Skip to main content
SpringerLink
Log in

Root subgroups on affine spherical varieties

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

Given a connected reductive algebraic group G and a Borel subgroup \(B \subseteq G\), we study B-normalized one-parameter additive group actions on affine spherical G-varieties. We establish basic properties of such actions and their weights and discuss many examples exhibiting various features. We propose a construction of such actions that generalizes the well-known construction of normalized one-parameter additive group actions on affine toric varieties. Using this construction, for every affine horospherical G-variety X we obtain a complete description of all G-normalized one-parameter additive group actions on X and show that the open G-orbit in X can be connected with every G-stable prime divisor via a suitable choice of a B-normalized one-parameter additive group action. Finally, when G is of semisimple rank 1, we obtain a complete description of all B-normalized one-parameter additive group actions on affine spherical G-varieties having an open orbit of a maximal torus \(T \subseteq B\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arzhantsev, I., Bragin, S., Zaitseva, Y.: Commutative algebraic monoid structures on affine spaces. Commun. Contemp. Math. 22(8), 1950064 (2020)

    Article  MathSciNet  Google Scholar 

  2. Arzhantsev, I., Flenner, H., Kaliman, S., Kutzschebauch, F., Zaidenberg, M.: Flexible varieties and automorphism groups. Duke Math. J. 162(4), 767–823 (2013)

    Article  MathSciNet  Google Scholar 

  3. Arzhantsev, I., Hausen, J., Herppich, E., Liendo, A.: The automorphism group of a variety with torus action of complexity one. Moscow Math. J. 14(3), 429–471 (2014)

    Article  MathSciNet  Google Scholar 

  4. Arzhantsev, I., Kotenkova, P.: Equivariant embeddings of commutative linear algebraic groups of corank one. Doc. Math. 20, 1039–1053 (2015)

    MathSciNet  MATH  Google Scholar 

  5. Arzhantsev, I.V., Zaidenberg, M.G., Kuyumzhiyan, K.G.: Flag varieties, toric varieties, and suspensions: three instances of infinite transitivity. Sb. Math. 203(7), 923–949 (2012)

    Article  MathSciNet  Google Scholar 

  6. Arzhantsev, I., Kuyumzhiyan, K., Zaidenberg, M.: Infinite transitivity, finite generation, and Demazure roots. Adv. Math. 351, 1–32 (2019)

    Article  MathSciNet  Google Scholar 

  7. Arzhantsev, I., Liendo, A.: Polyhedral divisors and SL(2)-actions on affine T-varieties. Michigan Math. J. 61(4), 731–762 (2012)

    Article  MathSciNet  Google Scholar 

  8. Arzhantsev, I., Romaskevich, E.: Additive actions on toric varieties. Proc. Amer. Math. Soc. 145(5), 1865–1879 (2017)

    Article  MathSciNet  Google Scholar 

  9. Avdeev, R.: On extended weight monoids of spherical homogeneous spaces. Transform. Groups 26(2), 403–431 (2021)

    Article  MathSciNet  Google Scholar 

  10. Avdeev, R., Cupit-Foutou, S.: New and old results on spherical varieties via moduli theory. Adv. Math. 328, 1299–1352 (2018)

    Article  MathSciNet  Google Scholar 

  11. Avdeev, R., Cupit-Foutou, S.: On the irreducible components of moduli schemes for affine spherical varieties. Transform. Groups 23(2), 299–327 (2018)

    Article  MathSciNet  Google Scholar 

  12. Avdeev, R., Zhgoon, V.: On the existence of \(B\)-root subgroups on affine spherical varieties, preprint (2021), see arXiv:2112.14268 [math.AG]

  13. Bravi, P., Van Steirteghem, B.: The moduli scheme of affine spherical varieties with a free weight monoid. Int. Math. Res. Not. IMRN 2016(15), 4544–4587 (2016)

    Article  MathSciNet  Google Scholar 

  14. Brion, M.: Classification des espaces homogènes sphériques. Compos. Math. 63(2), 189–208 (1987)

    MATH  Google Scholar 

  15. Brion, M., Pauer, F.: Valuations des espaces homogènes sphériques. Comment. Math. Helv. 62(2), 265–285 (1987)

    Article  MathSciNet  Google Scholar 

  16. Cox, D.: The homogeneous coordinate ring of a toric variety. J. Alg. Geom. 4(1), 17–50 (1995)

    MathSciNet  MATH  Google Scholar 

  17. Cox, D., Little, J., Schenck, H.: Toric Varieties, Graduate Studies in Mathematics, vol. 124. AMS, Providence, RI (2011)

    Google Scholar 

  18. Demazure, M.: Sous-groupes algébriques de rang maximum du groupe de Cremona. Ann. Sci. Éc. Norm. Supér. Sér. 4 3(4), 507–588 (1970)

  19. Dubouloz, A., Liendo, A.: Rationally integrable vector fields and rational additive group actions. Int. J. Math. 27(8), 1650060 (2016)

    Article  MathSciNet  Google Scholar 

  20. Dzhunusov, S., Zaitseva, Yu.: Commutative algebraic monoid structures on affine surfaces. Forum Math. 33(1), 177–191 (2021)

    Article  MathSciNet  Google Scholar 

  21. Freudenburg, G.: Algebraic Theory of Locally Nilpotent Derivations. Encyclopaedia of Mathematical Sciences, vol. 136, 2nd edn. Springer, Berlin (2017)

