Deformation spaces, rescaled bundles, and the generalized Kirillov formula

Abstract

In this paper, we construct a smooth vector bundle over the deformation to the normal cone $\mathrm{DNC}(V,M)$ through a rescaling of a vector bundle $E\to V$, which generalizes the construction of the spinor rescaled bundle over the tangent groupoid by Nigel Higson and Zelin Yi. We also provide an equivariant version of their construction. As the main application, we recover the Kirillov character formula for the equivariant index of Dirac-type operators. As another application, we get an equivariant generalization of the description of the Witten and the Novikov deformations of the de Rham–Dirac operator using the deformation to the normal cone obtained recently by O. Mohsen.