proposition 5.3 one can construct a holomorphic family of U(𝔤) invariant associative multiplications of the form

μ_{p,t}(a, b) = ab + (t/2)p(a, b) + ∑_{n≥2} t^{n}μ_{p,n}(a, b),  p ∈ X_{0}.  (5.2)

The same argument as in the proof of Proposition 5.6 shows that such a family has the universal property: any U(𝔤) invariant deformation quantization on M is equivalent to the pullback of (5.2) by a unique formal path in X_{0}.

6. The G invariant Φ_{h} associative quantization in two parameters

6.1. Proposition. Let 𝔤 be a simple Lie algebra, M a semisimple orbit in 𝔤^{∗}. Let v be the KKS Poisson bracket on M. Let f be an invariant bracket on M satisfying [[f, f]] = −φ_{M}, [[f, v]] = 0. Then there exists a two parameter multiplication μ_{t,h} on A

μ_{t,h}(a, b) = ab + (h/2)f(a, b) + (t/2)v(a, b) + ∑_{k+l≥2} h^{k}t^{l}μ_{k,l}(a, b)  (6.1)

which is U(𝔤) invariant and Φ_{h} associative.

Proof. The existence of a multiplication which is Φ_{h} associative up to and including h^{2} terms is nearly identical to the proof of Proposition 5.3.

So, suppose we have a multiplication defined to order n,

μ_{t,h}^{(n)}(a, b) = ab + h/2f(a, b) + t/2v(a, b) + ∑_{2≤k+l≤n} h^{k}t^{l}μ_{k,l}(a, b),

which is U(𝔤) invariant and Φ_{h} associative to order h^{n}. Consider the obstruction cochain,

obs_{n+1} = ∑_{k=0,... ,n+1} h^{k}t^{n+1−k}b_{k}.

The same argument as in the proof of Proposition 5.3 shows that obs_{n+1} is a Hochschild cocycle. This means that all coefficients b_{k} are Hochschild cocycles. Hence, b_{k} = ∂a_{k} + β_{k} for all k, where β_{k} ∈ (Λ^{3}𝔪)^{𝔩}. Therefore,

obs_{n+1} = ∂a + β,

where a = ∑ h^{k}t^{n+1−k}a_{k}, β = ∑ h^{k}t^{n+1−k}β_{k}. The element β is a cocycle from C^{3}(n + 1) (see Subsection 4.4). By Proposition 4.5 b) there exists α ∈ C^{2}(n) such that [[hf +tv, α]] = β. This shows that, as in the proof of Proposition 5.3, we can modify μ_{t,h}^{(n)} adding a multiple of α to get a new multiplication to order n with (n+1)-st obstruction cocycle obs_{n+1} being a Hochschild coboundary, ∂a. So, we are able to continue the recursive construction of the desired two parameter deformation.

6.2. Remark. Let π : Y → X_{0} be the projection of the variety of good brackets over M to the variety of nondegenerate Poisson brackets (see Remark 3.27). In the similar way as Proposition 6.1, one can prove the existence of a family of multiplications of the form

μ_{f,t,h}(a, b) = ab + (h/2)f(a, b) + (t/2)(πf)(a, b) + ∑_{k+l≥2} h^{k}t^{l}μ_{f,k,l}(a, b), f ∈ Y. (6.2)

This family satisfies the universal property for two parameter quantizations, i.e., any two parameter U(𝔤) invariant Φ_{h} associative quantization on M of the form (6.1) is the pullback of a two parameter formal path in Y.

7. Polarization

7.1. We retain notations from the previous sections. Recall that M = G/L where G is a complex connected simple Lie group and L is a Levi subgroup. It is known that the natural projection π : G → M is a holomorphic principal fiber bundle with structure group L.

