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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AL] H. Azad, J.-J. Loeb,Plurisubharmonic functions and the Kempf-Ness theorem, Bull. London Math. Soc.25 (1993), 162–168.
[BKV] J. Berndt, O. Kowalski, and L. Vanhecke,Geodesics in weakly symmetric spaces, Ann. Glob. Anal. Geom.15 (1997), 153–156.
[BV] J. Berndt, L. Vanhecke,Geometry of weakly symmetric spaces, J. Math. Soc. Japan48 (1996), 745–760.
[BPV] J. Berndt, F. Prüfer, and L. Vanhecke,Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Royal Soc. Edinburgh, Sect. A125 (1995), 265–282.
[Bi] F. Bien,Orbits, multiplicities and differential operators, Contemp. Math.145 (1993), 199–227.
[Br] M. Brion,Classification des espaces homogènes sphériques, Compositio Math.63 (1987), 189–208.
[BrPa] M. Brion, F. Pauer,Valuations des espaces homogènes sphériques, Comment. Math. Helv.62 (1987), 265–285.
[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.
[Go] R. Godement,A theory of spherical functions I, Trans. Amer. Math. Soc.73 (1952), 496–556.
[He1] S. Helgason,Differential Geometry, Lie Groups, and Symmetric Spaces, Acad. Press, 1978.
[He2] S. Helgason,Groups and Geometric Analysis, Acad. Press, 1984. Русский перевод: С. Хелгасон,ГруппьИ и геомериуескуй анализ, Мир, Москва, 1987.
[KoRa] B. Kostant, S. Rallis,Orbits and representations associated with symmetric spaces, Amer. J. Math.93 (1971), 753–809.
[Kr] M. Krämer,Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math.38 (1979), 129–153.
[L] J. Lauret,Commutative spaces which are not weakly symmetric, Bull. London Math. Society30 (1998), 29–37.
[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.
[Na] S. Nagai,Weakly symmetric spaces in complex and quaternionic space forms, Archiv Math.65 (1997), 342–351.
[Ng] H. Nguyêñ,Weakly symmetric spaces and bounded symmetric domains, Transformation Groups2 (1997), 351–374.
[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.
[Pa1] D. Panyushev,A. restriction theorem and the Poincaré series for U-invariants, Math. Ann.301 (1995), 655–675.
[Pa2] D. Panyushev,Reductive group actions on affine varieties and their doubling. Ann. Inst. Fourier45 (1995), 929–950.
[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.
[Ri] R. W. Richardson,On orbits of algebraic groups and Lie groups, Bull. Austral. Math. Soc.25 (1982), 1–28.
[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.
[St] R. Steinberg,Endomorphisms of Linear Algebraic Groups, Mem. Amer. Math. Soc.80 (1968).
[Sz] Z. I. Szabo,Spectral geometry for operator families on Riemannian manifolds, Proc. Symp. Pyre Math.54 (1993), 615–665.
[Th] E. G. F. Thomas,An infinitesimal characterization of Gelfand pairs, Contemporary Math. (AMS)26 (1984), 379–385.
[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.
[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.
Author information
Authors and Affiliations
Additional information
Supported by the Alexander von Humboldt Foundation and Russian Foundation for Basic Research, Grant No. 95-01-01263.
Supported by the U. S. Civilian Research and Development Foundation, Award No. 206, Russian Foundation for Basic Research, Grant No. 98-01-00598, and the Alexander von Humboldt Foundation.
Rights and permissions
About this article
Cite this article
Akhiezer, D.N., Vinberg, E.B. Weakly symmetric spaces and spherical varieties. Transformation Groups 4, 3–24 (1999). https://doi.org/10.1007/BF01236659
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF01236659