Remote Access Transactions of the Moscow Mathematical Society

Transactions of the Moscow Mathematical Society

ISSN 1547-738X(online) ISSN 0077-1554(print)

 
 

 

Excellent affine spherical homogeneous spaces of semisimple algebraic groups


Author: R. S. Avdeev
Translated by: E. Khukhro
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 71 (2010).
Journal: Trans. Moscow Math. Soc. 2010, 209-240
MSC (2010): Primary 20G05; Secondary 14M17, 20G20, 32M10
DOI: https://doi.org/10.1090/S0077-1554-2010-00183-5
Published electronically: December 21, 2010
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A spherical homogeneous space $ G/H$ of a connected semisimple algebraic group $ G$ is called excellent if it is quasi-affine and its weight semigroup is generated by disjoint linear combinations of the fundamental weights of the group $ G$. All the excellent affine spherical homogeneous spaces are classified up to isomorphism.


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

  • 1. È. B. Vinberg and A. L. Onishchik, Seminar on Lie groups and algebraic groups, 2nd ed., URSS, Moscow, 1995; English transl. of 1st ed.: A. L. Onishchik and È. B. Vinberg, Lie groups and algebraic groups, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1990. MR 1403378 (97d:22001)
  • 2. È. B. Vinberg and B. N. Kimel'fel'd, Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups, Funkts. Anal. Prilozh. 12 (1978), no. 3, 12-19; English transl., Funct. Anal. Appl. 12 (1979), 168-174. MR 0509380 (82e:32042)
  • 3. È. B. Vinberg, Commutative homogeneous spaces and co-isotropic symplectic actions, Uspekhi Mat. Nauk 56 (2001), no. 1, 3-62; English transl., Russian Math. Surveys 56 (2001), no. 1, 1-60. MR 1845642 (2002f:53088)
  • 4. D. Panyushev, Parabolic subgroups with Abelian unipotent radical as a testing site for invariant theory, Canad. J. Math. 51 (1999), no. 3, 616-635. MR 1701328 (2000e:14080)
  • 5. M. Krämer, Sphärische Untergruppen in kompakten zusammenhängenden Liegruppen, Compositio Math. 38 (1979), 129-153. MR 528837 (80f:22011)
  • 6. I. V. Mikityuk, On the integrability of invariant Hamiltonian systems with homogeneous configuration spaces, Mat. Sb. 129 (1986), no. 4, 514-534; English transl., Math. USSR-Sb. 57 (1987), 527-546. MR 842398 (88e:58032)
  • 7. M. Brion, Classification des espaces homogènes sphériques, Compositio Math. 63 (1987), 189-208. MR 906369 (89d:32068)
  • 8. O. S. Yakimova, Weakly symmetric spaces of semisimple Lie groups, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2002 (2002), no. 2, 57-60; English transl., Moscow Univ. Math. Bull. 57 (2002), no. 2, 37-40. MR 1934062 (2004b:53076)
  • 9. R. S. Avdeev, Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups, Izv. Ross. Akad. Nauk Ser. Mat., to appear.
  • 10. D. Panyushev, Complexity of quasiaffine homogeneous varieties, $ t$-decompositions, and affine homogeneous spaces of complexity 1, Adv. Soviet Math. 8 (1992), 151-166. MR 1155672 (93d:14076)
  • 11. D. Timashev, Homogeneous spaces and equivariant embeddings, arXiv: math/0602228v1.
  • 12. Th. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France. 102 (1974), 317-334. MR 0366941 (51:3187)
  • 13. Th. Vust, Plongements d'espaces symétriques algébriques: une classification, Ann. Scuola Norm. Super. Pisa, Scienze $ 4^e$ série, 17 (1990), no. 2, 165-195. MR 1076251 (91m:14079)
  • 14. E. B. Dynkin, Maximal subgroups of the classical groups, Trudy Moskov. Mat. Ob-va 1 (1952), 39-166. (Russian) MR 0049903 (14:244d)
  • 15. I. M. Gel'fand and M. L. Tsetlin, Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR 71 (1950), 1017-1020. (Russian) MR 0034763 (11:639e)

Similar Articles

Retrieve articles in Transactions of the Moscow Mathematical Society with MSC (2010): 20G05, 14M17, 20G20, 32M10

Retrieve articles in all journals with MSC (2010): 20G05, 14M17, 20G20, 32M10


Additional Information

R. S. Avdeev
Affiliation: Moscow State University
Email: suselr@yandex.ru

DOI: https://doi.org/10.1090/S0077-1554-2010-00183-5
Keywords: Spherical homogeneous space, semisimple algebraic group, Lie algebra, affine, highest weights
Published electronically: December 21, 2010
Additional Notes: This research was partially supported by the Russian Foundation for Basic Research (grant no.09-01-00648a), as well as by the grant NSh-1983.2008.1.
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society