The tangent space to M, T (M), is the associated vector bundle corresponding to the Adaction of L on 𝔪, a unique 𝔩 invariant subspace in 𝔤 complement to 𝔩 (see Subsection 3.5). According to Proposition 3.6, T (M) may be presented as a direct sum of subbundles, T (M) = ⊕_{ᾱ∈\overline{Ω}}Tᾱ(M) where Tᾱ(M) is the associated vector bundle corresponding to 𝔪ᾱ.

Assigning to each g ∈ G the horizontal subspace g𝔪 provides G with an invariant connection ∇. This connection defines a G invariant connection on any associated vector bundle over M.

7.2. Let us choose \overline{Ω}^{+}, a system of positive quasiroots in \overline{Ω}. Let the subbundle T^{+}(M) = T\overline{Ω}^{+}(M) (T^{−}(M) = T\overline{Ω}^{−}(M)) correspond to 𝔪^{+} = ⊕_{ᾱ∈\overline{Ω}^{+}}𝔪ᾱ (𝔪^{−} = ⊕_{ᾱ∈\overline{Ω}^{−}}𝔪ᾱ).

By a realization M as an orbit in 𝔤^{∗} the decomposition T (M) = T\overline{Ω}^{+}(M)⊕T\overline{Ω}^{−}(M) defines complement polarizations of M with respect to the KKS symplectic form on M. These polarizations define two complement foliations on M, which are fibrating with the natural projections M → G/P^{+} and M → G/P^{−}, respectively, where P^{+}, P^{−} are upper and low parabolic subgroups containing L.

7.3. Let A^{+} = A\overline{Ω}^{+} denote the sheaf of holomorphic functions a on M constant along the polarization defined by \overline{Ω}^{+}, i.e., such that ∇_{X}a = 0 for any vector field X ∈ T\overline{Ω}^{+}(M).

7.4. Proposition. Let ν be a U(𝔤) invariant bidifferential operator on M vanishing on constants. Let \overline{Ω}^{+} be a system of positive quasiroots. Then ν(a, b) = 0 for any sections a, b ∈ A\overline{Ω}^{+}.

Proof. The connection ∇ induces an equivariant isomorphism of A modules between the sheaf of differential operators on A and the sheaf ST^{−}(M) ⊗_{A} ST^{+}(M), where ST^{−}(M) and ST^{+}(M) denote the sheaves of symmetric tensors over T^{−}(M) and T^{+}(M). It provides an equivariant isomorphism between the space of invariant bidifferential operators on M and the space ((S𝔪^{−} ⊗_{ℂ}S𝔪^{+})^{⊗2})^{𝔩}. Thus one may regard ν as a sum of terms of the form A_{1}B_{1} ⊗A_{2}B_{2} where A_{1}, A_{2} ∈ S𝔪^{−}, B_{1}, B_{2} ∈ S𝔪^{+}. Since ν is invariant, A_{1}B_{1} ⊗A_{2}B_{2} must be of weight zero with respect to the Cartan subalgebra 𝔥 ⊂ 𝔩. Since A_{1}B_{1} ⊗ A_{2}B_{2} is vanishing on constants, either B_{1} or B_{2} must belong to a positive symmetric power of 𝔪^{+}, let such be B_{1}. But the corresponding to B_{1} differential operator takes the functions of A^{+} to zero, therefore, the bidifferential operator corresponding to A_{1}B_{1} ⊗ A_{2}B_{2} when applies to the pair a, b ∈ A^{+} gives zero, too.

7.5. Corollary. All U(𝔤) invariant Φ_{h} associative multiplications are trivial on A\overline{Ω}^{+} for any choice of \overline{Ω}^{+}. It means that for any of such a multiplication, μ, one has μ(a, b) = ab whenever a, b ∈ A\overline{Ω}^{+}.

Proof. Indeed, μ has the form μ(a, b) = ab+{bidifferential operators vanishing on constants}. So, the corollary follows from Proposition 7.4.

