Available in electronic format
Available in print format		 Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826 (e) ISSN 0002-9939 (p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

The point spectrum of the Dirac operator on noncompact symmetric spaces

Author(s): S. Goette; U. Semmelmann
Journal: Proc. Amer. Math. Soc. 130 (2002), 915-923.
MSC (2000): Primary 58C40; Secondary 53C35, 22E30
Posted: October 1, 2001
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

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:

[AS]
M. F. Atiyah, W. Schmid, A Geometric Construction of the Discrete Series for Semisimple Lie Groups, Inv. math. 42 (1977), 1-62; Erratum, Inv. math. 54 (1979), 189-192. MR 57:3310; MR 81d:22015

[Ba]
P. D. Baier, Über den Diracoperator auf Mannigfaltigkeiten mit Zylinderenden, Diplomarbeit, Universität Freiburg (1997).

[Baer]
C. Bär, The Dirac fundamental tone of the hyperbolic space, Geom. Dedicata 41 (1992), 103-107. MR 93c:58217

[BGV]
M. Berline, E. Getzler, N. Verne, Heat kernels and Dirac operators, Springer, Berlin-Heidelberg-New York (1992).

[BH]
A. Borel, F. Hirzebruch, Characteristic Classes and Homogeneous Spaces, II, Amer. J. Math. 81 (1959), 315-382. MR 22:988

[Bo]
R. Bott, The Index Theorem for Homogeneous Differential Operators, in S. S. Cairns: Differential and Combinatorial Topology, a Symposium in Honor of Marston Morse, Princeton Univ. Press (1965), 167-186. MR 31:6246

[Bu]
U. Bunke, The spectrum of the Dirac operator on the hyperbolic space, Math. Nachr. 153 (1991), 179-190. MR 92h:58196

[CG]
M. Cahen, S. Gutt, Spin Structures on Compact Simply Connected Riemannian Symmetric Spaces, Simon Stevin 62 (1988), 209-242. MR 90d:58161

[C]
R. Camporesi, The spinor heat kernel in maximally symmetric spaces, Comm. Math. Phys. 148 (1992), 283-308. MR 93g:58140
[CH]
R. Camporesi, A. Higuchi, On the eigenfunctions of the Dirac operator on spheres and real hyperbolic spaces, J. Geom. Phys. 20 (1996), 1-18. MR 98c:58169

[CM]
A. Connes, H. Moscovici, The $L^2$-Index Theorem for Noncompact Homogeneous Spaces of Lie Groups, Ann. of Math. 115 (1982), 291-330. MR 84f:58108

[FJO]
M. Flensted-Jensen, K. Okamoto, An explicit construction of the $K$-finite vectors in the discrete series for an isotropic semisimple symmetric space, Mem. Soc. Math. France, Nouv. sér. 15 (1984), 157-199. MR 87c:22025

[GV]
E. Galina, J. Vargas, Eigenvalues and eigenspaces for the twisted Dirac operator over $\mathrm{SU}(N,1)$ and $\text{Spin}(2N,1)$, Trans. Amer. Math. Soc. 345 (1994), 97-113. MR 95a:22009

[H]
S. Helgason Differential Geometry, Lie Groups and Symmetric Spaces, Academic Press, New York (1978). MR 80k:53081

[HS]
F. Hirzebruch, P. Slodowy, Elliptic genera, involutions and homogeneous spin manifolds, Geom. Ded. 35 (1990), 309-343. MR 92a:57028

[K]
A. W. Knapp, Representation Theory of Semisimple Groups, Princeton Univ. Press, Princeton, N. J. (1986). MR 87j:22022

[LM]
H. B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton Univ. Press, Princeton, N. J. (1989). MR 91g:53001

[P]
K. R. Parthasarathy, Dirac operator and the discrete series, Ann. of Math. 96 (1972), 1-30.

[S]
S. Seifarth, The discrete spectrum of the Dirac operators on certain symmetric spaces, Preprint No. 25 of the IAAS, Berlin (1992).

[W]
J. A. Wolf, Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1972/73), 611-640. MR 46:10340

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 58C40, 53C35, 22E30

Retrieve articles in all Journals with MSC (2000): 58C40, 53C35, 22E30

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: 10.1090/S0002-9939-01-06158-5
PII: S 0002-9939(01)06158-5
Received by editor(s): September 18, 2000
Posted: October 1, 2001
Additional Notes: Both authors were supported by a research fellowship of the DFG
Communicated by: Rebecca Herb
Copyright of article: Copyright 2001, American Mathematical Society

Forward Citation(s):

Information for authors on submitting citations

The following works have cited this article

Roberto Camporesi; Emmanuel Pedon., The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one, Proc. Amer. Math. Soc. (2) 130 (2002), http://www.ams.org/journal-getitem?pii=S0002-9939-01-06294-3, posted on 07/25/2001, 507-516.

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2009, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google