16 1. Geometry of Coadjoint Orbits
Indeed, this center consists of functions f such that
cijXk∂if k
= 0, 1 j n.
But this means exactly that f is annihilated by all Lie vector fields K∗(Xj),
1 j n, hence is K(G)-invariant.
4. The moment map
4.1. The universal property of coadjoint orbits.
We have seen that any coadjoint orbit is a homogeneous symplectic man-
ifold. The converse is “almost true”: up to some algebraic and topological
corrections (see below for details) any homogeneous symplectic manifold is
a coadjoint orbit.
This theorem looks more natural in the context of Poisson G-manifolds
(see Appendix II.3.2 for the introduction to Poisson manifolds).
In this section we always assume that G is connected.
Let us define a Poisson G-manifold as a pair (M, f(·)
M
) where M is
a Poisson manifold with an action of G and f(·)
M
: g
C∞(M)
: X
fX
M
is a homomorphism of Lie algebras such that the following diagram is
commutative:
(19)
g
L(·)
−−−→ V ect(M)
f
M
(·)

⏐s-grad
C∞(M)
where LX is the Lie field on M associated with X g and s-grad(f) denotes
the skew gradient of a function f, i.e. the vector field on M such that
s-grad(f)g = {f, g} for all g
C∞(M).
For a given Lie group G the collection of all Poisson G-manifolds forms
the category P(G) where a morphism α :
(
M, f(·)
M
)

(
N, f(·)
N
)
is a smooth
map from M to N which preserves the Poisson brackets: {α∗(φ), α∗(ψ)} =
α∗({φ,
ψ}) and makes the following diagram commutative:
(20)
C∞(N)
α∗
−−−→
C∞(M)
f
N
(·)



⏐f(M)·
g
id
−−−→ g
Observe that the last condition implies that α commutes with the G-
action.
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