LECTURE 1. INTRODUCTION TO HAMILTONIAN GROUP ACTIONS 299(c) If two symplectic manifolds M 1 and M2 are acted on in a Hamiltonian
fashion by a group G with moment maps μ 1 and μ 2 then the moment map for thediagonal action of G on M 1 x M2 with the product symplectic structure is μ 1 + μ 2.
Example 1.7. The standard symplectic form on S^2 is w = -dcosBd</> = -dzd</>
(where </> is the azimuthal angle and z is the height function). The associatedmoment map for the action of U(l) on S^2 by rotation about the z axis is μ(z, </>) = z.
Example 1.8. If JR.^2 = e has the symplectic structure w = dxdy, t he moment
map for the standard action of U(l) on JR.^2 with multiplicity m E Z is μ(x, y) =
- m(x^2 + y^2 )/2.
Example 1.9. Let U(n) act on e n in the standard way. Then the moment map
fl for the action of U(n) on epn-^1 is given by the formula
(1.1) μ '([ Z 1 , · · · , Zn ]) ij = '°'n ZiZj I
6k=l Zk^12 ·
The standard symplectic form on e n descends under reduction to the standard
symplectic form on epn-^1 (which corresponds to the Fubini-Study metric).
Example 1.10. Suppose a torus Tacts one preserving the standard symplectic
structure, and suppose t he action factors through a homomorphism B : T---+ U(l)
which can be written
B( expT X) = expu(l) (,B(X))
in terms of a linear map ,B E t * which maps the integer lattice oft into Z (in other
words a weight) and t he exponential maps
expT : t ---+ Tand
expU(l) : JR---+ U(l)
(the latter being normalized as expu(l)(t) = e^2 7rit). Then by Proposition 1.6 (b)
and Example 1.8 we see that t he moment map for the action of T on e is
1μ(z) = -2,Blzl2·
It follows t hat if T acts on en via a collection of weights ,81, ... , ,Bn E t * then t he
moment map is
1 n
μ(z1, ···,Zn)= -2 L lz11^2 ,Bj,
j=l
and the image of the moment map is the cone in t * spanned by {,81, ... , ,Bn}.
1.2. The symplectic quotient
Because the moment map μ is equivariant, we may form the symplectic quotient
(or Marsden-Weinstein reduction)
Mred =Mo= μ-^1 (0)/G.
The space Mo inherits a symplectic structure from the symplectic structure on M.
Corollary 1.4 implies that if 0 is a regular value of μ then Mo is an orbifold or
V-manifold (see [35]).