The Atiyah-Bott Lefschetz Formula Applied to the Based Loops on SU(2)

TL;DR

This paper extends the Atiyah-Bott Lefschetz Formula to the based loop group of SU(2), enabling new calculations of characters for Demazure modules using equivariant index theory.

Contribution

It introduces an analogue of the Atiyah-Bott Lefschetz Formula for the based loop group of SU(2), linking geometric analysis with representation theory.

Findings

Derived an explicit formula for the equivariant index on the based loop group

Provided a method to compute characters of Demazure modules

Extended classical index formulas to infinite-dimensional loop groups

Abstract

The Atiyah-Bott Lefschetz Formula is a well-known formula for computing the equivariant index of an elliptic operator on a compact smooth manifold. We provide an analogue of this formula for the based loop group $\Omega SU(2)$ with respect to the natural $(T \times S^1)$-action. From this result we also derive an effective formula for computing characters of certain Demazure modules.