Skip to main content
SpringerLink
Log in

Cross-sections, quotients, and representation rings of semisimple algebraic groups

  • Published:
Transformation Groups Aims and scope Submit manuscript

Abstract

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny \( \tau :\hat{G} \to G \) is bijective; this answers Grothendieck’s question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg’s theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck’s questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T ⇢ G/T where T is a maximal torus of G and W the Weyl group.

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. J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969. Russian transl.: Дж. Адамс, Лекции по группам Ли, Наука, M., 1979.

    MATH  Google Scholar 

  2. D. J. Benson, Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Note Series, Vol. 190, Cambridge University Press, Cambridge, 1993.

    Book  MATH  Google Scholar 

  3. A. Borel, Linear Algebraic Groups, 2nd enlarged ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, 1991.

  4. A. Borel, J. Tits, Compléments à l’article “Groupes réductifs”, Publ. math. IHES 41 (1972), 253–276.

    MathSciNet  MATH  Google Scholar 

  5. N. Bourbaki, Algèbre Commutative, Chap. V, VI, Hermann, Paris, 1964. Russian transl.: Н. Бурбаки, Коммутативная алгебра, Мир, M., 1971.

    MATH  Google Scholar 

  6. N. Bourbaki, Groupes et Algèbres de Lie, Chap. IV, V, VI, Hermann, Paris, 1968. Russian transl.: Н. Бурбаки, Группы и алгебры Ли. Группы Кокстера и системы Титса. Группы, порождениями. Системы корней, Мир, M., 1972.

    MATH  Google Scholar 

  7. J.-L. Colliot-Thélène, B. Kunyavskiĭ, V. L. Popov, Z. Reichstein, Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?, arXiv:0901.4358v1 (27 January, 2009).

  8. R. Friedman, J. W. Morgan, Automorphism sheaves, spectral covers, and the Kostant and Steinberg sections, in: Vector Bundles and Representation Theory (Columbia, MO, 2002), Contemp. Math., Vol. 322, Amer. Math. Soc., Providence, RI, 2003, pp. 217–244.

    Google Scholar 

  9. W. Fulton, Introduction to Toric Varieties, Princeton University Press, Princeton, New Jersey, 1993.

    MATH  Google Scholar 

  10. A. Grothendieck, Compléments de géométrie algébrique. Espaces de transformations, in: Séminaire C. Chevalley, 1956–1958. Classification de groupes de Lie algébriques, Vol. 1, Exposé no. 5, Secr. math. ENS, Paris, 1958.

  11. A. Grothendieck, EGA I, Publ. Math. IHES 4 (1960), 5–228.

    Google Scholar 

  12. A. Grothendieck et al., Revêtements Etales et Groupe Fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin, 1991.

    Google Scholar 

  13. Grothendieck–Serre Correspondence, Bilingual Edition, P. Colmez, J.-P. Serre, eds., American Mathematical Society, Société Mathématique de France, 2004.

  14. G. H. Hardy, S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75–115.

    Article  Google Scholar 

  15. J. Harris, Algebraic Geometry. A First Course, Graduate Texts in Mathematics, Vol. 133, Springer-Verlag, New York, 1995. Russian transl.: Дж. Харрис, Алгебраическая геометрия. Начальный курс, МЦНМО, M., 2006.

    Google Scholar 

  16. J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York, 1972. Russian transl.: Дж. Хамфрис, Введение в теорию алгебр Ли и их представлений, МЦНМО, M., 2003.

    MATH  Google Scholar 

  17. J. E. Humphreys, Linear Algebraic Groups, Graduate Texts in Mathematics, Vol. 21, Springer-Verlag, New York, 1975. Russian transl.: Дж. Хамфрис, Линейные алгебраические группы, Наука, M., 1980.

    MATH  Google Scholar 

  18. J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1995.

    MATH  Google Scholar 

  19. D. Husemoller, Fibre Bundles, McGraw-Hill Book Company, New York, 1966. Russian transl.: Д. Хьюзмоллер, Расслоенные пространства, Мир, M., 1970.

    MATH  Google Scholar 

  20. V. G. Kac, Root systems, representations of quivers and invariant theory, in: Invariant Theory, Proceedings, Montecatini 1982, Lecture Notes in Mathematics, Vol. 996, Springer-Verlag, Berlin, 1983, pp. 74–108.

    Google Scholar 

  21. B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.

    Article  MathSciNet  MATH  Google Scholar 

  22. M. Lorenz, Multiplicative invariants and semigroup algebras, Algebras and Representation Theory 4 (2001), 293–304.

    Article  MathSciNet  MATH  Google Scholar 

  23. M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 135, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VI, Springer, Berlin, 2005.

    MATH  Google Scholar 

  24. D. Mumford, J. Fogarty, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 34, Springer-Verlag, Berlin, 1982.

    MATH  Google Scholar 

  25. T. Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 15, Springer-Verlag, Berlin, 1988.

    MATH  Google Scholar 

  26. V. L. Popov, On the “Lemma of Seshadri”, in: Lie Groups, their Discrete Subgroups, and Invariant Theory, Advances in Soviet Mathematics, Vol. 8, Amer. Math. Soc., Providence, RI, 1992, 167–172.

    Google Scholar 

  27. V. L. Popov, Letter to A. Premet, July 5, 2009.

  28. R. W. Richardson, The conjugating representation of a semisimple group, Invent. Math. 54 (1979), 229–245.

    Article  MathSciNet  MATH  Google Scholar 

  29. R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), 287–312.

    Article  MathSciNet  MATH  Google Scholar 

  30. J.-P. Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Publ. Math. IHES 34 (1968), 37–52.

    MATH  Google Scholar 

  31. J.-P. Serre, Groupes finis d’automorphismes d’anneaux locaux réguliers, in: Colloque d’Algèbre, Secrétariat matheématique, Paris, 1968, pp. 8-01–8–11

    Google Scholar 

  32. P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  33. T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, Boston, 1998.

    Book  MATH  Google Scholar 

  34. R. Steinberg, Regular elements of semi-simple algebraic groups, Publ. Math. IHES 25 (1965), 49–80.

    MathSciNet  Google Scholar 

  35. R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, Conn., 1968.

    MATH  Google Scholar 

  36. R. Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177.

    Article  MathSciNet  MATH  Google Scholar 

  37. B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, Vol. 8, American Mathematical Society, Providence, Rhode Island, 1996.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow, 119991, Russia

    Vladimir L. Popov

Authors
  1. Vladimir L. Popov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vladimir L. Popov.

Additional information

To T. A. Springer on his 85th birthday

“Is Steinberg’s theorem […] only true for simply connected groups […] ? What happens for GP(1), for instance? Is there a rational section of G over I(G) (“invariants”) in this case? […] Is it true that I(G) is a rational variety […] ?”

A. Grothendieck, Letter to J.-P. Serre, January 15, 1969 [GS, pp. 240–241]

Rights and permissions

Reprints and permissions

About this article

Cite this article

Popov, V.L. Cross-sections, quotients, and representation rings of semisimple algebraic groups. Transformation Groups 16, 827–856 (2011). https://doi.org/10.1007/s00031-011-9137-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00031-011-9137-6

Keywords

Search

Navigation