Equivariant cohomology and localization for Lie algebroids and applications

Let A be a Lie algebroid on a differentiable manifold M, and assume that A is equipped with an infinitesimal G-action compatible with a G-action on M, where G is a compact Lie group. We define an equivariant cohomology associated with these data and prove a localization formula together with a Bott-type formula.