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The point spectrum of the Dirac operator on noncompact symmetric spaces


Authors: S. Goette and U. Semmelmann
Journal: Proc. Amer. Math. Soc. 130 (2002), 915-923
MSC (2000): Primary 58C40; Secondary 53C35, 22E30
DOI: https://doi.org/10.1090/S0002-9939-01-06158-5
Published electronically: October 1, 2001
MathSciNet review: 1866049
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Abstract: In this note, we consider the Dirac operator $D$on a Riemannian symmetric space $M$ of noncompact type. Using representation theory, we show that $D$has point spectrum iff the ${\hat A}$-genus of its compact dual does not vanish. In this case, if $M$ is irreducible, then $M=\mathrm{U}(p,q)/\mathrm{U}(p)\times \mathrm{U}(q)$ with $p+q$ odd, and  $\operatorname{Spec}_{p}(D)=\{0\}$.


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Additional Information

S. Goette
Affiliation: Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Email: goette@blaschke.mathematik.uni-tuebingen.de

U. Semmelmann
Affiliation: Mathematisches Institut, Universität München, Theresienstr. 39, D-80333 München, Germany
Email: semmelma@rz.mathematik.uni-muenchen.de

DOI: https://doi.org/10.1090/S0002-9939-01-06158-5
Received by editor(s): September 18, 2000
Published electronically: October 1, 2001
Additional Notes: Both authors were supported by a research fellowship of the DFG
Communicated by: Rebecca Herb
Article copyright: © Copyright 2001 American Mathematical Society

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