Glasnik Matematicki, Vol. 43, No.2 (2008), 337-362.


Tadashi Miyazaki

Department of Mathematical Sciences, University of Tokyo, Japan

Abstract.   We describe explicitly the structures of standard (g,K)-modules of SL(3,R).

2000 Mathematics Subject Classification.   22E46, 11F70.

Key words and phrases.   Semisimple Lie group, principal series representation, generalized principal series representation.

Full text (PDF) (free access)

DOI: 10.3336/gm.43.2.08


  1. D. Bump, Automorphic forms and representations, Cambridge Studies in Advanced Mathematics 55, Cambridge University Press, Cambridge, 1997.

  2. T. Fujimura, On some degenerate principal series representations of O(p,2), J. Lie Theory 11 (2001), 23-55.

  3. R. Howe, K-type structure in the principal series of GL3. I, in: Analysis on homogeneous spaces and representation theory of Lie groups, Okayama--Kyoto (1997), volume 26 of Adv. Stud. Pure Math., pages 77-98. Math. Soc. Japan, Tokyo, 2000.

  4. R. E. Howe and E.-C. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series representations, Bull. Amer. Math. Soc. (N.S.) 28 (1993), 1-74.
    MathSciNet     CrossRef

  5. H. Kraljevic, Representations of the universal convering group of the group SU(n,1), Glas. Mat. Ser. III 8(28) (1973), 23-72.

  6. S. T. Lee, Degenerate principal series representations of Sp(2n,R), Compositio Math. 103 (1996), 123-151.
    MathSciNet     Numdam

  7. S. T. Lee and H. Y. Loke, Degenerate principal series representations of Sp(p,q), Israel J. Math. 137 (2003), 355-379.
    MathSciNet     CrossRef

  8. H. Manabe, T. Ishii, and T. Oda, Principal series Whittaker functions on SL(3,R), Japan. J. Math. (N.S.) 30 (2004), 183-226.

  9. T. Miyazaki, Whittaker functions for generalized principal series representations of SL(3,R), Manuscripta Math., to appear.

  10. T. Miyazaki, The (g,K)-module structures of principal series representations of Sp(3,R), preprint.

  11. T. Oda, The standard (g,K)-modules of Sp(2,R) I, submitted.

  12. E. Thieleker, On the integrable and square-integrable representations of Spin(1,2m), Trans. Amer. Math. Soc. 230 (1977), 1-40.
    MathSciNet     CrossRef

Glasnik Matematicki Home Page