7.6. Since μ from Corollary 7.5 when restricted to A^{+} coincides with the usual multiplication, it is associative in the usual sense. On the other hand, μ is Φ_{h} associative, so the usual and Φ_{h} associativities coincide on A^{+}. We will prove this fact independently in a more general setting.

Let V be a representation of L. Denote by V (M) the corresponding associated vector bundle on M. When this does not lead to confusion we will use the same notation V (M) for the sheaf of holomorphic sections of the bundle V (M). Denote by V^{+}(M) = V\overline{Ω}^{+}(M) the sheaf of holomorphic sections v of V (M) constant along the polarization defined by \overline{Ω}^{+}, i.e., such that ∇_{X}v = 0 for any vector field X ∈ T\overline{Ω}^{+}(M).

It is clear that V (M) is an A module under the natural multiplication m : A ⊗ V (M) → V (M), m(a, v) = av, a ∈ A, v ∈ V (M). Since A is commutative, V (M) may be considered as a two-sided module. This multiplication is obviously associative, i.e., for sections a, b ∈ A and v ∈ V (M) one has (ab)v = a(bv) and (av)b = a(vb). The following proposition shows that this multiplication being restricted to to A^{+} and V^{+}(M) is also Φ_{h} associative.

7.7. Proposition. Let V be a representation of L. Let us choose a system of positive quasiroots \overline{Ω}^{+}. Then for any sections a, b ∈ A^{+}, v ∈ V^{+}(M) one has abv = mΦ_{h}(a⊗b⊗v) and avb = mΦ_{h}(a ⊗ v ⊗ b).

Proof. Let us prove the first relation. The second relations can be proven similarly. Let Ω^{+} be a system of positive roots for 𝔤 which projects on \overline{Ω}^{+}. Let 𝔭^{+} be the corresponding parabolic subalgebra of 𝔩 and 𝔲^{+} the radical of 𝔭^{+}, so 𝔭^{+} = 𝔩 ⊕ 𝔲^{+}. Let 𝔭^{−} and 𝔲^{−} denote the corresponding opposite subalgebras, in particular, 𝔤 = 𝔭^{+} ⊕ 𝔲^{−} and 𝔤 = 𝔭^{−} ⊕ 𝔲^{+}.

Let π : G → M be the natural projection. Let U be an open set in M. Sections of V (M) over U can be identified with functions f : π^{−1}(U) → V such that f(gl) = l^{−1}f(g) for l ∈ L, g ∈ G. Sections of V^{+}(M) must, in addition, satisfy the condition E_{α}f = 0, α ∈ 𝔲^{+}, and the root vector E_{α} acts on f as a complex left-invariant vector field on G. In particular, functions of A^{+} are identified with functions ψ over π^{−1} such that E_{α}ψ = 0 for all α ∈ 𝔭^{+}.

Let us write Φ_{h} in the form

Φ_{h} = 1 ⊗ 1 ⊗ 1 + ∑_{k≥2} h^{k}Φ_{k}^{1} ⊗ Φ_{k}^{2} ⊗ Φ_{k}^{3},  (7.1)

where each Φ_{k}^{i}, i = 1, 2, belongs to S𝔭^{+} ⊗ S𝔲^{−} and Φ_{k}^{3} belongs to S𝔭^{−} ⊗ S𝔲^{+} (S𝔭^{+} denote the space of symmetric tensors over 𝔭^{+}, and so on). The total degree of each Φ_{k}^{1} ⊗Φ_{k}^{2} ⊗Φ_{k}^{3} is greater than zero.

