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Degenerate principal series representations for quaternionic unitary groups

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Abstract

We give a complete description of all points of reducibility and the composition series of the degenerate principal series representations for quaternionic unitary groups which are induced from a character of the maximal parabolic subgroup with abelian unipotent radical. The case of even orthogonal groups is also included.

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References

  1. J. Alperin, Diagrams for modules, Journal of Pure and Applied Algebra 16 (1980), 111–119.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Ban and C. Jantzen, The Langlands classification for non-connected p-adic groups, Israel Journal of Mathematics 126 (2001), 239–261.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Ban and C. Jantzen, Degenerate principal series for even-orthogonal groups, Representation Theory 7 (2003), 440–480.

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, preprint.

  5. P. Feit, Poles and residues of Eisenstein series for symplectic and unitary groups, Memoirs of the American Mathematical Society 346 (1986), 1–89.

    MathSciNet  Google Scholar 

  6. W. T. Gan and W. Tantono, The local Langlands conjecture for GSp(4) II: The case of inner forms, preprint.

  7. R. Gustafson, The degenerate principal series for Sp(2n), Memoirs of the American Mathematical Society 248 (1981), 1–81.

    Google Scholar 

  8. M. Harris, S. Kudla and W. J. Sweet Jr., Theta dichotomy for unitary groups, Journal of the American Mathematical Society 9 (1996), 941–1004.

    Article  MathSciNet  MATH  Google Scholar 

  9. R. Howe, Transcending classical invariant theory, Journal of the American Mathematical Society 2 (1989), 535–552.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Ichino, A regularized Siegel-Weil formula for unitary groups, Mathematische Zeitschrift 247 (2004), 241–277.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Igusa, Some results on p-adic complex powers, American Journal of Mathematics 106 (1984), 1013–1032.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Igusa, On functional equations of complex powers, Inventiones Mathematicae 85 (1986), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Jantzen, Degenerate principal series for symplectic groups, Memoirs of the American Mathematical Society 488 (1986), 1–111.

    Google Scholar 

  14. C. Jantzen, Degenerate principal series for symplectic and odd-orthogonal groups, Memoirs of the American Mathematical Society 590 (1996), 1–100.

    MathSciNet  Google Scholar 

  15. K. Johnson, Degenerate principal series and compact groups, Mathematische Annalen 287 (1990), 703–718.

    Google Scholar 

  16. K. Johnson, Degenerate principal series for tube type domains, Contemporary Mathematics 138, American Mathematical Society, Providence, RI, 1992, pp. 175–187.

    Google Scholar 

  17. M. Karel, Functional equations of Whittaker functions on p-adic groups, American Journal of Mathematics 101 (1979), 1303–1325.

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel Journal of Mathematics 84 (1994), 361–401.

    Article  MathSciNet  Google Scholar 

  19. S. Kudla and S. Rallis, Degenerate principal series and invariant distributions, Israel Journal of Mathematics 69 (1990), 25–45.

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Kudla and S. Rallis, Ramified degenerate principal series representations for Sp(n), Israel Journal of Mathematics 78 (1992), 209–256.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Kudla and S. Rallis, A regularized Siegel-Weil formula: the first term identity, Annals of Mathematics. Second Series 140 (1994), 1–80.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Kudla and S. Rallis, On first occurrence in the local theta correspondence, in Automorphic Representations, L-Functions and Applications: Progress and Prospects, de Gruyter, Berlin, 2005, pp. 273–308.

    Chapter  Google Scholar 

  23. S. Kudla and W. J. Sweet Jr., Degenerate principal series representations for U(n, n), Israel Journal of Mathematics 98 (1997), 253–306.

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Lapid and S. Rallis, On the local factors of representations of classical groups, in Automorphic Representations, L-Functions and Applications: Progress and Prospects, de Gruyter, Berlin, 2005, pp. 309–359.

    Chapter  Google Scholar 

  25. S. T. Lee, On some degenerate principal series representations of U(n, n), Journal of Functional Analysis 126 (1994), 305–366.

    Article  MathSciNet  MATH  Google Scholar 

  26. S. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence, Transactions of the American Mathematical Society 350 (1998), 5017–5046.

    Article  MathSciNet  Google Scholar 

  27. S. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence II, Israel Journal of Mathematics 100 (1997), 29–59.

    Article  MathSciNet  MATH  Google Scholar 

  28. S. T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence III, Journal of Algebra 319 (2008), 336–359.

    Article  MathSciNet  MATH  Google Scholar 

  29. J.-S. Li, Singular unitary representations of classical groups, Inventiones Mathematicae 97 (1989), 237–255.

    Google Scholar 

  30. J.-S. Li, On the classification of irreducible low rank unitary representations of classical groups, Compositio Mathematica 71 (1989), 29–48.

