Transformation Groups

, Volume 4, Issue 1, pp 3–24

Weakly symmetric spaces and spherical varieties

  • D. N. Akhiezer
  • E. B. Vinberg

DOI: 10.1007/BF01236659

Cite this article as:
Akhiezer, D.N. & Vinberg, E.B. Transformation Groups (1999) 4: 3. doi:10.1007/BF01236659


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.

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