    Book  Google Scholar 

  22. Fulton, W.: Introduction to Toric Varieties, Annals of Mathematics Studies, vol. 131. Princeton University Press, Princeton, NJ (1993)

    Book  Google Scholar 

  23. Gaifullin, S., Shafarevich, A.: Flexibility of normal affine horospherical varieties. Proc. Amer. Math. Soc. 147(8), 3317–3330 (2019)

    Article  MathSciNet  Google Scholar 

  24. Gandini, J., Pezzini, G.: Orbits of strongly solvable spherical subgroups on the flag variety. J. Algebr. Comb. 47(3), 357–401 (2018)

    Article  MathSciNet  Google Scholar 

  25. Knop, F.: The Luna-Vust theory of spherical embeddings. In: Proceedings of the Hyderabad Conference on Algebraic Groups. (Hyderabad, India, 1989), pp. 225–249. Manoj Prakashan, Madras (1991)

  26. Knop, F.: Automorphisms, root systems, and compactifications of homogeneous varieties. J. Amer. Math. Soc. 9(1), 153–174 (1996)

    Article  MathSciNet  Google Scholar 

  27. Knop, F.: Some remarks on multiplicity free spaces. In: Representation Theories and Algebraic Geometry, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, pp. 301–317. Dordrecht, Springer Netherlands (1998)

  28. Langlois, K., Perepechko, A.: Demazure roots and spherical varieties: the example of horizontal \(\text{SL}_2\)-actions, preprint (2014), see arXiv:1406.5744v2 [math.AG]

  29. Liendo, A.: Affine \(\mathbb{T}\)-varieties of complexity one and locally nilpotent derivations. Transform. Groups 15(2), 389–425 (2010)

    Article  MathSciNet  Google Scholar 

  30. Liendo, A.: \(\mathbb{G}_a\)-actions of fiber type on affine \(\mathbb{T}\)-varieties. J. Algebra 324(12), 3653–3665 (2010)

    Article  MathSciNet  Google Scholar 

  31. Luna, D.: Grosses cellules pour les variétés sphériques. In: Algebraic Groups and Lie Groups, Austral. Math. Soc. Lect. Ser. vol. 9, pp. 267–280. Cambridge Univ. Press, Cambridge (1997)

  32. Luna, D., Vust, Th.: Plongements d’espaces homogènes. Comment. Math. Helv. 58(2), 186–245 (1983)

    Article  MathSciNet  Google Scholar 

  33. Losev, I.V.: Proof of the Knop conjecture. Ann. Inst. Fourier 59(3), 1105–1134 (2009)

    Article  MathSciNet  Google Scholar 

  34. Montagard, P.-L.: Une nouvelle propriété de stabilité du pléthysme. Comment. Math. Helv. 71(3), 475–505 (1996)

    Article  MathSciNet  Google Scholar 

  35. Matveev, D.A.: Commuting homogeneous locally nilpotent derivations. Sb. Math. 210(11), 1609–1632 (2019)

    Article  MathSciNet  Google Scholar 

  36. Nill, B.: Complete toric varieties with reductive automorphism group. Math. Z. 252(4), 767–786 (2006)

    Article  MathSciNet  Google Scholar 

  37. Oda, T.: Convex Bodies and Algebraic Geometry: An Introduction to Toric Varieties, A Series of Modern Surveys in Math, vol. 15. Springer Verlag, Berlin (1988)

  38. Panyushev, D.I.: Complexity and nilpotent orbits. Manuscripta Math. 83(3–4), 223–237 (1994)

    Article  MathSciNet  Google Scholar 

  39. Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic Geometry. IV: Linear Algebraic Groups, Invariant Theory, Encycl. Math. Sci., vol. 55, pp. 123–278 (1994)

  40. Regeta, A., van Santen, I.: Characterizing quasi-affine spherical varieties via the automorphism group. J. Éc. Polytech. Math. 8, 379–414 (2021)

    Article  MathSciNet  Google Scholar 

  41. Timashev, D.A.: Homogeneous spaces and equivariant embeddings, Encycl. Math. Sci., vol. 138, Springer-Verlag, Berlin Heidelberg (2011)

  42. Vinberg, E.B., Kimel’fel’d, B.N.: Homogeneous domains on flag manifolds and spherical subgroups of semisimple Lie groups. Funct. Anal. Appl. 12(3), 168–174 (1978)

    Article  MathSciNet  Google Scholar 

  43. Vinberg, E.B., Popov, V.L.: On a class of quasihomogeneous affine varieties. Math. USSR-Izv. 6(4), 743–758 (1972)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Vladimir Zhgoon for his interest in this work and useful discussions. Thanks are also due to the referee for a careful reading of a previous version of this paper and valuable comments.

Author information

Authors and Affiliations

  1. Faculty of Computer Science, HSE University, Pokrovsky boulevard 11, Moscow, Russia, 109028

    Ivan Arzhantsev & Roman Avdeev

Authors
  1. Ivan Arzhantsev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roman Avdeev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roman Avdeev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was supported by the Russian Science Foundation, Grant No. 19-11-00056.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arzhantsev, I., Avdeev, R. Root subgroups on affine spherical varieties. Sel. Math. New Ser. 28, 60 (2022). https://doi.org/10.1007/s00029-022-00775-1

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00029-022-00775-1

Keywords

Mathematics Subject Classification

Search

Navigation