Transformation Groups

, Volume 16, Issue 3, pp 827–856 | Cite as

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

  • Vladimir L. Popov


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 TG/T where T is a maximal torus of G and W the Weyl group.


Algebraic Group Toric Variety Maximal Torus Rational Section Representation Ring 
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  1. [Ada]
    J. F. Adams, Lectures on Lie Groups, Benjamin, New York, 1969. Russian transl.: Дж. Адамс, Лекции по группам Ли, Наука, M., 1979.zbMATHGoogle Scholar
  2. [Ben]
    D. J. Benson, Polynomial Invariants of Finite Groups, London Mathematical Society Lecture Note Series, Vol. 190, Cambridge University Press, Cambridge, 1993.zbMATHCrossRefGoogle Scholar
  3. [Bor]
    A. Borel, Linear Algebraic Groups, 2nd enlarged ed., Graduate Texts in Mathematics, Vol. 126, Springer-Verlag, 1991.Google Scholar
  4. [BT]
    A. Borel, J. Tits, Compléments à l’article “Groupes réductifs”, Publ. math. IHES 41 (1972), 253–276.MathSciNetzbMATHGoogle Scholar
  5. [Bou1]
    N. Bourbaki, Algèbre Commutative, Chap. V, VI, Hermann, Paris, 1964. Russian transl.: Н. Бурбаки, Коммутативная алгебра, Мир, M., 1971.zbMATHGoogle Scholar
  6. [Bou2]
    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.zbMATHGoogle Scholar
  7. [CTKPR]
    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).Google Scholar
  8. [FM]
    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. [Ful]
    W. Fulton, Introduction to Toric Varieties, Princeton University Press, Princeton, New Jersey, 1993.zbMATHGoogle Scholar
  10. [Gro1]
    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.Google Scholar
  11. [Gro2]
    A. Grothendieck, EGA I, Publ. Math. IHES 4 (1960), 5–228.Google Scholar
  12. [Gro3]
    A. Grothendieck et al., Revêtements Etales et Groupe Fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin, 1991.Google Scholar
  13. [GS]
    Grothendieck–Serre Correspondence, Bilingual Edition, P. Colmez, J.-P. Serre, eds., American Mathematical Society, Société Mathématique de France, 2004.Google Scholar
  14. [HR]
    G. H. Hardy, S. Ramanujan, Asymptotic formulae in combinatory analysis, Proc. London Math. Soc. 17 (1918), 75–115.CrossRefGoogle Scholar
  15. [Har]
    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. [Hum1]
    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.zbMATHGoogle Scholar
  17. [Hum2]
    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.zbMATHGoogle Scholar
  18. [Hum3]
    J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1995.zbMATHGoogle Scholar
  19. [Hus]
    D. Husemoller, Fibre Bundles, McGraw-Hill Book Company, New York, 1966. Russian transl.: Д. Хьюзмоллер, Расслоенные пространства, Мир, M., 1970.zbMATHGoogle Scholar
  20. [Kac]
    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. [Kos]
    B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [Lor1]
    M. Lorenz, Multiplicative invariants and semigroup algebras, Algebras and Representation Theory 4 (2001), 293–304.MathSciNetzbMATHCrossRefGoogle Scholar
  23. [Lor2]
    M. Lorenz, Multiplicative Invariant Theory, Encyclopaedia of Mathematical Sciences, Vol. 135, Subseries Invariant Theory and Algebraic Transformation Groups, Vol. VI, Springer, Berlin, 2005.zbMATHGoogle Scholar
  24. [MF]
    D. Mumford, J. Fogarty, Geometric Invariant Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 34, Springer-Verlag, Berlin, 1982.zbMATHGoogle Scholar
  25. [Oda]
    T. Oda, Convex Bodies and Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete 3. Folge, Band 15, Springer-Verlag, Berlin, 1988.zbMATHGoogle Scholar
  26. [Pop1]
    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. [Pop2]
    V. L. Popov, Letter to A. Premet, July 5, 2009.Google Scholar
  28. [Rich1]
    R. W. Richardson, The conjugating representation of a semisimple group, Invent. Math. 54 (1979), 229–245.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [Rich2]
    R. W. Richardson, Orbits, invariants, and representations associated to involutions of reductive groups, Invent. Math. 66 (1982), 287–312.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [Ser1]
    J.-P. Serre, Groupes de Grothendieck des schémas en groupes réductifs déployés, Publ. Math. IHES 34 (1968), 37–52.zbMATHGoogle Scholar
  31. [Ser2]
    J.-P. Serre, Groupes finis d’automorphismes d’anneaux locaux réguliers, in: Colloque d’Algèbre, Secrétariat matheématique, Paris, 1968, pp. 8-01–8–11Google Scholar
  32. [Slo]
    P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Mathematics, Vol. 815, Springer-Verlag, Berlin, 1980.zbMATHGoogle Scholar
  33. [Spr]
    T. A. Springer, Linear Algebraic Groups, 2nd ed., Birkhäuser, Boston, 1998.zbMATHCrossRefGoogle Scholar
  34. [Ste1]
    R. Steinberg, Regular elements of semi-simple algebraic groups, Publ. Math. IHES 25 (1965), 49–80.MathSciNetGoogle Scholar
  35. [Ste2]
    R. Steinberg, Lectures on Chevalley Groups, Yale University, New Haven, Conn., 1968.zbMATHGoogle Scholar
  36. [Stu3]
    R. Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [Stu]
    B. Sturmfels, Gröbner Bases and Convex Polytopes, University Lecture Series, Vol. 8, American Mathematical Society, Providence, Rhode Island, 1996.zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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