Local and global symbols on compact Lie groups

TL;DR

This paper explores the relationship between local and global symbols of pseudodifferential operators on compact Lie groups, extending classical results from the torus to more general settings, and characterizes key operator invariants.

Contribution

It generalizes the connection between local and global symbols from the torus to arbitrary compact Lie groups, including descriptions of principal symbols and traces.

Findings

Established relations between local and global symbols on compact Lie groups.

Described principal symbols, non-commutative residue, and canonical trace in terms of global symbols.

Extended classical results from the torus to general compact Lie groups.

Abstract

On the torus, it is possible to assign a global symbol to a pseudodifferential operator using Fourier series. In this paper we investigate the relations between the local and global symbols for the operators in the classical H\"ormander calculus and describe the principal symbols, the non-commutative residue and the canonical trace of an operator in terms of its global symbol. We also generalise these results to any compact Lie group.