## Journal of the Mathematical Society of Japan

### Generalized Whittaker functions on GSp(2,R) associated with indefinite quadratic forms

Tomonori MORIYAMA

#### Abstract

We study the generalized Whittaker models for G = GSp(2,R) associated with indefinite binary quadratic forms when they arise from two standard representations of G: (i) a generalized principal series representation induced from the non-Siegel maximal parabolic subgroup and (ii) a (limit of) large discrete series representation. We prove the uniqueness of such models with moderate growth property. Moreover we express the values of the corresponding generalized Whittaker functions on a one-parameter subgroup of G in terms of the Meijer G-functions.

#### Article information

Source
J. Math. Soc. Japan, Volume 63, Number 4 (2011), 1203-1262.

Dates
First available in Project Euclid: 27 October 2011

https://projecteuclid.org/euclid.jmsj/1319721140

Digital Object Identifier
doi:10.2969/jmsj/06341203

Mathematical Reviews number (MathSciNet)
MR2855812

Zentralblatt MATH identifier
1268.22018

#### Citation

MORIYAMA, Tomonori. Generalized Whittaker functions on GSp (2, R ) associated with indefinite quadratic forms. J. Math. Soc. Japan 63 (2011), no. 4, 1203--1262. doi:10.2969/jmsj/06341203. https://projecteuclid.org/euclid.jmsj/1319721140

