1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

298 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

for any YE TmM: in other words the function μx : M---+ IR defined by

μx(m) d~f μ(m)(X)

is the Hamiltonian function generating the vector field X#.

(b) μ: M---+ g * is equivariant (where G acts on g * by the coadjoint action).

Remark 1.1. In these lectures we shall only consider actions of compact connected

Lie groups, although the definition of Hamiltonian group action may be extended

to noncompact groups. In particular the term "torus" refers to the compact torus

T ~ U(l)n.

Remark 1.2. Existence of moment maps. One sees that Lx#W = d(ix#w) ,

so that ix#W is closed. The moment map μ x exists if and only if lx#W is also

exact. The moment map need not always exist: for example, if S^1 acts on T^2 by

eiX : ( ei81, ei82) 1---+ ( ei(8^1 +x), ei82)

we see that for the standard symplectic form w = dB 1 A dB2 we have ix#W = dB 2.

Since B2 is only defined mod 2n we see that the moment map does not exist. Con-

ditions guaranteeing the existence of a moment map (other than M being simply

connected) include the hypothesis that G is semisimple ([21], Theorem 26.1); con-

ditions on existence and uniqueness of the moment map can be formulated in terms

of Lie algebra cohomology.

1.1. Some elementary properties of moment maps

Proposition 1.3. (Guillemin-Sternberg [19])

Im(dμm)l_ = Lie(Stab(m)),

where .l_ denotes the annihilator under the canonical pairing g* ® g ---+ IR.

Proof: We have

w(Y;!t,Z) = dμy(Z) = (Y,dμm(Z))

for all Z E TmM. Thus Y annihilates all ~ E Im(dμm) if and only if Y E

Lie( Stab( m)).

Corollary 1.4. Zero is a regular valu e ofμ if and only if Stab( m) is finite for all

m E μ-^1 (0). In this situation μ-^1 (0) is a manifold and the action of G on it is

locally free.

Example 1.5. Let T be a torus acting on Mand let F c MT be a component of

the fixed point set. Then for any f E F we have dμ f = 0, so μ(F) is a point.

Proposition 1.6. (a) If H C G are two groups acting in a Hamiltonian fashion

on a symplectic manifold M, then μ H = n o μc where n : g ---+ h is the projection

map. In other words if XE h then μH(m)(X) = μc(m)(X) for any m EM. One

example that frequently arises is the case when H = T is a maximal torus of a

compact Lie group G.

(b) More generally if f : H ---+ G is a Lie group homomorphism, and the two

groups G and H act in a Hamiltonian fashion on a symplectic manifold M, in such

a way that the action is compatible with the homomorphism f, then μH = f* o μc

where f : g ---+ h * is induced from the homomorphism f. (The case (a) is the

special case where f is the inclusion map.)