Let us take a, b ∈ A^{+}, v ∈ V^{+}(M) and apply Φ_{k}^{1} ⊗ Φ_{k}^{2} ⊗ Φ_{k}^{3} to a ⊗ b ⊗ v Suppose Φ_{k}^{1}(a) and Φ_{k}^{2}(b) are not equal to zero. Then there are x_{1}, x_{2} ∈ S𝔲^{−} such that Φ_{k}^{1} = A ⊗ x_{1}, Φ_{k}^{2} = B ⊗ x_{2}, where A, B ∈ S𝔭^{+}. But since Φ_{k}^{1} ⊗ Φ_{k}^{2} ⊗ Φ_{k}^{3} is an invariant element, it must be of degree zero under the Cartan subalgebra. It follows that Φ_{k}^{3} has to be of the form Φ_{k}^{3} = C ⊗ x_{3} where C ∈ S𝔭^{−}, x_{3} ∈ 𝔲^{+} and x_{3} ≠ 0. Hence, Φ_{k}^{3}(v) = 0.

So, we have proven that in the expression (7.1) all terms except for the first when applying to a ⊗ b ⊗ v are equal to zero.

8. The real case

8.1. Let G be a real connected simple Lie group with complexification G^{ℂ}. Let 𝔤_{ℝ} be the Lie algebra of G with complexification 𝔤. Let L be the stabilizer of a semisimple element λ ∈ 𝔤_{ℝ}^{∗}, so that M = G/L may be identified with the coadjoint orbit passing through λ. It is well known that L is connected, therefore the complexification L^{ℂ} ⊂ G^{ℂ} is meaningful.

Denote by 𝔩_{ℝ} (𝔩) the Lie algebra of L (L^{ℂ}). Note that 𝔩 is a Levi subalgebra in 𝔤. The natural embedding M → M^{ℂ} = G^{ℂ}/L^{ℂ} may be regarded as complexification of M.

Let C^{a}(M) and C^{∞}(M) denote the sheaves of real analytic and smooth complex valued functions on M. The action of G on M defines a map of 𝔤_{ℝ} into the Lie algebra of real vector fields on M, 𝔤_{ℝ} → Vect_{ℝ}(M), that extends to a map 𝔤 → Vect(M) of 𝔤 into the Lie algebra of complex vector fields on M. It follows that U(𝔤) acts on the sections of C^{a}(M) and C^{∞}(M) as differential operators.

As a consequence we get that all the U(𝔤) invariant Φ_{h} associative multiplications constructed in the previous sections can be defined on the real manifold M.

8.2. Let us choose a system of positive roots, Ω^{+}, in 𝔤. Let P be the corresponding parabolic subgroup of G^{ℂ} with Levi factor L^{ℂ} and 𝔭 its Lie algebra. One has 𝔭 = 𝔩 ⊕ 𝔲^{+}, where 𝔲^{+} is the nilradical of 𝔭 We assume that 𝔭 is θ-stable parabolic, i.e., satisfies the condition

𝔤_{ℝ} ∩ 𝔭 = 𝔩_{ℝ}.  (8.1)

Then the natural map

M = G/L → G^{ℂ}/P  (8.2)

is an inclusion and the image is an open set. Thus the choice of Ω^{+} makes M into a complex manifold with holomorphic action of G. The corresponding system of positive quasiroots, \overline{Ω}^{+}, defines a complex polarization on M, whereas \overline{Ω}^{−} defines the complement polarization.

Note that for 𝔩_{ℝ} a θ-stable parabolic 𝔭 exists if 𝔩_{ℝ} is the centralizer of a semisimple element x ∈ 𝔤_{ℝ} such that ad(x) has imaginary eigenvalues.

One can prove, [Kn], that the smooth functions on M which are constant along the polarizations defined by \overline{Ω}^{+} and \overline{Ω}^{−} coincide with holomorphic and antiholomorphic functions on M.

8.3. Let S̃ denote the operator F_{h}σF_{h}^{−1}, where F_{h} is from Proposition 2.1.1 and σ is the usual permutation, acting on the tensor product of any two representations of U(𝔤). Let B_{h} be a quantized algebra of functions on M. We say that the multiplication on B_{h}, m_{h}, is S̃-commutative, if for any a, b ∈ B one has m_{h}(a ⊗ b) = m_{h}S̃(a ⊗ b).

