Transformation Groups

, Volume 4, Issue 1, pp 3–24

Weakly symmetric spaces and spherical varieties

  • D. N. Akhiezer
  • E. B. Vinberg
Article

Abstract

Weakly symmetric homogeneous spaces were introduced by A. Selberg in 1956. We prove that, for a real reductive algebraic group, they can be characterized as the spaces of real points of affine spherical homogeneous varieties of the complexified group. As an application, under the same assumption on the transitive group, we show that weakly symmetric spaces are precisely the homogeneous Riemannian manifolds with commutative algebra of invariant differential operators.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AL] H. Azad, J.-J. Loeb,Plurisubharmonic functions and the Kempf-Ness theorem, Bull. London Math. Soc.25 (1993), 162–168.Google Scholar
  2. [BKV] J. Berndt, O. Kowalski, and L. Vanhecke,Geodesics in weakly symmetric spaces, Ann. Glob. Anal. Geom.15 (1997), 153–156.Google Scholar
  3. [BV] J. Berndt, L. Vanhecke,Geometry of weakly symmetric spaces, J. Math. Soc. Japan48 (1996), 745–760.Google Scholar
  4. [BPV] J. Berndt, F. Prüfer, and L. Vanhecke,Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Royal Soc. Edinburgh, Sect. A125 (1995), 265–282.Google Scholar
  5. [Bi] F. Bien,Orbits, multiplicities and differential operators, Contemp. Math.145 (1993), 199–227.Google Scholar
  6. [Br] M. Brion,Classification des espaces homogènes sphériques, Compositio Math.63 (1987), 189–208.Google Scholar
  7. [BrPa] M. Brion, F. Pauer,Valuations des espaces homogènes sphériques, Comment. Math. Helv.62 (1987), 265–285.Google Scholar
  8. [Ca] É. Cartan,Sur la détermination d'un système orthogonal complet dans un espace de Riemann symétrique clos, Rend. Circ. Mat. Palermo53 (1929), 217–252.Google Scholar
  9. [Go] R. Godement,A theory of spherical functions I, Trans. Amer. Math. Soc.73 (1952), 496–556.Google Scholar
  10. [He1] S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Acad. Press, 1978.Google Scholar
  11. [He2] S. Helgason,Groups and Geometric Analysis, Acad. Press, 1984. Русский перевод: С. Хелгасон,ГруппьИ и геомериуескуй анализ, Мир, Москва, 1987.Google Scholar
  12. [KoRa] B. Kostant, S. Rallis,Orbits and representations associated with symmetric spaces, Amer. J. Math.93 (1971), 753–809.Google Scholar
  13. [Kr] M. Krämer,Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math.38 (1979), 129–153.Google Scholar
  14. [L] J. Lauret,Commutative spaces which are not weakly symmetric, Bull. London Math. Society30 (1998), 29–37.Google Scholar
  15. [Mi] И. В. Микитюк, Об инмегрируемосми инварианмиьИх гамильмоновьИх сисмем с однородньИми конфигурационньИми просмрансмвами, Матем. Сб.129 (1986), 514–534. English translation: I. V. Mikityuk,On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Math. USSR Sbornik57 (1987), 527–546.Google Scholar
  16. [Na] S. Nagai,Weakly symmetric spaces in complex and quaternionic space forms, Archiv Math.65 (1997), 342–351.Google Scholar
  17. [Ng] H. Nguyêñ,Weakly symmetric spaces and bounded symmetric domains, Transformation Groups2 (1997), 351–374.Google Scholar
  18. [OV] Э. Ъ. Винберт, А. Л. Онищик,Семинар по группам Ли и алгебраиуеским группам, Наука, Москва, 1988. English translation: A. L. Onishchik, E. B. Vinberg,Lie Groups and Algebraic Groups, Springer 1990.Google Scholar
  19. [Pa1] D. Panyushev,A. restriction theorem and the Poincaré series for U-invariants, Math. Ann.301 (1995), 655–675.Google Scholar
  20. [Pa2] D. Panyushev,Reductive group actions on affine varieties and their doubling. Ann. Inst. Fourier45 (1995), 929–950.Google Scholar
  21. [PV] Э. Ъ. Винберг, В. Л. Попов,Теория инварианмов, Соврем. провлемьИ математики. Фунд. направл., ВИНИТИ, Москва, т. 55, 1989, стр. 137–309. English translation: V. L. Popov, E. B. Vinberg,Invariant Theory, Algebraic Geometry IV, Encyclopedia Math. Sci., Springer, vol. 55, 1994, pp. 123–284.Google Scholar
  22. [Ri] R. W. Richardson,On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc.25 (1982), 1–28.Google Scholar
  23. [Se] 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.Google Scholar
  24. [St] R. Steinberg,Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc.80 (1968).Google Scholar
  25. [Sz] Z. I. Szabo,Spectral geometry for operator families on Riemannian manifolds, Proc. Symp. Pyre Math.54 (1993), 615–665.Google Scholar
  26. [Th] E. G. F. Thomas,An infinitesimal characterization of Gelfand pairs, Contemporary Math. (AMS)26 (1984), 379–385.Google Scholar
  27. [VK] Э. Ъ. Винберг, Ъ. Н. Кимельфельд, ОднородньИе обласми на флаговьИх многообразиях и сфериуеские подгруппьИ полипросмьИх групн Ли, Функ. анал. и его прил.12 (1978), 12–19. English translation: E. B. Vinberg, B. N. Kimel'feld,Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funct. Anal. Appl.12 (1978), 168–174.Google Scholar
  28. [Zi] W. Ziller,Weakly symmetric spaces, Topics in Geometry: In Memory of Joseph D'Atri, Progress in Nonlinear Diff. Eqs., vol. 20, 1996, Birkhäuser, pp. 355–368.Google Scholar

Copyright information

© Birkhäuser 1999

Authors and Affiliations

  • D. N. Akhiezer
    • 1
  • E. B. Vinberg
    • 2
  1. 1.MoscowRussia
  2. 2.Department of MathematicsMoscow State UniversityMoscowRussia

Personalised recommendations