Global symbolic calculus of pseudo-differential operators on homogeneous vector bundles
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 uni...
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Zusammenfassung: | 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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DOI: | 10.48550/arxiv.1911.01553 |