8.4. Theorem. Let G be a real connected simple Lie group, L a Lie subgroup which is a stabilizer of a semisimple element λ ∈ 𝔤_{ℝ}^{∗}, and M = G/L. Let r ∈ ∧^{2}𝔤 be an r-matrix and U_{h}(𝔤) the corresponding quantum group. Let X be the variety of φ-brackets on M, as in Proposition 5.3, for φ = [[r, r]]. Then there exists a universal family of multiplications on C^{∞}(M) of the form

m_{f,h}(a, b) = ab + (h/2)(f − r_{M})(a, b) + ∑_{n≥2} m_{f,n}(a, b), f ∈ X,  (8.3)

which is U_{h}(𝔤) invariant and associative.

Suppose the map (8.2) induces a complex structure on M. Then m_{f,h} being restricted to the sheaf of holomorphic functions is S̃ commutative.

Proof. According to Proposition 2.2.1 we put m_{f,h} = μ_{f,h}F_{h}^{−1}, where the multiplication μ_{f,h} is from Proposition 5.3. The functions of C^{a}(M) are restrictions to M of holomorphic functions on M^{ℂ}. It follows that m_{f,h} is a well defined multiplication on C^{a}(M). Since the coefficients by h^{n} in m_{f,h} are bidifferential operators, this multiplication is defined, actually, for smooth complex valued functions on M. The universality of m_{f,h} follows from the universality of μ_{f,h}, see Proposition 5.6. The S̃-commutativity of m_{f,h} for holomorphic functions follows directly from the commutativity of μ_{f,h} for such functions, see Corollary 7.5.

8.5. As a consequence, we obtain the statement reverse to Corollary 2.3.3: any Poisson bracket on M of the form (2.8) admits a U_{h}(𝔤) invariant quantization. The proof is analogous to Corollary 5.5.

The following theorem contains, in particular, the statement reverse to Corollary 2.3.6.

8.6. Theorem. Let M be as in Theorem 8.4. Let v be the KKS Poisson bracket on M. Let r ∈ ∧^{2}𝔤 be an r-matrix and U_{h}(𝔤) the corresponding quantum group. Let p be a Poisson bracket on M of the form p = f − r_{M}, where f satisfies [[f, f]] = [[r, r]]_{M} and [[f, r_{M}]] = 0. Then there exists a two parameter U_{h}(𝔤) invariant associative multiplication on C^{∞}(M) of the form

m_{t,h}(a, b) = ab + h/2(f − r_{M})(a, b) + t/2v(a, b) + ∑_{k+l≥2} h^{k}t^{l}m_{k,l}(a, b).

Suppose the map (8.2) induces a complex structure on M. Then m_{f,h} being restricted to the sheaf of holomorphic functions is S̃ commutative.

Proof. We put m_{t,h} = μ_{t,h}F_{h}^{−1} and use the argument as in the proof of Theorem 8.4.

8.7. Remark. As follows from Corollary 7.5, any U_{h}(𝔤) invariant multiplications, in particular the multiplications from Theorems 8.4 and 8.6, being restricted to holomorphic functions are equal to m_{0}F_{h}^{−1}, where m_{0} is the usual multiplication.

9. The quantization of vector bundles

9.1. Let ρ : L → GL(V) be a representation of L in a complex vector space V. Then ρ extends holomorphically to L^{ℂ}. Let ρ_{ℂ} denote such an extension. The vector bundle on M, V (M), associated with ρ is the restriction of the vector bundle on M^{ℂ}, V (M^{ℂ}), associated with ρ_{ℂ}. Let us choose a system of positive quasiroots, \overline{Ω}^{+}, and the corresponding parabolic subgroup, P ⊃ L^{ℂ}. Let us assume that the map (8.2) defines the complex structure on M. Let ρ_{P} denote the extension of ρ_{ℂ} to P, which is trivial on the unipotent radical of P. Then V (M) is the pullback of the vector bundle on G^{ℂ}/P associated by ρ_{P}. So, V (M) acquires the structure of a holomorphic vector bundles on M. The holomorphic sections of V (M) form a sheaf V^{+}(M) whose sections are the restrictions to M of sections of V^{+}(M^{ℂ}) constant along the polarization defined by \overline{Ω}^{+}.

