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arXiv · 1911.01553

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

Mitsuru Wilson
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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.

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Mitsuru Wilson. 2019-12-14. Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles. https://arxiv.org/abs/1911.01553

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