28 1. Geometry of Coadjoint Orbits

We show that for n ≥ 2 there is no algebraic polarization for any F ∈ Ω.

Indeed, it is easy to compute that rk BF = 2n for F ∈ Ω. So, the would-be

polarization h must have codimension n in g. It also must contain stab(F ),

hence, h/stab(F ) has to be an n-dimensional Stab(F )-invariant subspace in

TF Ω

∼

=

g/stab(F )

∼

=

K2n.

Consider the action of Stab(F ) on TF Ω

∼

=

K2n

in more detail. We

can write the matrix F in the form F =

vvtJ

for some column vector

v ∈

K2n.

Two vectors v and v define the same functional F if v = ±v .

Therefore Stab(F ) consists of matrices g satisfying gv = ±v. The connected

component of the unit in Stab(F ) is defined by the condition gv = v. It is

the so-called odd symplectic group Sp(2n − 1, R) (see Section 6.2.2).

The map v → F = vvtJ is G-equivariant and identifies the linear action

of Stab(F ) on TF Ω with the standard one on

K2n.

According to the Witt

theorem,8

this action admits only two non-trivial invariant subspaces: the

1-dimensional space Kv and the (2n − 1)-dimensional space

(Kv)⊥.

Hence,

an n-dimensional invariant subspace can exist only for n = 1. ♦

Now we explain the relation between the notions of geometric and alge-

braic polarization. As before, denote by pF the map from G onto Ω defined

by pF (g) = K(g)F , and by (pF )∗ its derivative at e which maps g onto TF Ω.

Theorem 5. There is a bijection between the set of G-invariant real polar-

izations P of a coadjoint orbit Ω ⊂

g∗

and the set of admissible real algebraic

polarizations h of a given element F ∈ Ω.

Namely, to a polarization P ⊂ T Ω there corresponds the algebraic polar-

ization h = (pF )∗ −1(P (F )).

Proof. We use the following general result

Proposition 6. Let M = G/K be a homogeneous manifold.

a) There is a one-to-one correspondence between G-invariant subbundles

P ⊂ TM and K-invariant subspaces h ⊂ g containing Lie(K).

b) The subbundle P is integrable if and only if the corresponding subspace

h is a subalgebra in g.

Proof. a) Choose an initial point m0 ∈ M and denote by p the projection

of G onto M acting by the formula p(g) = g ·m0. Let p∗(g) be the derivative

of p at the point g ∈ G. Define a subbundle Q ⊂ TG by

Q(g) := p∗

−1(g)(P

(g · m0)).

8The

Witt theorem claims that in a vector space V with a symmetric or antisymmetric

bilinear form B any partial B-isometry A0 : V0 → V can be extended to a global B-isometry

A : V → V .