LECTURE 4. THE DUISTERMAAT-HECKMAN THEOREM 319Thus we need to prove the result only for torus actions. A sketch of the proof
when G = U(l) [27] follows: We write
TJ= DC/~x)Suppose 0 is a regular value ofμ. Then μ-^1 (JR+) is a manifold with boundary
μ -^1 (0). One may show that
(4.6) f,
()rye iw f,. _
Resx-o = K(rye'w).
- μ - l(O) X - d() μ-l(O)/G
(since in the U(l) case the map K may be written as
(4.7)where p : μ -^1 (0) --+ μ-^1 (0)/ G is integration over the fibre). Applying the equi-
variant Stokes' theorem to μ-^1 (tR+) and then taking the residue at X = 0 we find
that
R esx=o f, _ ()ry '""' R iμ(F)X r ryeiw -
μ^1 (O) d() _ X - FEFμ(F)>O L., esx=oe J F F eF(X) - 0,(4.8)
which is exactly the U(l) case of the residue formula.
Nonabelian localization has had two major applications thus far. The first is
that the residue formula has been used in [25] to give a proof of formulas for inter-
section numbers in moduli spaces of vector bundles on Riemann surfaces: some of
the background underlying these results is described in Lecture 5. The second is
that nonabelian localization underlies some proofs (see e.g. [26], [37]) of a conjec-
ture of Guillemin and Sternberg [18] that "quantization commutes with reduction":
in other words that the G invariant part of the quantization (see Section 4.3) of
a symplectic manifold equipped with a Hamiltonian G action is isomorphic to the
quantization of the reduced space Mred. For an expository account and references
on results about this conjecture, see the recent survey article by Sjamaar [36].
4.4.2. The residue formula by induction
Guillemin and Kalkman [16] and independently Martin [32] have given an alterna-
tive version of the residue formula which uses the one-variable proof inductively:
Theorem 4.9. (Guillemin-Kalkman; Martin) Suppose M is a symplectic
manifold acted on by a torus Tin a Hamiltonian fashion, and TJ E HT(M). Then
1 K(TJ) = L, 1 Ki(ResiTJ).
Meed i (M;),·cdHere, M i is the fixed point set of a one parameter subgroup Ti of T (so that μr(Mi)
are critical values of μr ): it is a symplectic manifold equipped with a Hamiltonian
action of T/Ti and the natural map Ki : H:;,/T;(Mi) --+ H *((Mi)red)- The mapResi : Hf(M) --+ H:;,/T; (Mi ) is defined by
(4.9) ResiT/ = Resx;=o(iM;TJ)
where