24 1. Geometry of Coadjoint Orbits

Definition 3. Let (M, σ) be a symplectic manifold. A real polarization

of (M, σ) is an integrable subbundle P of the tangent bundle TM such that

each fiber P (m) is a maximal isotropic subspace in the symplectic vector

space (TmM, σ(m)). In particular, the dimension of P is equal to

1

2

dim M.

Recall that a subbundle P is called integrable if there exists a foliation

of M, i.e. a decomposition of M into disjoint parts, the so-called leaves,

such that the tangent space to a leaf at any point m ∈ M is exactly P (m).

To formulate the necessary and suﬃcient conditions for the integrability

of P we need some notation.

Let us call a vector field ξ on M P -admissible if ξ(m) ∈ P (m) for all

m ∈ M. The space of all P -admissible vector fields is denoted by V ectP (M).

The dual object is the ideal ΩP (M) of all P -admissible differential forms

ω on M which have the property:

ω(ξ1,...,ξk) = 0 for any P -admissible vector fields ξ1,...,ξk, k = deg ω.

Frobenius Integrability Criterion. The following are equivalent:

a) A subbundle P ⊂ TM is integrable.

b) The vector space V ectP (M) is a Lie subalgebra in V ect(M).

c) The vector space ΩP (M) is a differential ideal in the algebra Ω(M).

In practice only those polarizations that are actually fibrations of M are

used. In this case the set of leaves is itself a smooth manifold B and M is

a fibered space over B with leaves as fibers. These leaves are Lagrangian

(i.e. maximal isotropic) submanifolds of M.

Let CP

∞(M)

denote the space of smooth functions on M which are con-

stant along the leaves. In fact it is a subalgebra in

C∞(M)

which can also

be defined as the set of functions annihilated by all admissible vector fields.

Lemma 8. A subbundle P ⊂ TM of dimension

1

2

dim M is a polarization

iff CP

∞(M)

is a maximal abelian subalgebra in the Lie algebra

C∞(M)

with

respect to Poisson brackets.

Proof. Assume that P is a polarization. The space VectP (M) consists of

vector fields tangent to the fibers of P . Therefore, for f ∈ CP

∞(M)

we

have df ∈ ΩP

1

(M). It follows that s-grad f(m) is σ-orthogonal to P (M),

hence belongs to P (M). So, for any f1,f2 ∈ CP

∞(M)

we have {f1, f2} =

(s-grad f1)f2 = 0.

Moreover, if (s-grad f1)f2 = 0 for all f2 ∈ CP

∞(M),

then s-grad f(m) ∈

P (m) and f1 is constant along the fibers, hence belongs to CP

∞(M).

We

have shown that CP ∞(M) is a maximal abelian Lie subalgebra in C∞(M).