    MathSciNet  MATH  Google Scholar 

  31. J.-S. Li, A. Paul, E.-C. Tan and C.-B. Zhu, The explicit duality correspondence of (Sp(p, q), O*(2n)), Journal of Functional Analysis 200 (2003), 71–100.

    Article  MathSciNet  MATH  Google Scholar 

  32. H. Y. Loke (with an appendix by S. T. Lee), Howe quotients of unitary characters and unitary lowest weight modules, Representation Theory 10 (2006), 21–47.

    Article  MathSciNet  MATH  Google Scholar 

  33. C. Moeglin, M.-F. Vignera and J.-L. Waldspurger, Correspondence de Howe sur un corps p-adique, Springer Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin, 1987.

    Google Scholar 

  34. G. I. Ol’šanskiĭ, Intertwining operators and complementary series in the class of representations of the general group of matrices over a locally compact division algebra, induced from parabolic subgroups, Math. USSR Sb. 22 (1974), no 2., 218–255.

    Google Scholar 

  35. B. Ørsted and G. Zhang, Generalized principal series representations and tube domains, Duke Mathematical Journal 78 (1995), 335–357.

    Article  MathSciNet  Google Scholar 

  36. I. Piatetski-Shapiro and S. Rallis, Rankin triple L-functions, Compositio Mathematica 64 (1987), 31–115.

    MathSciNet  MATH  Google Scholar 

  37. S. Rallis and G. Schiffmann, Distributions invariantes par le groupe orthogonal, Springer Lectures Notes in Mathematics 497, Springer-Verlag, Berlin, 1975, pp. 494–642.

    Google Scholar 

  38. S. Sahi, Unitary representations on the Shilov boundary of a symmetric tube domain, Contemporary Mathematics 145, American Mathematical Society, Providence, RI, 1993, pp. 275–286.

    Google Scholar 

  39. S. Sahi, Jordan algebras and degenerate principal series, Journal für die Reine und Angewandte Mathematik 462 (1995), 1–18.

    Article  MathSciNet  MATH  Google Scholar 

  40. W. Scharlau, Quadratic and Hermitian Forms, Springer-Verlag, Berlin, 1985.

    MATH  Google Scholar 

  41. F. Shahidi, Twisted endoscopy and reducibility of induced representations for p-adic groups, Duke Mathematical Journal 66 (1992), 1–41.

    Article  MathSciNet  MATH  Google Scholar 

  42. G. Shimura, Euler Products and Eisenstein Series, CBMS Regional Conference Series in Mathematics 93, American Mathematical Society, Providence, RI, 1997.

    MATH  Google Scholar 

  43. G. Shimura, Some exact formulas on quaternion unitary groups, Journal für die Reine und Angewandte Mathematik 509 (1999), 67–102.

    Article  MathSciNet  MATH  Google Scholar 

  44. G. Shimura, Arithmetic and Analytic Theories of Quadratic Forms and Clifford Groups, Mathematical Surveys and Monographs 109, American Mathematical Society, Providence, RI, 2004.

    MATH  Google Scholar 

  45. W. J. Sweet, Jr., A computation of the gamma matrix of a family of p-adic zeta integrals, Journal of Number Theory 55 (1995), 222–260.

    Article  MathSciNet  MATH  Google Scholar 

  46. M. Tadić, On reducibility of parabolic induction, Israel Journal of Mathematics 107 (1998), 29–91.

    Article  MathSciNet  MATH  Google Scholar 

  47. J.-L. Waldspurger, Démonstration d’une conjecture de dualité de Howe dans le cas padique, p ≠ 2, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Mathematical Conference Proceedings 2, Weizmann, Jerusalem, 1990, pp. 267–324.

  48. T. Yasuda, The residual spectrum of inner forms of Sp(2), Pacific Journal of Mathematics 232 (2007), 471–490.

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Zhang, Jordan algebras and generalized principal series representations, Mathematische Annalen 302 (1995), 773–786.

    Article  MathSciNet  MATH  Google Scholar 

  50. C.-B. Zhu, Invariant distributions of classical groups, Duke Mathematical Journal 65 (1992), 85–119.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Science, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan

    Shunsuke Yamana

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  1. Shunsuke Yamana
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Correspondence to Shunsuke Yamana.

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The author is supported by the Grant-in-Aid for JSPS Fellows.

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Yamana, S. Degenerate principal series representations for quaternionic unitary groups. Isr. J. Math. 185, 77–124 (2011). https://doi.org/10.1007/s11856-011-0102-9

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  • DOI: https://doi.org/10.1007/s11856-011-0102-9

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