Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T05:30:21.045Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogeneous bundles and the trace of the heat kernel

Published online by Cambridge University Press:  09 April 2009

H. D. Fegan
H. D. Fegan
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

We study the heat equation on a homogeneous bundle over a compact Lie group. The trace of the heat kernel is explicitly calculated. By comparing this with the formula constructed form the eigenvalues (with multiplicities) of the Laplacian we obtain and unusual formula involving the Clebsch-Gordan numbers. The main method is to use invariance under conjugation to pass from the group to its maximal torus, where a direct calculation can be carried out.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 40 , Issue 1 , February 1986 , pp. 71 - 82
Copyright
Copyright © Australian Mathematical Society 1986

References

[1]Fegan, H. D., ‘The heat equation on a compact Lie group’, Trans. Amer. Math. Soc. 246 (1978), 339–357.CrossRefGoogle Scholar
[2]Fegan, H. D., ‘The spectrum of the Laplacian on forms over a Lie group’, Pacific J. Math. 90 (1980), 373–387.CrossRefGoogle Scholar
[3]Fegan, H. D., ‘Differential equations on Lie groups and tori, the wave equation and Huygen's Principle’, Rocky Mountain J. Math. 14 (1984), 699–704.Google Scholar
[4]Fegan, Howard D. and Gilkey, Peter B., ‘Invariants of the heat equation’, Pacific J. Math. 117 (1985), 233–254.CrossRefGoogle Scholar
[5]Patodi, V. K., ‘Curvature and the eigenforms of the Laplace operator’, J. Differential Geom. 5 (1971), 233–249.CrossRefGoogle Scholar
[6]Wallach, N. R., Harmonic analysis on homogeneous spaces (M. Dekker, New York, 1973).Google Scholar