DEFORMATION SPACES, GETZLER'S RESCALING, AND AN EQUIVARIANT INDEX THEOREM FOR LIE GROUPOIDS

In recent work, Higson and Yi developed a new perspective on Getzler's symbol calculus, reinterpreting the latter in terms of a convolution algebra of sections of the rescaled bundle over the tangent groupoid of a spin manifold. We generalize the construction of the rescaled bundle to a deformation to the normal cone, which leads to generalizations of the Getzler method to both the equivariant setting and the case of the adiabatic groupoid of any Lie groupoid. Among applications, we obtain an equivariant longitudinal index theorem for Lie groupoids.