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

2019/11/05 by Mitsuru Wilson, Wilson, Mitsuru
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Nonlinear Waves and Solitons #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1911.01553

openalex publication_date 2019/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞(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 \mathbbT\hookrightarrow SU(2)→ℂP¹, which is known as the Hopf fibration. Lastly, we outline how functional calculus of a pseudo-differential operator can be computed using our calculus.

Citations

• On the heat equation and the index theorem

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