Annals of Global Analysis and Geometry

, Volume 41, Issue 2, pp 161–186 | Cite as

Orthogonal basis for spherical monogenics by step two branching

Original Paper

Abstract

Spherical monogenics can be regarded as a basic tool for the study of harmonic analysis of the Dirac operator in Euclidean space \({{\mathbb R}^m}\). They play a similar role as spherical harmonics do in case of harmonic analysis of the Laplace operator on \({{\mathbb R}^m}\). Fix the direct sum \({{\mathbb R}^m={\mathbb R}^p \oplus {\mathbb R}^q}\). In this article, we will study the decomposition of the space \({{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}\) of spherical monogenics of order n under the action of Spin(p) × Spin(q). As a result, we obtain a Spin(p) × Spin(q)-invariant orthonormal basis for \({{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}\). In particular, using the construction with p = 2 inductively, this yields a new orthonormal basis for the space \({{\mathcal M}_n({\mathbb R}^m, {\mathbb C}_m)}\).

Keywords

Clifford analysis Dirac operators Representations Branching rules Spin groups 

Mathematics Subject Classification (2000)

30G35 33C45 22E70 

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Mathematical InstituteCharles UniversityPrahaCzech Republic
  2. 2.Department of Engineering SciencesUniversity College Ghent, Member of Ghent University AssociationGhentBelgium

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