Hostname: page-component-7c8c6479df-8mjnm Total loading time: 0 Render date: 2024-03-28T22:26:53.171Z Has data issue: false hasContentIssue false

The Explicit Fourier Decomposition of L2SO(n)/SO(n - m))

Published online by Cambridge University Press:  20 November 2018

Robert S. Strichartz*
Affiliation:
Cornell University, Ithaca, New York
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The decomposition of L2SO(n)/SO(n - m)) into a direct sum of irreducible representations of SO(n) is given abstractly by the branching theorem and the Frobenius reciprocity theorem [1]. The goal of this paper is to obtain this decomposition explicitly, generalizing the theory of spherical harmonics (m = 1). The case m = 2 has been studied in Levine [6], and the case 2mn in Gelbart [3]. Our results shed more light on these cases as well as revealing new phenomena which only occur when 2m > n.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Boerner, H., Representations of groups(North-Holland, Amsterdam, 1963).Google Scholar
2. Bowman, F., An introduction to determinants and matrices (The English University Press, London, 1962).Google Scholar
3. Gelbart, S., A theory of Stiefel harmonics, Trans. Amer. Math. Soc. 192 (1974), 29-50.Google Scholar
4. Helgason, S., Invariants and fundamental functions, Acta Math. 109 (1963), 241 258.Google Scholar
5. Kostant, B., Lie group representations on polynomial rings, Amer. J.Math. 85 (1963), 327 404.Google Scholar
6. Levine, D., Systems of singular integrals on spheres, Trans. Amer. Math. Soc. 144 (1969), 493 522.Google Scholar
7. Sugiura, M., Representations of compact groups realized by spherical functions on symmetric spaces, Proc. Japan Acad. 38 (1962), 111 113.Google Scholar
8. Takeuchi, M., Polynomial representations associated with symmetric bounded domains, Osaka J. Math. 10 (1973), 441 475.Google Scholar
9. Ton-That, Tuong, Lie group representations and harmonic polynomials of a matrix variable (preprint).Google Scholar