Advertisement

Classification of minimal representations of real simple Lie groups

  • Hiroyoshi TamoriEmail author
Article
  • 21 Downloads

Abstract

Based on an idea by Gan and Savin (Represent Theory 9:46–93, 2005), we give a classification of minimal representations of connected simple real Lie groups not of type A. Actually, we prove that there exist no new minimal representations up to infinitesimal equivalence.

Keywords

Minimal representation Reductive group 

Mathematics Subject Classification

22E46 22E47 20G05 15A72 

Notes

Acknowledgements

The author is deeply grateful to his advisor Prof. Toshiyuki Kobayashi for helpful comments and warm encouragement. The author expresses his sincere thanks to Dr. Yoshiki Oshima and Dr. Masatoshi Kitagawa for inspiring discussions. This work was supported by JSPS KAKENHI Grant Number 17J01075 and the Program for Leading Graduate Schools, MEXT, Japan.

References

  1. 1.
    Barchini, L., Sepanski, M., Zierau, R.: Positivity of zeta distributions and small unitary representations. Contemp. Math. 398, 1–46 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Binegar, B., Zierau, R.: Unitarization of a singular representation of \({SO}(p, q)\). Commun. Math. Phys. 138, 245–258 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourbaki, N.: Groupes et algèbres de Lie, Chapitres \(4, 5\) et \(6\). Hermann, Paris (1968)Google Scholar
  4. 4.
    Braverman, A., Joseph, A.: The minimal realization from deformation theory. J. Algebra 205, 13–36 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Brylinski, R.: Geometric quantization of real minimal nilpotent orbits. Differ. Geom. Appl. 9(1–2), 5–58 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brylinski, R., Kostant, B.: Differential operators on conical Laglangian manifolds. In: Brylinski, J.L., Brylinski, R., Guillemin, V., Kac, V. (eds.) Lie Theory and Geometry: In Honor of Bertram Kostant, Progr. Math., vol. 123. Birkhäuser, Boston (1994)CrossRefGoogle Scholar
  7. 7.
    Brylinski, R., Kostant, B.: Minimal reprerentations of \({E}_6\), \({E}_7\), and \({E}_8\) and the generalized Capelli identity. Proc. Natl. Acad. Sci. USA 91(7), 2469–2472 (1994)CrossRefzbMATHGoogle Scholar
  8. 8.
    Brylinski, R., Kostant, B.: Minimal representations, geometric quantization, and unitarity. Proc. Natl. Acad. Sci. USA 91(13), 6026–6029 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Brylinski, R.: Kostant B (1995) Lagrangian models of minimal representations of \({E}_6\), \({E}_7\) and \({E}_8\). In: Gindikin, S., Lepowsky, J., Wilson, R. (eds.) Functional Analysis on the Eve of the 21st Century, Progr. Math., 131, vol. 1, pp. 13–63. Birkhäuser, Boston (1995)Google Scholar
  10. 10.
    Duflo, M.: Représentations unitaires irréductibles des groupes simples complexes de rang deux. Bull. Soc. Math. Fr. 107, 55–96 (1979)CrossRefzbMATHGoogle Scholar
  11. 11.
    Enright, T., Howe, R., Wallach, N.: A classification of unitary highest weight modules. In: Representation Theory of Reductive Groups, Progr. Math., vol. 40, pp. 97–143. Birkhäuser, Boston (1983)Google Scholar
  12. 12.
    Enright, T., Parthasarathy, R., Wallach, N., Wolf, J.: Unitary derived functor modules with small spectrum. Acta Math. 154(1–2), 105–136 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Folland, G.B.: Harmonic Analysis in Phase Space, Annals of Mathematics Studies, vol. 122. Princeton University Press, Princeton (1989)Google Scholar
  14. 14.
    Gan, W.T., Savin, G.: Uniqueness of Joseph ideal. Math. Res. Lett. 11, 589–597 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gan, W.T., Savin, G.: On minimal representations definitions and properties. Represent. Theory 9, 46–93 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Garfinkle, D.: A new construction of the Joseph ideal. Ph.D. thesis, M.I.T. (1982)Google Scholar
  17. 17.
    Goncharov, A.B.: Constructions of Weil representations of some simple Lie algebras. Funktsional. Anal. i Prilozhen 16(2), 70–71 (1982)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Gross, B., Wallach, N.: A distinguished family of unitary representations for the exceptional groups of real rank \(=\) \(4\). In: Brylinski, J.L., Brylinski, R., Guillemin, V., Kac, V. (eds.) Lie Theory and Geometry: In Honor of Bertram Kostant, Progr. Math., vol. 123, pp. 289–304. Birkhäuser, Boston (1994)Google Scholar
  19. 19.
    Gross, B., Wallach, N.: On quaternionic discrete series representations, and their continuations. J. Reine Angew. Math. 481, 73–123 (1996)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Hilgert, J., Kobayashi, T., Möllers, J., Ørsted, B.: Fock model and Segal–Bargmann transform for minimal representations of Hermitian Lie groups. J. Funct. Anal. 263(11), 3492–3563 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hilgert, J., Kobayashi, T., Möllers, J.: Minimal representations via Bessel operators. J. Math. Soc. Jpn. 66(2), 349–414 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Huang, J.S.: Minimal representations, shared orbits, and dual pair correspondences. Int. Math. Res. Notices 1995(6), 309–323 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Huang, J.S., Li, J.S.: Unipotent representations attached to spherical nilpotent orbits. Am. J. Math. 121(3), 497–517 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)Google Scholar
