  Let $D$ be a (generalized) Dirac operator on a non-compact complete
Riemannian manifold $M$ acted on by a compact Lie group $G$. Let $v:M -->
Lie(G)$ be an equivariant map, such that the corresponding vector field on $M$
does not vanish outside of a compact subset. These data define an element of
$K$-theory of the transversal cotangent bundle to $M$. Hence a topological
index of the pair $(D,v)$ is defined as an element of the completed ring of
characters of $G$.
  We define an analytic index of $(D,v)$ as an index space of certain
deformation of $D$ and we prove that the analytic and topological indexes
coincide.
  As a main step of the proof, we show that index is an invariant of a certain
class of cobordisms, similar to the one considered by Ginzburg, Guillemin and
Karshon. In particular, this means that the topological index of Atiyah is also
invariant under this class of non-compact cobordisms.
  As an application we extend the Atiyah-Segal-Singer equivariant index theorem
to our non-compact setting. In particular, we obtain a new proof of this
theorem for compact manifolds.
