The index theory on non-compact manifolds with proper group action

We construct a regularized index of a generalized Dirac operator on a complete Riemannian manifold endowed with a proper action of a unimodular Lie group. We show that the index is preserved by a certain class of non-compact cobordisms and prove a gluing formula for the regularized index. The results of this paper generalize our previous construction of index for compact group action and the recent paper of Hochs and Mathai who studied the case of a Hamiltonian action on a symplectic manifold. As an application of the cobordism invariance of the index we give an affirmative answer to a question of Hochs and Mathai about the independence of the Hochs–Mathai quantization of the metric, connection and other choices.