A Lie-algebraic approach to the local index theorem on compact homogeneous spaces

TL;DR

This paper presents a Lie algebra-based method to prove the local index theorem on compact homogeneous spaces, simplifying previous approaches by utilizing Kostant's cubic Dirac operator and the quantum Weil algebra.

Contribution

It introduces a novel Lie algebraic approach to the local index theorem, replacing the Riemannian Dirac operator with Kostant's cubic Dirac operator for simplification.

Findings

Simplified proof of the local index theorem using Lie algebra methods.

Demonstrated the effectiveness of Kostant's cubic Dirac operator in this context.

Connected the theorem to the quantum Weil algebra of Alekseev and Meinrenken.

Abstract

Using a K-theory point of view, Bott related the Atiyah-Singer index theorem for elliptic operators on compact homogeneous spaces to the Weyl character formula. This article explains how to prove the local index theorem for compact homogenous spaces using Lie algebra methods. The method follows in outline the proof of the local index theorem due to Berline and Vergne. But the use of Kostant's cubic Dirac operator in place of the Riemannian Dirac operator leads to substantial simplifications. An important role is also played by the quantum Weil algebra of Alekseev and Meinrenken.