Proceedings of the Japan Academy, Series A, Mathematical Sciences

Secondary Whittaker functions for $P_J$-principal series representations of $Sp(3,\mathbf {R})$

Miki Hirano and Takayuki Oda

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In this paper, we give explicit formulas for the secondary Whittaker functions for $P_J$-principal series representations of $Sp(3,{\mathbf R})$, which are power series solutions of a holonomic system of rank 24.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 81, Number 6 (2005), 105-109.

First available in Project Euclid: 2 August 2005

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Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Whittaker functions Whittaker models


Hirano, Miki; Oda, Takayuki. Secondary Whittaker functions for $P_J$-principal series representations of $Sp(3,\mathbf {R})$. Proc. Japan Acad. Ser. A Math. Sci. 81 (2005), no. 6, 105--109. doi:10.3792/pjaa.81.105.

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  • I. M. Gelfand and A. Zelevinsky, Canonical basis in irreducible representations of $\mathfrak{gl}_3$ and its applications, in Group theoretical methods in physics, Vol. II (Yurmala, 1985), 127–146, VNU Sci. Press, Utrecht, 1986.
  • Harish-Chandra, Spherical functions on a semi-simple Lie group I, II, Amer. J. Math. 80 (1958), 241-310, 553–613.
  • M. Hirano, T. Ishii and T. Oda, Confluence from Siegel-Whittaker functions to Whittaker functions on $Sp(2,\textbf{R})$, Math. Proc. Cambridge Philos. Soc. (To appear).
  • M. Hirano and T. Oda, Integral switching engine for special Clebsch-Gordan coefficients for the representations of $\mathfrak{gl}_3$ with respect to Gelfand-Zelevinsky basis. (Preprint).
  • T. Ishii, On principal series Whittaker functions on $Sp(2,\mathbf{R})$. (Preprint).
  • B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101–184.
  • H. Maass, Siegel's modular forms and Dirichlet series, Lecture Notes in Math., 216, Springer, Berlin, 1971.
  • H. Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), no. 3-4, 183–241.
  • R. Miatello and N. R. Wallach, Automorphic forms constructed from Whittaker vectors, J. Funct. Anal. 86 (1989), no. 2, 411–487.
  • T. Oda and M. Tsuzuki, Automorphic Green functions associated with the secondary spherical functions, Publ. Res. Inst. Math. Sci. 39 (2003), no. 3, 451–533.
  • T. Oshima, A definition of boundary values of solutions of partial differential equations with regular singularities, Publ. Res. Inst. Math. Sci. 19 (1983), no. 3, 1203–1230.
  • L. J. Slater, Generalized hypergeometric functions, Cambridge Univ. Press, Cambridge, 1966.
  • E. Stade, $\mathrm{GL}(4,\textbf{R})$-Whittaker functions and ${}_4F_3(1)$ hypergeometric series, Trans. Amer. Math. Soc. 336 (1993), no. 1, 253–264.