  In this paper, an equality between the Hochs-Mathai type index and the
Atiyah-Patodi-Singer type index is established when the manifold and the group
action are both non-compact, which generalizes a result of Ma and Zhang for
compact group actions. As a technical preparation, a problem concerning the
Fredholm property of the global elliptic boundary value problems of the
Atiyah-Patodi-Singer type on a non-compact manifold is studied.
