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First Order Operators on Manifolds With a Group Action

Published online by Cambridge University Press:  20 November 2018

H. D. Fegan  and
B. Steer
H. D. Fegan
Affiliation:
Department of Mathematics Lehigh University Bethlehem, PA 18015-3174 U.S.A.
B. Steer
Affiliation:
Hertford College Oxford OX1 3BW U.K.
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Abstract

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We investigate questions of spectral symmetry for certain first order differential operators acting on sections of bundles over manifolds which have a group action. We show that if the manifold is in fact a group we have simple spectral symmetry for all homogeneous operators. Furthermore if the manifold is not necessarily a group but has a compact Lie group of rank 2 or greater acting on it by isometries with discrete isotropy groups, and let D be a split invariant elliptic first order differential operator, then D has equivariant spectral symmetry.

Keywords

58G3557S25
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 48 , Issue 4 , 01 August 1996 , pp. 758 - 776
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Atiyah, M.F., Patodi, V.K. and Singer, I.M., Spectral Asymmetry and Riemannian Geometry IJIJII, Math. Proc. Cambridge Philos. Soc. 77(1975), 43–69. 78(1976), 405–432. and 79(1976), 71–99.Google Scholar
2. Atiyah, M.F. and Smith, L., Compact Lie Groups and the Stable Homotopy of Spheres,, Topology 13(1974), 135–142.Google Scholar
3. Fegan, H.D., Conformally Invariant First Order Differential Operators, Quart. J. Math. Oxford, (2) 27(1976), 371–378.Google Scholar
4. Fegan, H.D., Second Order Operators with Non-zero Eta Invariant, Canad. Math., Bull. 35(1992), 341–353.Google Scholar
5. Fegan, H.D. and Steer, B., Spectral Symmetry of the Dirac Operator in the Presence of a Group Action, Trans. Amer. Math., Soc. 335(1993), 631–647.Google Scholar
6. Hitchin, N.J., Harmonic Spinors, Adv., Math. 14(1974), 1–55.Google Scholar
7. Kostant, B., Lie Group Representation on Polynomial Rings, Amer. J., Math. 85(1963), 327–404.Google Scholar
8. Lang, S., Algebra, Addison-Wesley, Menlo Park, 1984.Google Scholar
9. Palais, R.S., Slices and Equivariant Imbeddings, Seminar on Transformation Groups by A. Borel, Princeton University Press, Princeton, New Jersey.Google Scholar
10 Roe, J., Elliptic Operators Toplogy and Asymptotic Methods Pitman Research Notes, Longman Scientific and Technical, Harlow, EssexGoogle Scholar
11 Sternberg, S., Lectures on Differential Geometry, Chelsea Pub. Co, New York, 1983.Google Scholar
12 Wallach, N. R., Harmonic Analysis on Homogeneous Spaces, Marcel Dekker, New York, 1973.Google Scholar