In the following we fix a system \overline{Ω}^{+} and the corresponding complex structure on M. Given a representation V of G, we denote by V (M) and V^{+}(M) the sheaves of smooth and holomorphic sections of the associated to V vector bundle on M.

9.2. Definition. Let A ⊂ C^{∞}(M) be a subsheaf of algebras and E a sheaf of A modules with a U(𝔤) action. Let m_{0} : A ⊗ A → A and n_{0} : A ⊗ E → E denote the multiplication in A and the action of A on E. Let A_{h} be a U_{h}(𝔤) invariant quantization of A with the deformed multiplication m_{h} = m_{0} + hm_{1} + o(h). We say that E[[h]] is a quantizqtion of E as a A_{h} module if a deformed U_{h}(𝔤) invariant action n_{h} = n_{0} + hn_{1} + o(h) : A ⊗ E → E[[h]] is given, which makes E[[h]] into an A_{h} module. In particular, it means that the associativity holds: n_{h}(m_{h}(a, b), x) = n_{h}(a, n_{h}(b, x)) for a, b ∈ A, x ∈ E.

Since A is commutative, E is, in fact, a two-sided module. So, in the similar way one defines a quantization of E as a two-sided A_{h} module.

9.3. Example. Let A^{+} be the sheaf of holomorphic functions on M. Then, as follows from Remark 8.7, there exists a unique U_{h}(𝔤) invariant quantization of A^{+}, A_{h}^{+}, and it has the multiplication of the form m_{h} = m_{0}F_{h}^{−1}. Let V be a representation of L. Then Proposition 7.7 and the argument of Theorem 8.4 show that the sheaf of holomorphic sections V^{+}(M) can be uniquely quantized as a left and even as a two-sided module. We denote this quantization by V_{h}^{+}(M). The left (right) multiplication by elements of A has the form n_{h}(a ⊗ x) = n_{0}F_{h}^{−1}(a ⊗ x) (n_{h}(x ⊗ a) = n_{0}F_{h}^{−1}(x ⊗ a)) for a ∈ A, x ∈ E.

The following proposition shows that the sheaf of smooth sections V (M) admits a U_{h}(𝔤) invariant quantization.

9.4. Theorem. Let A = C^{∞}(M) and A_{h} be a U_{h}(𝔤) invariant quantization of A. Let V be a representation of L. Then there exists a U_{h}(𝔤) invariant quantization of V (M) as a left A_{h} module.

Proof. We have V (M) = A ⊗_{A^{+}} V^{+}(M). Let V_{h}(M) = A_{h} ⊗_{A_{h}^{+}} V_{h}^{+}(M), where A_{h} is considered as a right and V_{h}^{+}(M) as a left A_{h}^{+} module (see Example 9.3). It is clear that V_{h}(M) is the required quantization.

9.5. The two-sided quantization. In general, it is not clear whether a two-sided quantization of V (M) exists. However we will show that there is a quantization, A_{h}, of the sheaf of smooth functions on M such that for any representation V there exist a quantization of V (M) as a two-sided A_{h} module.

