318 L. C. JEFFREY, HAMILTONIAN GROUP ACTIONS
4.4.1. The residue formula
A related result is the residue formula, Theorem 8.1 of [23]:
Theorem 4. 7. {[23], corrected as in [25])
Let T/ E H(; ( M) induce T)o E H * ( Mred). Then we have
(4.3) 1 "'(TJ)eiw,.d = n 0 CcRes (v^2 (X) L h'fa,(X)[dX]),
Meed FEFwhere n 0 is the order of the stabilizer in G of a generic element of μ-^1 (0), and the
constant cc is defined by
(4.4)G (-l)s+n+
C = -IW____,l-vo-1-(T-)"We have introduced s =dim G and l =dim T; here n+ = (s - l)/2 is the number of
positive roots.^1 Also, F denotes the set of components of the fixed point set of T,
and if F is one of these components then the meromorphic function h fa, on t 0 <C is
defined by
(4.5) hri (X) = eiμ(F)(X) ( i'FTJ(X)eiw
F j F ep(X)
and the polynomial V : t--+ IR is defined by V(X) = TI,>o 1(X), where/ runs over
the positive roots of G.
The residue map Res is defined on (a subspace of) the meromorphic differential
forms on t 0 <C: its definition depends on some choices, but the sum of the residues
over all F E F is independent of these choices. When T = U(l) we define the
residue on meromorphic functions of the form ei>-.X / X N when .A "I- 0 (for N E Z)
by
= 0, if .A< 0.More generally the residue is specified by certain axioms (see [23], Proposi-
tion 8.11), and may be defined as a sum of iterated multivariable residues
Resx,=>-.i ... Resxi=>-.i for a suitably chosen basis of t yielding coordinates
X1,... , X1 (see [26]).
The main ingredients in the proof of Theorem 4. 7 are the normal form theorem
(Proposition 1.14) and the abelian localization theorem (Theorem 3.10). We outline
a proof as follows. First (following S. Martin [32]) we may reduce to symplectic
quotients by the action of the maximal torus T:
Proposition 4.8. [32] We have
f,
"'(TJeiw) = _1_ f, "'(VTJeiw) = (-l)n+ f, "'(V2·ryeiw).
μ -^1 (0)/G JWI μ -^1 (0)/T IWI μ:y^1 (0)/T(^1) Here, the roots of G a r e the nonzero weights of its complexified adjoint action. We fix the
convention that weights f3 Et* satisfy f3 E Hom(A^1 ,Z) rather than f3 E Hom(A^1 ,27rZ) (where
A^1 = Ker( exp : t -> T) is the integer lattice). This definition of roots differs by a factor of 27r
from the definition used in [23].