On traces of operators, associated with actions of compact Lie groups

TL;DR

This paper investigates the properties of operator traces associated with compact Lie group actions on manifolds, revealing their localization at specific submanifolds and analyzing their structural characteristics.

Contribution

It introduces a detailed analysis of traces of operators linked to compact Lie group actions, highlighting their localization and structural properties on submanifolds.

Findings

Traces are localized at special submanifolds within the ambient manifold.

The structure of these traces is characterized on the localized submanifolds.

The study provides a framework for understanding operator traces in the context of Lie group actions.

Abstract

Given a pair $(M,X)$, where $X$ is a smooth submanifold in a closed smooth manifold $M$, we study the operation, which takes each operator $D$ on the ambient manifold to a certain operator on the submanifold. The latter operator is called the trace of $D$. More precisely, we study traces of operators, associated with actions of compact Lie groups on $M$. We show that traces of such operators are localized at special submanifolds in $X$ and study the structure of the traces on these submanifolds.