The Cubic, the Quartic, and the Exceptional Group G2

  • Anthony van Groningen
  • Jeb F. WillenbringEmail author
Part of the Developments in Mathematics book series (DEVM, volume 38)


We study an example first addressed in a 1949 paper of J. A. Todd, in which the author obtains a complete system of generators for the covariants in the polynomial functions on the eight-dimensional space of the double binary form of degree (3,1), under the action of SL2 ×SL2. We reconsider Todd’s result by examining the complexified Cartan complement corresponding to the maximal compact subgroup of simply connected split G 2. A result of this analysis involves a connection with the branching rule from the rank two complex symplectic Lie algebra to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra. Special cases of this branching rule are related to covariants for cubic and quartic binary forms.

Key words

Binary form Branching rule Double binary form G2 Harmonic polynomials Principal \(\mathfrak{s}\mathfrak{l}_{2}\) Symmetric space 

Mathematics Subject Classification (2010):

20G05 22E45 17B10. 



We thank Allen Bell, Nolan Wallach and Gregg Zuckerman for helpful conversations about the results presented here. The first author’s Ph.D. thesis [17] contains a more thorough treatment of the graded K-multiplicities associated with the symmetric pair (\(G_{2},\mathfrak{s}\mathfrak{l}_{2} \oplus \mathfrak{s}\mathfrak{l}_{2}\)), which was jointly directed by Allen Bell and the second author. The problem concerning the graded decomposition of \(\mathcal{H}_{\mathfrak{p}}\) as a K-representation was suggested by Nolan Wallach (see [19]), while the problem of studying the restriction of a finite-dimensional representation to a principally embedded \(\mathfrak{s}\mathfrak{l}_{2}\)-subalgebra was suggested by Gregg Zuckerman (see [20]).

Finally, we would like to thank the referee for many helpful suggestions, and corrections.


  1. [1]
    Ilka Agricola, Old and new on the exceptional group G 2, Notices Amer. Math. Soc. 55 (2008), no. 8, 922–929.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Michel Brion, On the general faces of the moment polytope, Internat. Math. Res. Notices 4 (1999), 185–201, DOI 10.1155/S1073792899000094.Google Scholar
  3. [3]
    David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993.zbMATHGoogle Scholar
  4. [4]
    E. B. Dynkin, Semisimple subalgebras of semisimple lie algebras, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates).Google Scholar
  5. [5]
    Roe Goodman and Nolan R. Wallach, Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, Vol. 255, Springer, New York, 2009.Google Scholar
  6. [6]
    È. B. Vinberg, V. V. Gorbatsevich, and A. L. Onishchik, Structure of Lie groups and Lie algebras, Current problems in mathematics. Fundamental directions, Vol. 41 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990, pp. 5–259 (Russian).Google Scholar
  7. [7]
    Roger Howe, The classical groups and invariants of binary forms, The mathematical heritage of Hermann Weyl (Durham, NC, 1987), Proc. Sympos. Pure Math., Vol. 48, Amer. Math. Soc., Providence, RI, 1988, pp. 133–166.Google Scholar
  8. [8]
    Roger Howe, Eng-Chye Tan, and Jeb F. Willenbring, Stable branching rules for clas- sical symmetric pairs, Trans. Amer. Math. Soc. 357 (2005), no. 4, 1601–1626, DOI 10.1090/S0002-9947-04-03722-5.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Bertram Kostant, The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group, Amer. J. Math. 81 (1959), 973–1032.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753–809.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    Yannis Yorgos Papageorgiou, SL(2)(C), the cubic and the quartic, ProQuest LLC, Ann Arbor, MI, 1996. Thesis (Ph.D.), Yale University.Google Scholar
  12. [12]
    Yannis Y. Papageorgiou, sl2, the cubic and the quartic, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 29–71 (English, with English and French summaries).Google Scholar
  13. [13]
    Ivan Penkov and Vera Serganova, Bounded simple \((\mathfrak{g},\textrm{ sl}(2))\) -modules for \(\textrm{rk}\,\mathfrak{g} = 2\). J. Lie Theory, 20 (2010), no. 3, 581–615.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Roger Penrose and Wolfgang Rindler, Spinors and space-time. Vol. 1, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1987. Two-spinor calculus and relativistic fields.Google Scholar
  15. [15]
    A. V. Smirnov, Decomposition of symmetric powers of irreducible representations of semisimple lie algebras, and the brion polytope, Tr. Mosk. Mat. Obs. 65 (2004), 230–252. (Russian, with Russian summary); English transl., Trans. Moscow Math. Soc. (2004), 213–234.Google Scholar
  16. [16]
    J. A. Todd, The geometry of the binary (3, 1) form, Proc. London Math. Soc. (2) 50 (1949), 430–437.Google Scholar
  17. [17]
    Anthony Paul van Groningen, Graded multiplicities of the nullcone for the algebraic symmetric pair of type G. ProQuest LLC, Ann Arbor, MI, 2007. Thesis (Ph.D.), The University of Wisconsin, Milwaukee.Google Scholar
  18. [18]
    David A. Vogan Jr., The unitary dual of G 2, Invent. Math. 116 (1994), no. 1–3, 677–791.Google Scholar
  19. [19]
    N. R. Wallach and J. Willenbring, On some q-analogs of a theorem of kostant-rallis, Canad. J. Math. 52 (2000), no. 2, 438–448, DOI 10.4153/CJM-2000-020-0.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    Jeb F. Willenbring and Gregg J. Zuckerman, Small semisimple subalgebras of semisimple lie algebras, Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 12, World Sci. Publ., Hackensack, NJ, 2007, pp. 403–429.Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Milwaukee School of EngineeringMilwaukeeUSA
  2. 2.Department of Mathematical SciencesUniversity of Wisconsin-MilwaukeeMilwaukeeUSA

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