  We prove a local index theorem of Atiyah-Singer type for Dirac operators on
manifolds with a Lie structure at infinity (Lie manifolds for short). With the
help of a renormalized supertrace, defined on a suitable class of regularizing
operators, the proof of the index theorem relies on a rescaling technique
similar in spirit to Getzler's rescaling. With a given Lie manifold we
associate an appropriate integrating Lie groupoid. We then describe the heat
kernel of a geometric Dirac operator via a functional calculus with values in
the convolution algebra of sections of a rescaled bundle over the adiabatic
groupoid. Finally, we calculate the right coefficient in the heat kernel
expansion by deforming the Dirac operator into a polynomial coefficient
operator over the rescaled bundle and applying the Lichnerowicz theorem to the
fibers of the groupoid and the Lie manifold.