Let us construct the quantization A_{h}. Let

R = F_{h}e^{ht/2}F_{h}^{−1} = R′_{i} ⊗ R′′_{i} ∈ U_{h}(𝔤) ⊗ U_{h}(𝔤),

where t ∈ 𝔤 ⊗ 𝔤 is the split Casimir, be the R-matrix (summation by i is assumed). It satisfies the property, [Dr2],

Δ′(x) = RΔ(x)R^{−1}, x ∈ U_{h}(𝔤),

where Δ is the comultiplication in U_{h}(𝔤) and Δ′ the opposite one,

(Δ ⊗ 1)R = R^{13}R^{23} = R′_{i} ⊗ R′_{j} ⊗ R′′_{i}R′′_{j}
(1 ⊗ Δ)R = R^{13}R^{12} = R′_{i}R′_{j} ⊗ R′′_{j} ⊗ R′′_{i},

and

(1 ⊗ ε)R = (ε ⊗ 1)R = 1 ⊗ 1,

where ε is the counit in U_{h}(𝔤).

The element R defines the U_{h}(𝔤) equivariant map S : E ⊗ F → F ⊗ E, a ⊗ b ↦ σR(a ⊗ b), σ is the usual permutation, for any U_{h}(𝔤) modules E and F.

Let A′_{h} = A_{h}^{+}⊗_{ℂ}A_{h}^{−}, where A_{h}^{−} is a unique U_{h}(𝔤) invariant quantization of the sheaf A^{−} of antiholomorphic functions on M. We provide A′_{h} with the structure of a sheaf of algebras in the following way. For a = a_{1} ⊗ b_{1}, b = a_{2} ⊗ b_{2} ∈ A′_{h}, we put m_{h}(a, b) = a_{1}a′_{2} ⊗ b′_{1}b_{2}, where a′_{2} ⊗ b′_{1} = S(b_{1} ⊗ a_{2}) and a_{1}a′_{2} and b′_{1}b_{2} means the multiplications in A_{h}^{+} and A_{h}^{−}, respectively. It easily follows from the above properties of R that the multiplication m_{h} is U_{h}(𝔤) invariant and associative on A′_{h} and is presented as a power series in h with coefficients being bidifferential operators on A. Since bidifferential operators on smooth functions are fully defined by their values on A^{+}⊗A^{−}, this multiplication can be extended to the whole algebra A of smooth functions on M.

One can show that the Poisson bracket of the obtained quantization A_{h} is the bracket reduced to M from the Poisson bracket r′ − r′′ on the group G, see Remark 2.3.4.

9.6. Theorem. Let A_{h} be the quantization of the sheaf of smooth functions on M constructed above. Let V be a representation of L. Then there exists a quantization of the sheaf of smooth sections of V (M) as a two-sided A_{h} module.

Proof. Let V′_{h}(M) = V_{h}^{+}(M)⊗_{ℂ}A_{h}^{−}. Let us define left and right multiplications of elements of V′_{h}(M) by elements of A′_{h} = A_{h}^{+} ⊗_{ℂ}A_{h}^{−}. Let a = a_{1} ⊗b_{1} ∈ A′_{h} and x = x_{1} ⊗b_{2} ∈ V′_{h}(M). Put n_{h}^{left}(a ⊗ x) = a_{1}x′_{1} ⊗ b′_{1}b_{2}, where x′_{1} ⊗ b′_{1} = S(b_{1} ⊗ x_{1}), and n_{h}^{right}(x ⊗ a) = x_{1}a′_{1} ⊗ b′_{2}b_{1}, where a′_{1}⊗b′_{2} = S(b_{2}⊗a_{1}). Here a_{1}x′_{1} means, for example, the multiplication when V_{h}^{+}(M) is considered as a left A_{h}^{+} module. It easily follows from the above properties of R that the multiplications n_{h}^{left} and n_{h}^{right} make V′_{h}(M) into a two-sided A′_{h} module. The same argument as in the proof of Theorem 9.4 shows that those multiplications define, in fact, the structure of a A_{h} module on V (M)[[h]].

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Dept. of Math. Bar-Ilan University, 52900 Ramat-Gan, Israel

Current address: Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn

E-mail address: donin@mpim-bonn.mpg.de,  donin@macs.biu.ac.il