  We develop a Chern-Weil theory for compact Lie group action whose generic
stabilizers are finite in the framework of equivariant cohomology. This
provides a method of changing an equivariant closed form within its
cohomological class to a form more suitable to yield localization results. This
work is motivated by our work on reproving wall crossing formulas in
Seiberg-Witten theory, where the Lie group is the circle. As applications, we
derive two localization formulas of Kalkman type for G = SU(2) or SO(3)-actions
on compact manifolds with boundary. One of the formulas is then used to yield a
very simple proof of a localization formula due to Jeffrey-Kirwan in the case
of G = SU(2) or SO(3).
