Advertisement

ÉTALE REPRESENTATIONS FOR REDUCTIVE ALGEBRAIC GROUPS WITH FACTORS Sp n OR SO n

  • DIETRICH BURDE
  • WOLFGANG GLOBKE
  • ANDREI MINCHENKO
Article

Abstract

An étale module for a linear algebraic group G is a complex vector space V with a rational G-action on V that has a Zariski-open orbit and dim G = dim V . Such a module is called super-étale if the stabilizer of a point in the open orbit is trivial. Popov (2013) proved that reductive algebraic groups admitting super-étale modules are special algebraic groups. He further conjectured that a reductive group admitting a super-étale module is always isomorphic to a product of general linear groups. Our main result is a construction of counterexamples to this conjecture, namely, a family of super-étale modules for groups with a factor Sp n for arbitrary n ≥ 1. A similar construction provides a family of étale modules for groups with a factor SO n , which shows that groups with étale modules with non-trivial stabilizer are not necessarily special. Both families of examples are somewhat surprising in light of the previously known examples of étale and super-étale modules for reductive groups. Finally, we show that the exceptional groups F4 and E8 cannot appear as simple factors in the maximal semisimple subgroup of an arbitrary Lie group with a linear étale representation.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    O. Baues, Left-symmetric algebras for \( \mathfrak{gl}(n) \), Trans. Amer. Math. Soc. 351 (1999), no. 7, 2979–2996.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    D. Burde, Left-invariant affine structures on reductive Lie groups, J. Algebra 181 (1996), no. 3, 884–902.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Burde, Left-symmetric algebras, or pre-Lie algebras in geometry and physics, Cent. Eur. J. Math. 4 (2006), no. 3, 323–357.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Burde, W. Globke, Étale representations for reductive algebraic groups with one-dimensional center, J. Algebra 487 (2017), 200–216.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    C. Chevalley, Algebraic Lie algebras, Ann. of Math. (2) 48 (1947), 91–100.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    A. Grothendieck, Torsion homologique et sections rationnelles, Séminaire Claude Chevalley 3 (1958), 1–29.Google Scholar
  7. 7.
    J. Helmstetter, Algèbres symétriques à gauche, C. R. Acad. Sci. Paris Sér. A–B 272 (1971), 1088–1091.zbMATHGoogle Scholar
  8. 8.
    T. Kimura, Introduction to Prehomogeneous Vector Spaces, Translations of Mathematical Monographs, Vol. 215, American Mathematical Society, Providence, RI, 2003.Google Scholar
  9. 9.
    V. L. Popov, Some subgroups of the Cremona groups, in: Affine Algebraic Geometry, World Sci. Publ., Hackensack, NJ, 2013, pp. 213–242.Google Scholar
  10. 10.
    M. Sato, T. Kimura, A classification of irreducible prehomogeneous vector spaces and their relative invariants, Nagoya Math. J. 65 (1977), 1–155.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    J.-P. Serre, Espaces fibrés algébriques, Séminaire Claude Chevalley 3 (1958), 1–37.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • DIETRICH BURDE
    • 1
  • WOLFGANG GLOBKE
    • 2
  • ANDREI MINCHENKO
    • 1
  1. 1.Faculty of MathematicsUniversity of ViennaViennaAustria
  2. 2.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia

Personalised recommendations