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).
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