Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-02T23:29:06.874Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix coefficients of the large discrete series representations of Sp(2; R) as hypergeometric series of two variables

Published online by Cambridge University Press:  11 January 2016

Takayuki Oda
Takayuki Oda*
Affiliation:
Graduate School of Mathematical Sciences, University of Tokyo, Komaba Meguro-Ku, Tokyo 153-8914Japan, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate the radial part of the matrix coefficients with minimal K-types of the large discrete series representations of Sp(2; R). They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

Keywords

22E3022E4543A9033C65
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 208: Memorial Volume for Professor Hiroshi Saito , December 2012 , pp. 201 - 263
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

References

[1] Debiard, A. and Gaveau, B., Représentation intégrale de certaines séries de fonctions sphériques d’un système de racines BC, J. Funct. Anal. 96 (1991), 256–296.Google Scholar
[2] Debiard, A. and Gaveau, B., Integral formulas for the spherical polynomials of a root system of type BC2 , J. Funct. Anal. 119 (1994), 401–454.Google Scholar
[3] Hayata, T., Koseki, H., and Oda, T., Matrix coefficients of the middle discrete series of SU(2, 2), J. Funct. Anal. 185 (2001), 297–341.CrossRefGoogle Scholar
[4] Hayata, T., Koseki, H., and Oda, T., Matrix coefficients of the middle discrete series of SU(2, 2), II: The explicit asymptotic expansion, J. Funct. Anal. 259 (2010), 301–307.Google Scholar
[5] Heckman, G. J., Root systems and hypergeometric functions, II, Compos. Math. 64 (1987), 353–373.Google Scholar
[6] Heckman, G. J., Lectures on hypergeometric and spherical functions, preprint, 1992.Google Scholar
[7] Heckman, G. J. and Opdam, E. M., Root systems and hypergeometric functions, I, Compos. Math. 64 (1987), 329–352.Google Scholar
[8] Hirai, T., “Explicit form of the characters of discrete series representations of semisimple Lie groups” in Harmonic Analysis on Homogeneous Spaces (Williamstown, Mass., 1972), Proc. Sympos. Pure Math. 26, Amer. Math. Soc., Providence, 1973, 281–287.Google Scholar
[9] Iida, M., Spherical functions of the principal series representations of Sp(2, R) as hypergeometric functions of C2-type, Publ. Res. Inst. Math. Sci. 32 (1996), 689–727.Google Scholar
[10] Miyazaki, T. and Oda, T., Principal series Whittaker functions on Sp(2,R) II, Tohoku Math. J. (2) 50 (1998), 243–260; Correction, Tohoku Math. J. (2) 54 (2002), 161–162.CrossRefGoogle Scholar
[11] Oda, T., An explicit integral representation of Whittaker functions on Sp(2; R) for the large discrete series representations, Tohoku Math. J. (2) 46 (1994), 261–279.Google Scholar
[12] Oda, T., “Matrix coefficients of the large discrete series representations of Sp(2,R) as hypergeometric functions of two variables, I” in Automorphic Forms on Sp(2,R) and SU(2,2), I (Kyoto, 1995), RIMS Kôkyûroku 909, Kyoto, 1995, 90–101.Google Scholar
[13] Oda, T., “Matrix coefficients of the large discrete series representations of Sp(2,R) as hypergeometric functions of two variables, II” in Automorphic Forms on Sp(2,R) and SU(2,2), II (Kyoto, 1998), RIMS Kôkyûroku 1094, Kyoto, 1999, 60–82.Google Scholar
[14] Opdam, E. M., Root systems and hypergeometric functions, III, Compos. Math. 67 (1988), 21–49.Google Scholar
[15] Opdam, E. M., Root systems and hypergeometric functions, IV, Compos. Math. 67 (1988), 191–209.Google Scholar
[16] Schmid, W., On the realization of the discrete series of a semisimple Lie group, Rice Univ. Studies 56 (1970), 99–108.Google Scholar
[17] Takayama, N., Propagation of singularities of solutions of the Euler-Darboux equation and a global structure of the space of holonomic solutions, II, Funkcial. Ekvac. 36 (1993), 187–234.Google Scholar