§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.