LECTURE 1. INTRODUCTION TO HAMILTONIAN GROUP ACTIONS 301μ-^1 (0) on which the symplectic form is given as follows. Let P ~ μ-^1 (0) .!!._., Mred be
the orbifold principal G-bundle given by the projection map q: μ-^1 (0)---> μ-^1 (0)/G,
and let e E !1^1 (P) 0 g be a connection for it. Let Wo denote the induced symplectic
form on Mred, in other words q*wo = i 0 w. Then if we define a 1-form T on
0 C P x g* by Tp,z = z(B) (for p E P and z E g* } , the symplectic form on 0 is
given by(1.2) w = q*wo + dT.
Further, the moment map on 0 is given by μ(p, z) = z.
Corollary 1.15. Lett be a regular value for the moment map for the Hamiltonian
action of a torus T on a symplectic manifold M. Then in a neighbourhood oft all
symplectic quotients Mt are diff eomorphic to Mt 0 by a diff eomorphism under whichWt= Wt 0 + (t - to, dB) where 8 E !1^1 (μ-^1 (to)) 0 t is a connection for the action of
T on μ-^1 (to).
Corollary 1.16. Suppose G acts in a Hamiltonian fashion on a symplectic mani-
fold M, and suppose 0 is a regular value for the moment mapμ. Then the reduced
space M>. = μ-^1 (0>.)/G at the orbit 0>. fibres over Mo= μ-^1 (0)/G with fibre the
orbit 0>.; furthermore, if 7f : M>. --->Mo is the projection map, then the symplecticform W>. on μ-^1 (0>.)/G is given as W>. = n*wo + !l>., where Wo is the symplectic
form on Mo and !l>. restricts to the standard Kirillov-Kostant symplectic form on
the fibre.