  25. 25.
    Jakobsen, H.P.: Hermitian symmetric spaces and their unitary highest weight modules. J. Funct. Anal. 52(3), 385–412 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Joseph, A.: The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Sci. École. Norm. Sup. (4) 9(1), 1–29 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kazhdan, D.: The minimal representation of \({D}_{4}\). In: Connes, A., Duflo, M., Joseph, A., Rentschler, R. (eds.) Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math., vol. 92, pp. 125–158. Birkhäuser, Boston (1990)Google Scholar
  28. 28.
    Kazhdan, D., Savin, G.: The smallest representation of simply laced groups. In: Gelbert, S., Howe, R., Sarnak, P. (eds.) Festschrift in Honor of I. I. Piatetski-Shapiro on the Occasion of his Sixtieth Birthday. Part I., Israel Math. Conf. Proc., vol. 2, pp. 209–223. Weizmann Science Press of Israel, Jerusalem (1990)Google Scholar
  29. 29.
    Knapp, A.W.: Lie Groups Beyond an Introduction, Progr. Math., vol. 140, 2nd edn. Birkhäuser, Boston (2002)Google Scholar
  30. 30.
    Kobayashi, T.: Algebraic analysis of minimal representations. Publ. Res. Inst. Math. Sci. 47(2), 585–611 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Kobayashi, T., Mano, G.: The Schrödinger model for the minimal representation of the indefinite orthogonal group \({O}(p, q)\). Mem. Am. Math. Soc. 213, 1000 (2011)zbMATHGoogle Scholar
  32. 32.
    Kobayashi, T., Ørsted, B.: Conformal geometry and branching laws for unitary representatins attached to minimal nilpotent orbits. C. R. Acad. Sci. Paris Sér. I Math. 326(8), 925–930 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kobayashi, T., Ørsted, B.: Analysis on the minimal representation of \({O}(p, q)\), I. Realization via conformal geometry. Adv. Math. 180, 486–512 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kobayashi, T., Ørsted, B.: Analysis on the minimal representation of \({O}(p, q)\). II. Branching laws. Adv. Math. 180, 513–550 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Kobayashi, T., Ørsted, B.: Analysis on the minimal representation of \({O}(p, q)\), III. Ultrahyperbolic equations on \({\mathbb{R}}^{p-1, q-1}\). Adv. Math. 180, 551–595 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Kobayashi, T., Oshima, Y.: Classification of symmetric pairs with discretely decomposable restrictions of \((\mathfrak{g}, k)\)-modules. J. Reine Angew. Math. 703, 201–223 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Kostant, B.: The vanishing scalar curvature and the minimal unitary representation of \({SO}(4,4)\). In: Connes, A., Duflo, M., Joseph, A., Rentschler, R. (eds.) Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory, Progr. Math., vol. 92, pp. 85–124. Birkhäuser, Boston (1990)Google Scholar
  38. 38.
    Lepowsky, J., McCollum, G.W.: On the determination of irreducible modules by restriction to a subalgebra. Trans. Am. Math. Soc. 176, 45–57 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Li, J.S.: Minimal representations & reductive dual pairs. In: Representation theory of Lie groups (Park City, UT, 1998), IAS/Park City Math. Ser., vol. 8, pp. 293–340. Amer. Math. Soc., Providence (2000)Google Scholar
  40. 40.
    Loke, H.Y., Savin, G.: The smallest representations of nonlinear covers of odd orthogonal groups. Am. J. Math. 130(3), 763–797 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Okuda, T.: Smallest complex nilpotent orbits with real points. J. Lie Theory 25(2), 507–533 (2015)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Sabourin, H.: Une représentation unipotente associée à l’orbite minimale: Le cas de \(so(4,3)\). J. Funct. Anal. 137(2), 394–465 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Salmasian, H.: Isolatedness of the minimal representation and minimal decay of exceptional groups. Manuscr. Math. 120(1), 39–52 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Torasso, P.: Méthode des orbites de Kirillov-Duflo et représentations minimales des groupes simples sur un corps local de caractéristique nulle. Duke Math. J. 90(2), 261–377 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Vogan, D.: Singular unitary representations. In: Noncommutative harmonic analysis and Lie groups, Lecture Notes in Math., vol. 880, pp. 506–535. Springer, Berlin (1981)Google Scholar
  46. 46.
    Vogan, D.: The unitary dual of \({G}_2\). Invent. Math. 116(1–3), 677–791 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Graduate School of Mathematical SciencesThe University of TokyoTokyoJapan

Personalised recommendations