Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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



Compact weakly symmetric spaces and spherical pairs

Author: H. D. Nguyêñ
Journal: Proc. Amer. Math. Soc. 128 (2000), 3425-3433
MSC (2000): Primary 53C35; Secondary 32M15
Published electronically: May 18, 2000
MathSciNet review: 1676304
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $(G,H)$ be a spherical pair and assume that $G$ is a connected compact simple Lie group and $H$ a closed subgroup of $G$. We prove in this paper that the homogeneous manifold $G/H$ is weakly symmetric with respect to $G$ and possibly an additional fixed isometry $\mu$. It follows that M. Krämer's classification list of such spherical pairs also becomes a classification list of compact weakly symmetric spaces. In fact, our proof involves a case-by-case study of the isotropy representations of all spherical pairs on Krämer's list.

References [Enhancements On Off] (What's this?)

  • 1. D.N. Akhiezer and E.B. Vinberg, Weakly symmetric spaces and spherical varieties, preprint (1998).
  • 2. M. Flensted-Jensen, Spherical functions on a simply connected semisimple Lie group, Amer. J. Math. 99 (1977), 341-361. MR 56:16266
  • 3. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics 80, Academic Press, 1978.
  • 4. S. Helgason, Groups and Geometric Analysis, Pure and Applied Mathematics 113, Academic Press, 1984. MR 86c:22017
  • 5. A. Korányi and J.A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. 81 (1965), 265-288. MR 30:4980
  • 6. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979) No. 2, 129-153. MR 80f:22011
  • 7. J. Lauret, Commutative spaces which are not weakly symmetric, Bull. London Math. Soc. 30 (1998) No. 1, 29-36. MR 99a:22016
  • 8. S. Murakami, Exceptional simple Lie groups and related topics in recent differential geometry, Proceedings: Differential Geometry and Topology (Tianjin 1986-87), Lecture Notes in Mathematics 1369, Springer-Verlag, 1989. MR 90g:22009
  • 9. H.D. Nguyêñ, Weakly symmetric spaces and bounded symmetric domains, Transform. Groups 2 (1997) No. 4, 351-374. MR 98k:53067
  • 10. H.D. Nguyêñ, Characterizing weakly symmetric spaces as Gelfand pairs, J. Lie Theory (to appear). CMP 99:09
  • 11. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. 20 (1956), 47-87. MR 19:531g
  • 12. H. Tamaru, Riemannian geodesic orbit metrics on fiber bundles, preprint (1997).
  • 13. E.G.F. Thomas, An infinitesimal characterization of Gelfand pairs, Contemp. Math. 26 (1984), 379-385. MR 85i:22015
  • 14. M. Wang and W. Ziller, Symmetric spaces and strongly isotropy irreducible spaces, Math. Ann. 296 (1993), 285-326. MR 94g:53042
  • 15. J.A. Wolf, Spaces of Constant Curvature, McGraw-Hill, 1967.
  • 16. J.A. Wolf and A. Korányi, Working notes, August 1983.
  • 17. W. Ziller, Weakly symmetric spaces, Topics in Geometry: In Memory of Joseph D' Atri, Progress in Nonlinear Diff. Eqs. 20, Birkhauser, 1996, 355-368. MR 97c:53081

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C35, 32M15

Retrieve articles in all journals with MSC (2000): 53C35, 32M15

Additional Information

H. D. Nguyêñ
Affiliation: Department of Mathematics, Rowan University, Glassboro, New Jersey 08028

Keywords: Weakly symmetric spaces, spherical pairs, Gelfand pairs
Received by editor(s): April 17, 1998
Received by editor(s) in revised form: January 4, 1999
Published electronically: May 18, 2000
Communicated by: Roe Goodman
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society