Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A singular representation of $ E\sb 6$


Authors: B. Binegar and R. Zierau
Journal: Trans. Amer. Math. Soc. 341 (1994), 771-785
MSC: Primary 22E47
DOI: https://doi.org/10.1090/S0002-9947-1994-1139491-1
MathSciNet review: 1139491
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Algebraic properties of a singular representation of $ {{\mathbf{E}}_6}$ are studied. This representation has the Joseph ideal as its annihilator and it remains irreducible when restricted to $ {{\mathbf{F}}_4}$.


References [Enhancements On Off] (What's this?)

  • [1] W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III, Invent. Math. 80 (1985), 1-68. MR 784528 (87i:22045)
  • [2] J.T. Chang, Remarks on localization and standard modules: the duality theorem on a generalized flag variety, M.S.R.I preprint, 1988. MR 1145942 (93d:22016)
  • [3] G. Elkington, Centralizers of unipotent elements in semisimple algebraic groups, J. Algebra 23 (1972), 137-163. MR 0308228 (46:7342)
  • [4] T. Enright, R. Howe, and N. Wallach, A classification of unitary highest weight modules, Representation Theory of Reductive Groups, Progr. Math., vol. 40, Birkhäuser, Boston, Mass., 1983, pp. 97-143. MR 733809 (86c:22028)
  • [5] T. Enright and A. Joseph, An intrinsic analysis of unitary highest weight modules, Math. Ann. 288 (1990), 571-594. MR 1081264 (91m:17005)
  • [6] D. Garfinkle, A new construction of the Joseph ideal, Ph.D. Thesis, M.I.T., 1982.
  • [7] S. Helgason, Differential geometry, Lie groups and symmetric spaces, Academic Press, New York, 1978. MR 514561 (80k:53081)
  • [8] -, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [9] A. Joseph, The minimal orbit in a simple Lie algebra and its associated maximal ideal, Ann. Sci. École Norm. Sup. (4) 9 (1976), 1-30. MR 0404366 (53:8168)
  • [10] B. Kostant, The principle three-dimensional subgroup and the Betti numbers of a complex semisimple Lie group, Amer. J. Math. 81 (1959), 973-1032. MR 0114875 (22:5693)
  • [11] -, The vanishing of scalar curvature and the minimal representation of $ SO(4,4)$, M.I.T. preprint.
  • [12] W. Schmid, Die Randwerte Holomorphic Functionen auf Hermitesch Symmetrischen Räumen, Invent. Math. 9 (1969), 61-80. MR 0259164 (41:3806)
  • [13] J. Schwartz, The determination of the admissible orbits in the real classical groups, Ph.D. dissertation, MIT, Cambridge, Mass., 1987.
  • [14] D. Vogan, Jr., Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98. MR 0506503 (58:22205)
  • [15] -, Singular unitary representations, Non-Commutative Harmonic Analysis, (Proc., 1980), Lecture Notes in Math., vol. 880, Springer-Verlag, Berlin, Heidelberg, and New York, 1981. MR 644845 (83k:22036)
  • [16] -, Representations of real reductive Lie groups, Birkhäuser, Boston, Mass., 1981.
  • [17] -, Unitary representations of reductive Lie groups, Princeton Univ. Press, Princeton, NJ, 1987. MR 908078 (89g:22024)
  • [18] -, Associated varieties and unipotent representations, preprint.
  • [19] D. Vogan, Jr. and G. Zuckerman, Unitary representations with non-zero cohomology, Compositio Math. 53 (1984), 51-90. MR 762307 (86k:22040)
  • [20] N. Wallach, The asymptotic behavior of holomorphic representations, Mém. Soc. Math. France No. 15 (1984), 291-305. MR 789089 (86j:22025)
  • [21] R. Zierau, Geometric constructions of unitary highest weight representations, Ph.D. dissertation, Univ. of California, Berkeley, 1984.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E47

Retrieve articles in all journals with MSC: 22E47


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1139491-1
Keywords: Reductive Lie groups, unitary representations, singular representations, metaplectic representation, exceptional groups
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society