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.