Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles

TL;DR

This paper develops a symbolic calculus for pseudo-differential operators on sections of homogeneous vector bundles over compact homogeneous spaces, enabling new analysis tools for such operators.

Contribution

It introduces a novel symbolic calculus framework for pseudo-differential operators on homogeneous vector bundles over compact spaces, linking symbols to irreducible representations.

Findings

Realization of symbols as operators on irreducible representations

Explicit action of $SU(2)$ vector fields on bundle sections

Outline of functional calculus computation for these operators

Abstract

A symbolic calculus for a pseudo-differential operators acting on sections of a homogeneous vector bundle over a compact homogeneous space $G/H$ with compact $G$ and $H$ is developed. We realize the symbol of a pseudo-differential operator as a linear operator acting on corresponding irreducible unitary representations of $H$ valued in the algebra $C^\infty(G)$ of smooth functions. We write down how left invariant vector fields of $SU(2)$ act on the sections of homogeneous vector bundles associated to the fibration $\mathbb{T}\hookrightarrow SU(2)\rightarrow\mathbb{C}P^1$, which is known as the Hopf fibration. Lastly, we outline how functional calculus of a pseudo-differential operator can be computed using our calculus.