Glasnik Matematicki, Vol. 44, No.2 (2009), 349-399.

INTERTWINING OPERATORS AND COMPOSITION SERIES OF GENERALIZED AND DEGENERATE PRINCIPAL SERIES FOR Sp(4,R)

Goran Muić

Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia
e-mail: gmuic@math.hr

Abstract.   In this paper we study analytic properties of intertwining operators and apply them to the determination of the composition series of degenerate and generalized principal series for Sp(4,R). We expect that that some of the methods developed here will extend to higher rank groups in order to extend the formalism of degenerate Eisenstein series given in our previous papers. In higher rank cases we expect to be more dependent on the algebraic theory of representation theory of the real reductive groups as developed by Vogan.

2000 Mathematics Subject Classification.   22E50.

Key words and phrases.   Intertwining operators, composition series, generalized principal series.

Full text (PDF) (free access)

DOI: 10.3336/gm.44.2.08

References:

  1. A. M. Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d'un groupe réductif p-adique, Trans. Amer. Math. Soc. 347 (1995), 2179-2189 (and Erratum, Trans. Amer. Math. Soc 348 (1996), 4687-4690.)
    MathSciNet     CrossRef

  2. J. Arthur, Intertwining operators and residues. I. Weighted characters, J. Funct. Anal. 84 (1989), 19-84.
    MathSciNet     CrossRef

  3. N. Bourbaki, Groupe et Algébres de Lie, Chap. 6, Masson, Paris, 1981.
    MathSciNet

  4. D. Goldberg, Reducibility of induced representations for Sp(2n) and SO(n), Amer. J. Math. 116 (1994), 1101-1151.
    MathSciNet     CrossRef

  5. R. Herb, Elliptic representations for Sp(2n) and SO(n), Pacific J. Math. 161 (1993), 347-358.
    MathSciNet

  6. C. Jantzen, H. Kim, Parametrization of the image of normalized intertwining operators, Pacific J. Math. 199 (2001), 367-415.
    MathSciNet

  7. H. Kim, The residual spectrum of Sp4, Compositio Math. 99 (1995), 129-151.
    MathSciNet

  8. H. Kim, The residual spectrum of G2, Canad. J. Math. 48 (1996), 1245-1272.
    MathSciNet

  9. H. Kim, Residual spectrum of odd orthogonal groups, Internat. Math. Res. Notices 17 (2001), 873-906.
    MathSciNet     CrossRef

  10. A. W. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series 36, Princeton University Press, Princeton, NJ, 1986.
    MathSciNet

  11. A. W. Knapp, D. Vogan, Cohomological induction and unitary representations, Princeton Mathematical Series 45, Princeton University Press, Princeton, NJ, 1995.
    MathSciNet

  12. D. Milicic, Algebraic D-modules and representation theory of semisimple Lie groups, Contemp. Math. 154, Amer. Math. Soc., Providence, RI, 1993, 133-168.
    MathSciNet

  13. C. Moeglin, Orbites unipotentes et spectre discret non ramifie: le cas des groupes classiques dèployès, Compositio Math. 77 (1991), 1-54.
    MathSciNet

  14. C. Moeglin, Reprèsentations unipotentes et formes automorphes de carrè intègrable, Forum Math. 6 (1994), 651-744.
    MathSciNet

  15. C. Moeglin, J. L. Waldspurger, Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4) 22 (1989), 605-674.
    MathSciNet

  16. C. Moeglin, J. L. Waldspurger, Spectral decomposition and Eisenstein series, Cambridge Tracts in Mathematics 113, Cambridge University Press, Cambridge, 1995.
    MathSciNet

  17. G. Muic, The unitary dual of p-adic G2, Duke Math. J. 90 (1997), 465-493.
    MathSciNet     CrossRef

  18. G. Muic, Reducibility of generalized principal series, Canad. J. Math. 57 (2005), 616-647.
    MathSciNet

  19. G. Muic, Composition series of generalized principal series; the case of strongly positive discrete series, Israel J. Math 140 (2004), 157-202.
    MathSciNet     CrossRef

  20. G. Muic, Reducibility of standard representations, Pacific J. Math. 222 (2005), 133-168.
    MathSciNet     CrossRef

  21. G. Muic, On certain classes of unitary representations for split classical groups, Canad. J. Math. 59 (2007), 148-185.
    MathSciNet

  22. G. Muic, Some applications of degenerate Eisenstein series on Sp2n, J. Ramanujan Math. Soc. 23, No.3 (2008) 222-257.
    MathSciNet

  23. G. Muic, M.Tadic, Unramified Unitary Duals for Split Classical p-adic Groups; The topology and Isolated Representations, Shahidi's volume (to appear).

  24. P. J. Sally, M. Tadic, Induced representations and classifications for G Sp(2,F) and Sp(2,F), Mém. Soc. Math. France (N.S.) 52 (1993), 75-133.
    MathSciNet

  25. F. Shahidi, Local coefficients as Artin factors for real groups, Duke Math. J. 52 (1985), 973-1007.
    MathSciNet     CrossRef

  26. B. Speh, Unitary representations of GL(n,R) with nontrivial (g,K)-cohomology, Invent. Math. 71 (1983), 443-465.
    MathSciNet     CrossRef

  27. B. Speh, D. A. Vogan, Reducibility of generalized principal series representations, Acta. Math. 145 (1980), 227-299.
    MathSciNet     CrossRef

  28. M. Tadic, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29-91.
    MathSciNet     CrossRef

  29. D. A. Vogan, Gelfand-Kirillov dimension for Harish-Chandra modules, Invent. Math. 48 (1978), 75-98.
    MathSciNet     CrossRef

  30. D. A. Vogan, Representations of real reductive Lie groups, Progress in Mathematics 15, Birkhäuser, Boston, 1981.
    MathSciNet

  31. D. A. Vogan, The unitary dual of G2, Invent. Math. 116 (1994), 677-791.
    MathSciNet     CrossRef

  32. T. Watanabe, Residual spectrum representations of Sp4, Nagoya Math J. 127 (1992), 15-47.
    MathSciNet

  33. G. Zuckerman, Tensor products of finite and infinite dimensional representations of semisimple Lie groups, Ann. Math. (2) 106 (1977), 295-308.
    MathSciNet     CrossRef

  34. A. V. Zelevinsky, Induced representations of reductive p-adic groups. On irreducible representations of $ GL(n), Ann. Sci. École Norm. Sup. (4) 13 (1980), 165-210.
    MathSciNet

Glasnik Matematicki Home Page