#### References

• A. N. Andrianov, Dirichlet series with Euler product in the theory of Siegel modular forms of genus two, Trudy Mat. Inst. Steklov., 112 (1971), 73–94.
• A. N. Andrianov and V. Kalinin, Analytic properties of standard zeta-functions of Siegel modular forms, Mat. Sb. (N.S.), 106 (1978), 323–339.
• E. W. Barnes, The Asymptotic Expansion of Integral Functions Defined by Generalized Hypergeometric Series, Proc. London Math. Soc., 5 (1907), 59–116.
• A. Borel, Automorphic forms on $SL \sb 2(\R)$, Cambridge Tracts in Math., 130, Cambridge University Press, Cambridge, 1997.
• D. Bump, Automorphic forms and representations, Cambridge University Press, 1997.
• D. Bump, S. Friedberg and M. Furusawa, Explicit formulas for the Waldspurger and Bessel models, Israel J. Math., 102 (1997), 125–177.
• C. Casselman and J. A. Shalika, The unramified principal series of $p$-adic groups, II, The Whittaker function, Compositio Math., 41 (1980), 207–231.
• A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher transcendental functions, I, Based on notes left by Harry Bateman, McGraw-Hill Book Company Inc., New York-Toronto-London, 1953.
• M. Furusawa, On $L$-functions for $GSp(4)\times GL(2)$ and their special values, J. Reine Angew. Math., 438 (1993), 187–218.
• B. H. Gross and D. Prasad, On irreducible representations of $SO_{2n+1}\times SO_{2m}$, Canad. J. Math., 46 (1994), 930–950.
• Harish-Chandra, Discrete series for semisimple Lie groups, II, Acta Math., 166 (1966), 1–111.
• M. Harris, Occult period invariants and critical values of the degree four $L$-function of $GSp(4)$, In: Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins Univ. Press, Baltimore, MD, 2004, pp.,331–354.
• T. Ishii, Siegel-Whittaker functions on $Sp(2,\R)$ for principal series representations, J. Math. Sci. Univ. Tokyo, 9 (2002), 303–346.
• T. Ishii and T. Moriyama, Spinor $L$-functions for generic cusp forms on $GSp(2)$ belonging to principal series representations, Trans. Amer. Math. Soc., 360 (2008), 5683–5709.
• H. Jacquet and J. A. Shalika, On Euler products and the classification of automorphic forms, II, Amer. J. Math., 103 (1981), 777–815.
• A. W. Knapp, Representation theory of semisimple groups, An overview based on examples, Princeton Mathematical Series, 36, Princeton University Press, Princeton NJ, 1986.
• S. Kudla, S. Rallis and D. Soudry, On the degree 5 $L$-function for $Sp(2)$, Invent. Math., 107 (1992), 483–541.
• F. Lemma, Régulateurs supérieurs, périodes et valeurs spéciales de la fonction $L$ de degré 4 de $GSp(4)$, (French, English, French summary) [Higher regulators, periods and special values of the degree-4 $L$-function of $GSp(4)$], C. R. Math. Acad. Sci. Paris, 346 (2008), 1023–1028.
• J. S. Li, Nonexistence of singular cusp forms, Compositio Math., 83 (1992), 43–51.
• H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math., 216, Springer-Verlag, 1976.
• C. S. Meijer, On the $G$-functions I-II, Indag. Math., 8 (1946), 124–134, 213–225.
• T. Miyazaki, The generalized Whittaker functions for $Sp(2,\R)$ and the gamma factor of the Andrianov $L$-functions, J. Math. Sci. Univ. Tokyo., 7 (2000), 241–295.
• T. Miyazaki, Nilpotent orbits and Whittaker functions for derived functor modules of $Sp(2,\R)$, Canad. J. Math., 54 (2002), 769–794.
• T. Moriyama, Entireness of the spinor $L$-functions for certain generic cusp forms on $GSp(2)$, Amer. J. Math., 126 (2004), 899–920.
• S. Niwa, On generalized Whittaker functions on Siegel's upper half space of degree 2, Nagoya Math. J., 121 (1991), 171–184.
• M. E. Novodvorsky, Automorphic $L$-functions for symplectic group $GSp(4)$, In: Automorphic Forms, Representations and $L$-Functions, (Eds. A. Borel and W. Classelman), Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence RI, 1979, pp.,87–95.
• M. E. Novodvorsky and I. I. Piatetski-Shapiro, Generalized Bessel models for a symplectic group of rank 2, Math. USSR-Sb., 19 (1973), 243–255.
• I. I. Piatetski-Shapiro, $L$-functions for $GSp \sb 4$, Olga Taussky-Todd: in memoriam, Pacific J. Math., Special Issue (1997), 259–275.
• I. I. Piatetski-Shapiro and S. Rallis, A new way to get an Euler product, J. Reine Angew. Math., 392 (1988), 110–124.
• A. Pitale and R. Schmidt, Bessel models for lowest weight representations of $GSp(4,R)$, Int. Math. Res. Not., (2009), 1159–1212.
• D. Prasad and R. Takloo-Bighash, Bessel models for $GSp(4)$, J. Reine Angew. Math., 655 (2011), 189–243.
• F. Rodier, Modèles pour les représentations du groupe $Sp(4,k)$ où $k$ est un corps local, [Models for the representations of $Sp(4,k)$ where $k$ is a local field], Reductive groups and automorphic forms, I (Paris, 1976/1977), Publ. Math. Univ. Paris VII, 1, Univ. Paris VII, Paris, 1978, pp.,123–145.
• T. Sugano, On holomorphic cusp forms on quaternion unitary groups of degree 2, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1985), 521–568.
• R. Takloo-Bighash, Spinor $L$-functions, theta correspondence, and Bessel coefficients, with an appendix by Philippe Michel, Forum Math., 19 (2007), 487–554.
• N. Wallach, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, In: Lie group representations, I, Lecture Notes in Math., 1024, Springer-Verlag, 1983, pp.,287–369.
• W. Wasow, Asymptotic expansions for ordinary differential equations, Pure and Appl. Math., XIV, Interscience Publishers John Wiley & Sons, Inc., New York-London-Sydney, 1965.
• H. Yamashita, Multiplicity one theorems for generalized Gel'fand-Graev representations of semisimple Lie groups and Whittaker models for the discrete series, In: Representations of Lie Groups, Kyoto, Hiroshima, 1986, (eds. K. Okamoto and T. Oshima), Adv. Stud. Pure Math., 14, Academic Press, Boston, MA, 1988, pp.,31–121.