Index Theory of Equivariant Dirac Operators on Non-compact Manifolds

We define a regularized version of an equivariant index of a (generalized) Dirac operator on a non-compact complete Riemannian manifold M acted on by a compact Lie group G. Our definition requires an additional data – an equivariant map v : M → g = LieG, such that the corresponding vector field on M does not vanish outside of a compact subset. For the case when M = C and G is the circle group acting on M by linear transformations, our definition coincides with the classical Bargmann-Fock construction. We show that our index is an invariant of a certain class of cobordisms, similar to the one considered by Ginzburg, Guillemin and Karshon. We use this result to extend the AtiyahSegal-Singer equivariant index theorem and the Guillemin-Sternberg “quantization commutes with reduction” theorem to our non-compact setting. In particular, we obtain new geometric proofs of these theorems on compact manifolds. We apply our results to the study of compact manifolds by constructing cobordism with simpler, but non-compact, manifolds. In particular, the equivariant index theorem reduces to the statement that the manifold is cobordant to the normal bundle to the set of points fixed by the action of G.