Localized Index and $L^2$-Lefschetz fixed point formula for orbifolds

TL;DR

This paper develops localized index formulas for Dirac operators on orbifolds with group actions, extending Atiyah's $L^2$-index and Lefschetz fixed point formulas to orbifold settings, providing new topological invariants.

Contribution

It introduces a new class of localized indices for Dirac operators on orbifolds, generalizing existing $L^2$-index and Lefschetz formulas, and derives cohomological formulas for these indices.

Findings

Derived an $L^2$-version of the Lefschetz fixed point formula for orbifolds.

Established cohomological formulas for localized indices.

Produced refined topological invariants for quotient orbifolds.

Abstract

We study a class of localized indices for the Dirac type operators on a complete Riemannian orbifold, where a discrete group acts properly, co-compactly and isometrically. These localized indices, generalizing the $L^2$-index of Atiyah, are obtained by taking certain traces of the higher index for the Dirac type operators along conjugacy classes of the discrete group. Applying the local index technique, we also obtain an $L^2$-version of the Lefschetz fixed point formula for orbifolds. These cohomological formulae for the localized indices give rise to a class of refined topological invariants for the quotient orbifold.