1549055384-Symplectic_Geometry_and_Topology__Eliashberg_

304 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS

map are the images μ r (Wj) of the fixed point sets Wj of one parameter subgroups
Sj of T. These critical values form hyperplanes ("walls") which subdivide the
moment polytope: the complement of the walls is a collection of open regions
consisting of regular values of the moment map.^1
Example 2.4. The group SU(3) has maximal torus T ~ U(1)^2. For .A E t* the
Weyl group images of .A are the six vertices of a hexagon: the "walls" in the moment
polytope for the action of T are the edges of the hexagon (exterior walls) and the
three lines connecting opposite vertices (interior walls).

2.3. The Duistermaat-Heckman theorem, version I

One version of the Duistermaat-Heckman theorem asserts that the pushforward

of the symplectic or Liouville measure by the moment map for a torus action is a

piecewise polynomial function. (More precisely it is a polynomial function on regions

consisting of regular values of μ: one sees from Proposition 1.3 that the critical

values of μ consist of "walls" which are the images under μ of fixed point sets of

one parameter subgroups of T.) We shall meet another version of the Duistermaat-

Heckman theorem in Lecture 4 (Theorem 4.1), which is most easily treated using

equivariant cohomology. Duistermaat and Heckman extracted Theorem 4.1 from

the version of the theorem presented in this section via an approach involving

stationary phase: see [13], Theorem 4.1.

Theorem 2.5. (Duistermaat-Heckman, version I) Let M be a symplectic

manifold of dimension 2n acted on in a Hamiltonian fashion by a torus T. As-

sume thatμ is proper (although M need not be compact). Define the pushforward

μ* (wn /n!) of the Liouville measure wn /n! on M by

1

w~ h(μ(m)) = r. μ*t~ )h(t) :

mEM n. ltEt n.

where h E C^00 (t*) is any function oft*. Then the pushforward is given by

μ *(wn/n!) = f(t)dt

where dt is Lebesgue measure on t and the function f : t -t IR is polynomial on

any region consisting of regular values ofμ.

Lemma 2.6. Suppose t is a regular value ofμ. Then f(t) = vol (Mt) where Mt=

μ -^1 (t)/T is the symplectic quotient at t.

Proof: This follows from the normal form theorem (Proposition 1.14).

Because of Lemma 2.6, an application of Corollary 1.15 yields the Theorem 2.5

as an immediate consequence.

The following Lemma links the version of the Duistermaat-Heckman theorem

presented here with the version presented in Section 4.

Lemma 2.7. Let M be a compact symplectic manifold. Define g: t -t C by

g(X) = 1 eiw eiμ(m)(X).

mEM

Then g = in J where J is the Fourier transform off : t* -t IR and the function f

was defined in Theorem 2. 5.

(^1) Note that not all points in the preimage μ- (^1) (μ(Wj)) of a wall will be fixed by a subtorus of
dimension 2': 1.