§5. Polarizations 27
We say that h is a real algebraic polarization of F if in addition the
condition
(ii) codimg h =
1
2
rk BF (i.e. h has maximal possible dimension
dim g+rk g
2
)
is satisfied.
The notion of a complex algebraic polarization is defined in the
same way: we extend F to gC by complex linearity and consider complex
subalgebras h
gC
that satisfy the equivalent conditions (i) or (i ) and the
condition (ii).
An algebraic polarization h is called admissible if it is invariant under
the adjoint action of Stab(F ). Note, that any polarization contains the Lie
algebra stab(F ), hence is invariant under the adjoint action of
Stab0(F
), the
connected component of unity in Stab(F ).
The relation of these “algebraic” polarizations to “geometric” ones de-
fined earlier is very simple and will be explained later (see Theorem 5). It
can happen that there is no real G-invariant polarization for a given F g∗.
The most visual example is the case G = SU(2) where g has no subalgebras
of dimension 2.
However, real G-invariant polarizations always exist for nilpotent and
completely solvable Lie algebras while complex polarizations always exist for
solvable Lie algebras. It follows from a remarkable observation by Michele
Vergne.
Lemma 9 (see [Ver1, Di2]). Let V be a real vector space endowed with a
symplectic bilinear form B. Consider any filtration of V :
{0} = V0 V1 · · · Vn = V
where dim Vk = k. Denote by Wk the kernel of the restriction B
Vk
. Then
a) The subspace W =

k
Wk is maximal isotropic for B.
b) If in addition V is a Lie algebra, B = BF for some F V

and all
Vk are ideals in V , then W is a polarization for F .
Note that in [Di2] it is also shown that for a Lie algebra g over an
algebraically closed field K the set of functionals F
g∗
that admit a
polarization over K contains a Zariski open subset, hence is dense in
g∗.
Example 10. Let G = Sp(2n, K), K = R or C. The Lie algebra g
consists of matrices of the form SJ where S is a symmetric matrix with
elements from K and J =
0n −1n
1n 0n
. The dual space
g∗
can be identified
with g using the pairing (X, Y ) = tr (XY ). Consider the subset Ω
g∗
given by the condition rk X = 1. This set is a single G-orbit in the case
K = C and splits into two G-orbits Ω± in the case K = R.
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