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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The point spectrum of the Dirac operator on noncompact symmetric spaces
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by S. Goette and U. Semmelmann
Proc. Amer. Math. Soc. 130 (2002), 915-923
DOI: https://doi.org/10.1090/S0002-9939-01-06158-5
Published electronically: October 1, 2001

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 $\operatorname {Spec}_{p}(D)=\{0\}$.
References
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Bibliographic 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
  • 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
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1866049