2.1. Convexity theorems
Lecture 2. The Geometry of the Moment Map
Theorem 2.1. (Atiyah [1]; Guillemin-Sternberg [19]) Suppose M is a con-
nected compact symplectic manifold equipped with a Hamiltonian action of a torus
T. Then the image μ(M) is a convex polytope, the convex hull of {μ(F)} where F
are the components of the fixed point set of T in M.
Example 2.2. Consider the orbits 0-' of SU(2) in su(2) ~ JR^3 through).. E JR+z.
The image of the moment map for the action of the maximal torus T ~ U(l) is the
interval [->., >.].
Example 2.3. When 0-' is the coadjoint orbit (through).. Et* ) for a compact Lie
group G with maximal torus T, the image μr(O>-) of the moment map μr for the
action of the maximal torus T is the convex hull Conv{ w>. : w E W} where W is
the Weyl group.
The convexity theorem above can be generalized to actions of nonabelian
groups. If M is a connected compact symplectic manifold equipped with a Hamil-
tonian action of a compact Lie group G with maximal torus T and positive Weyl
chamber t+, then the intersection of the image μ(M) of the moment map with the
positive Weyl chamber t+ is a convex polytope [19, 29].
I shall not attempt to give a proof of the convexity theorem in these lectures:
the proofs of Atiyah and of Guillemin-Sternberg both rely heavily on Morse theory
applied to the moment map.
2.2. The moment polytope
Given a compact symplectic manifold M equipped with the Hamiltonian action of
a torus T , we see that there is an associated polytope P , the moment polytope. The
fibres of the moment map μ are preserved by the action of T , so the value ofμ
parametrizes a family {Mt} of symplectic quotients.
By Proposition 1.3, we see that the moment polytope is decomposed according
to the stabilizers of points in the preimage, and the critical values of the moment
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