Journal of the Mathematical Society of Japan

The $(\mathfrak{g},K)$-module structures of principal series of $SU(2,2)$

Gombodorj BAYARMAGNAI

Full-text: Open access

Abstract

We explicitly describe the $(\mathfrak{g}_{\mbi{C}},K)$-module structures of the principal series representations of $SU(2,2)$ associated with a maximal parabolic subgroup.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 3 (2009), 661-686.

Dates
First available in Project Euclid: 30 July 2009

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1248961475

Digital Object Identifier
doi:10.2969/jmsj/06130661

Zentralblatt MATH identifier
05603959

Mathematical Reviews number (MathSciNet)
MR2552912

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 22E46: Semisimple Lie groups and their representations

Keywords
principal series representation Harish-Chandra module Clebsch-Gordan coefficients

Citation

BAYARMAGNAI, Gombodorj. The ( g , K ) -module structures of principal series of SU ( 2 , 2 ) . J. Math. Soc. Japan 61 (2009), no. 3, 661--686. doi:10.2969/jmsj/06130661. http://projecteuclid.org/euclid.jmsj/1248961475.


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References

  • G. Bayarmagnai, Explicit evaluation of certain Jacquet integrals on $SU(2,2)$, preprint, 2008.
  • D. A. Vogan, Representations of real reductive Lie groups, Progr. Math., 15, Birkhäuser, Boston, Basel, Stuttgart, 1981.
  • T. Hayata, Differential equations for principal series Whittaker functions on $SU(2,2)$, Indag. Math. (N.S.), 8 (1997), 493–528.
  • R. Howe, $K$-type structure in the principal series of $GL(3)$. I, : Analysis on homogeneous spaces and representation theory of Lie groups, Okayama–Kyoto (1997), Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000, pp. 77–98.
  • T. Ishii, On principal series Whittaker functions on $Sp(2,\mbi{R})$, J. Func. Anal., 225 (2005), 1–32.
  • A. U. Klimyk and B. Gruber, Structure and matrix elements of the degenerate series representations of $U(p+q)$ and $U(p,\,q)$ in a $U(p)\times U(q)$ basis, J. Math. Phys.,23 (1982), 1399–1408.
  • A. U. Klimyk and B. Gruber, Infinitesimal operators and structure of the most degenerate representationas of the groups $Sp(p+q)$ and $Sp(p,\,q)$ in a $U(p)\times U(q)$ basis, J. Math. Phys., 25 (1984), 743–750.
  • S.-T. Lee and H. Y. Loke, Degenerate principal series representations of $U(p,q)$ and $\textit{0}(p,q)$, Compositio Math., 132 (2002), 311–348.
  • T. Miyazaki, The $(\mathfrak{g},K)$-module structures of principal series representations $Sp(3,\mbi{R})$, 2007, arXiv: 0706.0066v1.
  • T. Miyazaki and T. Oda, Principal series Whittaker functions on $Sp(2,\mbi{R})$, Explicit formulae of differential equations, Proceeding of the 1993 Workshop, Automorphic forms and related topics, Pyungsan Inst. Math. Sci., Seoul, pp. 59–92.
  • V. F. Molchanov, Representations of pseudo-orthogonal group associated with a cone, Math. USSR Sbornik, 10 (1970), 333–347.
  • T. Oda, Standard $(\mathfrak{g},K)$-modules of $Sp(2,\mbi{R})$, Preprint Series, The University of Tokyo, UTMS 2007–3.
  • E. Thieleker, On the quasi-simple irreducible representations of the Lorentz groups, Trans. Amer, Math. Soc., 179 (1973), 465–505.
  • N. R. Wallach, Real reductive groups, I, Pure and Applied Mathematics, 132, Academic Press Inc., Boston, MA, 1988.
  • H. Yamashita, Embedding of discrete series into induced representations of semisimple Lie groups, I, General theory and the case of $SU(2,2)$, Japan J. Math., 16 (1990), 31–95.
  • H. Yamashita, Embedding of discrete series into induced representations of semi-simple Lie groups, II, Generalized Whittaker models for $SU(2,2)$, J. Math. Kyoto Univ., 31 (1991), 543–571.