§5. Polarizations 29

The following diagram is clearly commutative:

(32)

Q(e) = h −−−→ g

(Lg)∗(e)

− −−−− → Tg(G) ←−−− Q(g)

p∗(e)

⏐

⏐

p∗(e)

⏐

⏐

⏐

⏐

p∗(g)

⏐

⏐

p∗(g)

P (m0) −−−→ Tm0 M

(g ·)∗(m0)

− −−−−− → Tg·m0 M ←−−− P (g · m0)

(Here Lg is a left shift by g ∈ G and an asterisk as a lower index means the

derivative map.)

It follows that Q is a left-invariant subbundle of TG. Conversely, every

left-invariant subbundle Q ⊂ TG with the property h := Q(e) ⊃ Lie(K) can

be obtained by this procedure from a G-invariant subbundle P ⊂ TM: we

just define P (g · m0) as p∗(g)Q(g).

b) Being G-invariant, P is spanned by G-invariant vector fields X, X ∈

h. By the Frobenius criterion P is integrable iff the space V ectP (M) is a

Lie subalgebra. But the last condition is equivalent to the claim that h is a

Lie subalgebra in g.

Let us return to the proof of Theorem 5.

For a given subbundle P ⊂ T Ω we define h as in Proposition 6 (with

Ω in the role of M and F in the role of m0). We saw in Section 2.1 that

p∗(e)σ(F

) = BF . Therefore P (F ) is maximal isotropic with respect to σ

iff the same is true for h with respect to BF . The remaining statements of

Theorem 5 follow from Proposition 6.

Remark 6. Let h be a real polarization of F ∈

g∗,

let P be the real

polarization of Ω which corresponds to h, and let H = exp h. Then the leaves

of the G-invariant foliation of Ω determined by P have the form K(gH)(F ).

In particular, the leaf passing through F is an orbit of coadjoint action of

H. ♥

In conclusion we note that Theorem 5 can be easily reformulated and

proved in the complex case. It looks as follows.

Theorem 5 . There is a bijection between the set of all G-invariant complex

polarizations P of a coadjoint orbit Ω ⊂

g∗

and the set of all complex alge-

braic polarizations h of a given element F ∈ Ω. As before, to a polarization

P ⊂ TΩ

C

there corresponds the subalgebra h = p∗(F

)−1(P

(F )) ⊂

gC.