§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
C
there corresponds the subalgebra h = p∗(F
)−1(P
(F ))
gC.
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