Deformation Spaces, Rescaled Bundles and the Kirillov Character Formula

TL;DR

This paper develops a new geometric construction of vector bundles over deformation spaces, generalizing existing methods, and applies it to derive the Kirillov character formula and other equivariant index theorems.

Contribution

It introduces a generalized construction of rescaled bundles over deformation to the normal cone, extending prior work and providing new tools for equivariant index theory.

Findings

Recovered the Kirillov character formula for equivariant Dirac indices.

Provided an equivariant extension of Witten and Novikov deformations.

Established a new geometric framework for deformation and index calculations.

Abstract

In this paper, we construct a smooth vector bundle over the deformation to the normal cone $\text{DNC}(V,M)$ 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.