Secondary Whittaker functions for \(P_J\)-principal series representations of \(\text{Sp}(3,\mathbb R)\). (English) Zbl 1109.22009

For \(\text{Sp}(3,\mathbb R)\) the authors construct explicit formulas for the power series solutions at the regular singularity of the holonomic system coming from the principal series representation induced from the second parabolic subgroup \(P_J\). These secondary Whittaker functions are confluent hypergeometric series in three variables which are not simple \(\Gamma\)-series.


22E46 Semisimple Lie groups and their representations
11F70 Representation-theoretic methods; automorphic representations over local and global fields
33C80 Connections of hypergeometric functions with groups and algebras, and related topics
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[1] 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. · Zbl 0668.17006
[2] Harish-Chandra, Spherical functions on a semi-simple Lie group I, II, Amer. J. Math. 80 (1958), 241-310, 553-613. · Zbl 0093.12801
[3] 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). · Zbl 1097.22006
[4] 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).
[5] T. Ishii, On principal series Whittaker functions on \(Sp(2,\mathbf{R})\). (Preprint). · Zbl 1078.11031
[6] B. Kostant, On Whittaker vectors and representation theory, Invent. Math. 48 (1978), no. 2, 101-184. · Zbl 0405.22013
[7] H. Maass, Siegel’s modular forms and Dirichlet series , Lecture Notes in Math., 216, Springer, Berlin, 1971. · Zbl 0224.10028
[8] H. Matumoto, Whittaker vectors and the Goodman-Wallach operators, Acta Math. 161 (1988), no. 3-4, 183-241. · Zbl 0723.22019
[9] R. Miatello and N. R. Wallach, Automorphic forms constructed from Whittaker vectors, J. Funct. Anal. 86 (1989), no. 2, 411-487. · Zbl 0692.10029
[10] 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. · Zbl 1044.11033
[11] 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. · Zbl 0559.35007
[12] L. J. Slater, Generalized hypergeometric functions , Cambridge Univ. Press, Cambridge, 1966. · Zbl 0135.28101
[13] 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. · Zbl 0786.11027
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