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Functions on the model orbit in $E_{8}$

Authors: Jeffrey Adams, Jing-Song Huang and David A. Vogan Jr.
Journal: Represent. Theory 2 (1998), 224-263
MSC (1991): Primary 20G15, 22E46
Published electronically: June 2, 1998
MathSciNet review: 1628031
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Abstract: We decompose the ring of regular functions on the unique coadjoint orbit for complex $E_{8}$ of dimension 128, finding that every irreducible representation appears exactly once. This confirms a conjecture of McGovern. We also study the unique real form of this orbit.

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  • [1] D. Barbasch and D. Vogan, Unipotent representations of complex semisimple Lie groups, Ann. of Math. 121 (1985), 41-110. MR 86i:22031
  • [2] I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Models of representations of compact Lie groups, Funct. Anal. Appl. 9 (1975), 61-62. MR 54:2884
  • [3] R. Bott, Homogeneous vector bundles, Ann. of Math. 66 (1957), 203-248. MR 19:681d
  • [4] N. Bourbaki, Groupes et algèbres de Lie. Chapitres 4, 5, et 6, Masson, Paris, 1981.
  • [5] R. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, John Wiley & Sons Ltd, Chichester, England, 1985. MR 94k:20020
  • [6] D. Collingwood and W. McGovern, Nilpotent orbits in semisimple Lie algebras, Chapman and Hall, New York, 1993. MR 94j:17001
  • [7] D. Djokovic, Classification of nilpotent elements in simple exceptional real Lie algebras of inner type and description of their centralizers, J. Algebra 112 (1988), 503-524. MR 89b:17010
  • [8] M. Duflo, Théorie de Mackey pour les groupes de Lie algébriques, Acta Math. 149 (1982), 153-213. MR 85h:22022
  • [9] E. B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transl., Ser. 2, 6 (1957), 111-245.
  • [10] W. Graham, Functions on the universal cover of the principal nilpotent orbit, Invent. Math. 108 (1992), 15-27. MR 93h:22026
  • [11] Harish-Chandra, Harmonic analysis on reductive groups I. The theory of the constant term, J. Functional Anal. 19 (1975), 104-204. MR 53:3201
  • [12] S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, San Francisco, London, 1978. MR 80k:53081
  • [13] V. Hinich, On the singularities of nilpotent orbits, Israel J. Math 73 (1991), 297-308. MR 92m:14005
  • [14] J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Springer-Verlag, Berlin-Heidelberg-New York, 1972. MR 48:2197
  • [15] A. Knapp, Representation Theory of Real Semisimple Groups: an Overview Based on Examples, Princeton University Press, Princeton, New Jersey, 1986. MR 87j:22022
  • [16] B. Kostant, Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. 74 (1961), 329-387.
  • [17] B. Kostant and S. Rallis, Orbits and representations associated with symmetric spaces, Amer. J. Math. 93 (1971), 753-809. MR 47:399
  • [18] I. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, Clarendon Press, Oxford University Press, New York, 1995. MR 96h:05207
  • [19] W. McGovern, Rings of regular functions on nilpotent orbits and their covers, Invent. Math. 97 (1989), 209-217. MR 90g:22022
  • [20] W. McGovern, Rings of regular functions on nilpotent orbits II: model algebras and orbits, Comm. Alg. 22(3) (1994), 765-772. MR 95b:22035
  • [21] W. Schmid, Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen, Invent. Math. 9 (1969-1970), 61-80. MR 41:3806
  • [22] J. Schwartz, The determination of the admissible orbits in the real classical groups, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1987.
  • [23] J. Sekiguchi, Remarks on nilpotent orbits of a symmetric pair, J. Math. Soc. Japan 39 (1987), 127-138. MR 88g:53053
  • [24] P. Torasso, Quantification géométrique, opérateurs d'entrelacement et représentations unitaires de $\widetilde{\mathrm{SL}}_{3}({\mathbf{R}})$, Acta Math. 150 (1983), 153-242. MR 86b:22026
  • [25] D. Vogan, The algebraic structure of the representations of semisimple Lie groups I, Ann. of Math. 109 (1979), 1-60. MR 81j:22020
  • [26] D. Vogan, Representations of Real Reductive Lie Groups, Birkhäuser, Boston-Basel-Stuttgart, 1981. MR 83c:22022
  • [27] D. Vogan, Unitary Representations of Reductive Lie Groups, Annals of Mathematics Studies, Princeton University Press, Princeton, NJ, 1987. MR 96c:22023
  • [28] D. Vogan, Dixmier algebras, sheets, and representation theory, Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (A. Connes, M. Duflo, A. Joseph, and R. Rentschler, eds.), Birkhäuser, Boston, MA, 1990, pp. 333-395. MR 92h:22034
  • [29] D. Vogan, Associated varieties and unipotent representations, Harmonic Analysis on Reductive Groups (W. Barker and P. Sally, eds.), Birkhäuser, Boston-Basel-Berlin, 1991. MR 93k:22012
  • [30] M. Vergne, Instantons et correspondance de Kostant-Sekiguchi, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), 901-906. MR 96c:22026

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Additional Information

Jeffrey Adams
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742

Jing-Song Huang
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

David A. Vogan Jr.
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Received by editor(s): March 24, 1998
Received by editor(s) in revised form: April 17, 1998
Published electronically: June 2, 1998
Additional Notes: The first author was supported in part by NSF grant DMS-94-01074. The second author was partially supported by RGC-CERG grant number HKUST588/94P and HKUST713/96P. The third author was supported in part by NSF grant DMS-94-02994.
Article copyright: © Copyright 1998 American Mathematical Society

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