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An Analogue of the Kostant–Rallis Multiplicity Theorem for θ-Group Harmonics

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Representation Theory, Number Theory, and Invariant Theory

Part of the book series: Progress in Mathematics ((PM,volume 323))

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Abstract

The main result in this paper is the generalization of the Kostant–Rallis multiplicity formula to general θ-groups (in the sense of Vinberg). The special cases of the two most interesting examples one for E6 (three qubits) and one for E8 are given explicit formulas.

To Roger Howe with admiration

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References

  1. N. Bourbaki, Groupes et algebres de Lie, Chapitre 4,5,6, Herman, Paris, 1968.

    Google Scholar 

  2. C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778–782.

    Article  MathSciNet  Google Scholar 

  3. Jiri Dodak and Victor Kac, Polar representations, J. Algebra 92 (1985), 504–524.

    Article  MathSciNet  Google Scholar 

  4. A. G. Elashvili and E. B. Vinberg, Classification of trivectors of a 9-dimensional space, Sel. Math. Sov. 7 (1988), 63–98.

    MATH  Google Scholar 

  5. Roe Goodman and Nolan Wallach, Symmetry, Representations and Invariants, GTM 255, Springer, New York, 2009.

    Book  Google Scholar 

  6. George Kempf and Linda Ness, The length of vectors in representation spaces, Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, 1978), Lecture Notes in Math. 732, Berlin, New York: Springer-Verlag, 1979 pp. 233–243.

    Google Scholar 

  7. Bertram Kostant and Stephen Rallis, Orbits and Representations Associated with Symmetric Spaces, Amer. J. Math. 93 (1971), 753–809.

    Article  MathSciNet  Google Scholar 

  8. Hanspeter Kraft and Gerald W. Schwarz, Representations with reduced null cone, Progress in Mathematics, Volume 257, Birkhäuser, Springer, New York, 2014.

    Google Scholar 

  9. D. I. Panyushev, On the orbit spaces of finite and connected linear groups, Math. USSR. Isv. 20 (1983), 97–101.

    Article  Google Scholar 

  10. Igor R. Shafarevich, Basic Algebraic Geometry II, Springer-Verlag, Berlin, 1994.

    Google Scholar 

  11. T. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274–304.

    Article  MathSciNet  Google Scholar 

  12. E. B. Vinberg, The Weyl group of a graded Lie algebra. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 3, 488–526, English translation, Math USSR–Izv. 10 (1976), 463–493.

    Article  Google Scholar 

  13. Nolan R. Wallach, Geometric invariant theory over the real and complex numbers. To Appear in Springer UTX, 2018.

    Google Scholar 

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Authors and Affiliations

  1. University of California, San Diego, CA, USA

    Nolan R. Wallach

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  1. Nolan R. Wallach
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Correspondence to Nolan R. Wallach .

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Editors and Affiliations

  1. Department of Mathematics, Ohio State University, Columbus, OH, USA

    Jim Cogdell

  2. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Ju-Lee Kim

  3. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Chen-Bo Zhu

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Wallach, N.R. (2017). An Analogue of the Kostant–Rallis Multiplicity Theorem for θ-Group Harmonics. In: Cogdell, J., Kim, JL., Zhu, CB. (eds) Representation Theory, Number Theory, and Invariant Theory. Progress in Mathematics, vol 323. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-59728-7_20

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