Advertisement

Representations with a reduced null cone

  • Hanspeter KraftEmail author
  • Gerald W. Schwarz
Chapter
Part of the Progress in Mathematics book series (PM, volume 257)

Abstract

Let G be a complex reductive group and V a G-module. Let \(\pi: V \rightarrow V/\!\!/G\) be the quotient morphism defined by the invariants and set \(\mathcal{N}(V ):=\pi ^{-1}(\pi (0))\). We consider the following question. Is the null cone \(\mathcal{N}(V )\) reduced, i.e., is the ideal of \(\mathcal{N}(V )\) generated by G-invariant polynomials? We have complete results when G is \(\mathop{\mathrm{SL}}\nolimits _{2}\), \(\mathop{\mathrm{SL}}\nolimits _{3}\) or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.

Keywords

Null cone Null fiber Quotient morphism Semisimple groups Representations 

Mathematics Subject Classification

20G20 22E46 

References

  1. [BK79]
    Walter Borho and Hanspeter Kraft, Über Bahnen und deren Deformationen bei linearen Aktionen reduktiver Gruppen, Comment. Math. Helv. 54 (1979), no. 1, 61–104.CrossRefMathSciNetzbMATHGoogle Scholar
  2. [Bou68]
    N. Bourbaki, Eléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968.Google Scholar
  3. [Bri85]
    Michel Brion, Représentations exceptionnelles des groupes semi-simples, Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 2, 345–387.Google Scholar
  4. [Dix81]
    Jacques Dixmier, Sur les invariants des formes binaires, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 23, 987–990.MathSciNetzbMATHGoogle Scholar
  5. [Dyn52a]
    E. B. Dynkin, Maximal subgroups of the classical groups, Trudy Moskov. Mat. Obšč. 1 (1952), 39–166.MathSciNetzbMATHGoogle Scholar
  6. [Dyn52b]
    E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates).Google Scholar
  7. [GV78]
    V. Gatti and E. Viniberghi, Spinors of 13-dimensional space, Adv. in Math. 30 (1978), no. 2, 137–155.CrossRefMathSciNetzbMATHGoogle Scholar
  8. [Gro67]
    A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967), no. 32, 361.Google Scholar
  9. [Hes80]
    Wim H. Hesselink, Characters of the nullcone, Math. Ann. 252 (1980), no. 3, 179–182.CrossRefMathSciNetzbMATHGoogle Scholar
  10. [KPV76]
    Victor G. Kac, Vladimir L. Popov, and Ernest B. Vinberg, Sur les groupes linéaires algébriques dont l’algèbre des invariants est libre, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 12, Ai, A875–A878.Google Scholar
  11. [Kno86]
    Friedrich Knop, Über die Glattheit von Quotientenabbildungen, Manuscripta Math. 56 (1986), no. 4, 419–427.CrossRefMathSciNetzbMATHGoogle Scholar
  12. [Kos63]
    Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404.CrossRefMathSciNetzbMATHGoogle Scholar
  13. [KR71]
    Bertram Kostant and Stephen Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.CrossRefMathSciNetzbMATHGoogle Scholar
  14. [Kra84]
    Hanspeter Kraft, Geometrische Methoden in der Invariantentheorie, Aspects of Mathematics, D1, Friedr. Vieweg & Sohn, Braunschweig, 1984.Google Scholar
  15. [KW06]
    Hanspeter Kraft and Nolan R. Wallach, On the null cone of representations of reductive groups, Pacific J. Math. 224 (2006), no. 1, 119–139.CrossRefMathSciNetzbMATHGoogle Scholar
  16. [Lun75]
    D. Luna, Adhérences d’orbite et invariants, Invent. Math. 29 (1975), no. 3, 231–238.CrossRefMathSciNetzbMATHGoogle Scholar
  17. [Mat89]
    Hideyuki Matsumura, Commutative Ring Theory, second ed., Cambridge Studies in Advanced Mathematics, Vol. 8, Cambridge University Press, Cambridge, 1989, Translated from the Japanese by M. Reid.Google Scholar
  18. [Pan85]
    D. I. Panyushev, Regular elements in spaces of linear representations. II, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 5, 979–985, 1120.Google Scholar
  19. [Pan99]
    Dmitri I. Panyushev, Actions of “nilpotent tori” on G-varieties, Indag. Math. (N.S.) 10 (1999), no. 4, 565–579.Google Scholar
  20. [Pop76]
    V. L. Popov, Representations with a free module of covariants, Funkcional. Anal. i Priložen. 10 (1976), no. 3, 91–92.Google Scholar
  21. [Pro07]
    Claudio Procesi, Lie Groups, Universitext, Springer, New York, 2007, An approach through invariants and representations.Google Scholar
  22. [Ric89]
    R. W. Richardson, Irreducible components of the nullcone, Invariant theory (Denton, TX, 1986), Contemp. Math., Vol. 88, Amer. Math. Soc., Providence, RI, 1989, pp. 409–434.Google Scholar
  23. [Sch68]
    Issai Schur, Vorlesungen über Invariantentheorie, Bearbeitet und herausgegeben von Helmut Grunsky. Die Grundlehren der mathematischen Wissenschaften, Band 143, Springer-Verlag, Berlin, 1968.Google Scholar
  24. [Sch78]
    Gerald W. Schwarz, Representations of simple Lie groups with regular rings of invariants, Invent. Math. 49 (1978), no. 2, 167–191.CrossRefMathSciNetzbMATHGoogle Scholar
  25. [Sch87]
    Gerald W. Schwarz, On classical invariant theory and binary cubics, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 3, 191–216.CrossRefMathSciNetzbMATHGoogle Scholar
  26. [Sch88]
    Gerald W. Schwarz, Invariant theory of G 2 and Spin7, Comment. Math. Helv. 63 (1988), no. 4, 624–663.MathSciNetzbMATHGoogle Scholar
  27. [Sch79]
    Gerald W. Schwarz, Representations of simple Lie groups with a free module of covariants, Invent. Math. 50 (1978/79), no. 1, 1–12.Google Scholar
  28. [Ste98]
    John R. Stembridge, The partial order of dominant weights, Adv. Math. 136 (1998), no. 2, 340–364.CrossRefMathSciNetzbMATHGoogle Scholar
  29. [vLCL92]
    M. A. A. van Leeuwen, A. M. Cohen, and B. Lisser, LiE: A package for Lie Group Computations, CAN, Computer Algebra Netherland, Amsterdam, 1992.Google Scholar
  30. [Vin76]
    È. B. Vinberg, The Weyl group of a graded Lie algebra, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, 709.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematisches Institut der Universität BaselBaselSwitzerland
  2. 2.Department of MathematicsBrandeis UniversityWalthamUSA

Personalised recommendations