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
Consider the action of Stab(F ) on TF Ω
in more detail. We
can write the matrix F in the form F =
for some column vector
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
According to the Witt
this action admits only two non-trivial invariant subspaces: the
1-dimensional space Kv and the (2n − 1)-dimensional space
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 ⊂
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∗
(g · m0)).
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 .