Compact weakly symmetric spaces and spherical pairs

Nguyêñ, H.

doi:10.1090/S0002-9939-00-05425-3

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
DOI: https://doi.org/10.1090/S0002-9939-00-05425-3
Published electronically: May 18, 2000
MathSciNet review: 1676304
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Abstract:

Let be a spherical pair and assume that is a connected compact simple Lie group and a closed subgroup of . We prove in this paper that the homogeneous manifold is weakly symmetric with respect to 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.

1. D.N. Akhiezer and E.B. Vinberg, Weakly symmetric spaces and spherical varieties, preprint (1998).
2. Mogens Flensted-Jensen, Spherical functions on a simply connected semisimple Lie group, Amer. J. Math. 99 (1977), no. 2, 341–361. MR 0458063, https://doi.org/10.2307/2373823
3. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Pure and Applied Mathematics 80, Academic Press, 1978.
4. Sigurdur Helgason, Groups and geometric analysis, Pure and Applied Mathematics, vol. 113, Academic Press, Inc., Orlando, FL, 1984. Integral geometry, invariant differential operators, and spherical functions. MR 754767
5. Adam Korányi and Joseph A. Wolf, Realization of hermitian symmetric spaces as generalized half-planes, Ann. of Math. (2) 81 (1965), 265–288. MR 0174787, https://doi.org/10.2307/1970616
6. Manfred Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), no. 2, 129–153 (German). MR 528837
7. Jorge Lauret, Commutative spaces which are not weakly symmetric, Bull. London Math. Soc. 30 (1998), no. 1, 29–36. MR 1479033, https://doi.org/10.1112/S0024609397003925
8. Shingo Murakami, Exceptional simple Lie groups and related topics in recent differential geometry, Differential geometry and topology (Tianjin, 1986–87) Lecture Notes in Math., vol. 1369, Springer, Berlin, 1989, pp. 183–221. MR 1001187, https://doi.org/10.1007/BFb0087534
9. Hiêú Nguyêñ, Weakly symmetric spaces and bounded symmetric domains, Transform. Groups 2 (1997), no. 4, 351–374. MR 1486036, https://doi.org/10.1007/BF01234540
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. Erik G. F. Thomas, An infinitesimal characterization of Gel′fand pairs, Conference in modern analysis and probability (New Haven, Conn., 1982) Contemp. Math., vol. 26, Amer. Math. Soc., Providence, RI, 1984, pp. 379–385. MR 737415, https://doi.org/10.1090/conm/026/737415
14. McKenzie Wang and Wolfgang Ziller, Symmetric spaces and strongly isotropy irreducible spaces, Math. Ann. 296 (1993), no. 2, 285–326. MR 1219904, https://doi.org/10.1007/BF01445107
15. J.A. Wolf, Spaces of Constant Curvature, McGraw-Hill, 1967.
16. J.A. Wolf and A. Korányi, Working notes, August 1983.
17. Wolfgang Ziller, Weakly symmetric spaces, Topics in geometry, Progr. Nonlinear Differential Equations Appl., vol. 20, Birkhäuser Boston, Boston, MA, 1996, pp. 355–368. MR 1390324