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On generalized Whittaker functions on Siegel’s upper half space of degree 2

Published online by Cambridge University Press:  22 January 2016

S. Niwa
S. Niwa*
Affiliation:
Nagoya City College of Child Education, Owariasahi, 488, Japan
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In [5], H. Maass showed that the dimension of a space of generalized Whittaker functions satisfying certain system of differential equations on Siegel’s upper half space H2 of degree 2 is three. First of all, we shall investigate the structure of a space of generalized Whittaker functions which are eigen functions for the algebra of invariant differential operators on H2. The theory of generalized Whittaker functions is discussed in Yamashita [12], [13], [14], [15] with full generality. But, we will get an outlook of the space of generalized Whittaker functions by using elementary calculus instead of representation theory of Lie groups.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 121 , March 1991 , pp. 171 - 184
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1991

References

[1] Eichler, M., Quadratische Formen und orthogonale Groupen, Springer.Google Scholar
[2] Friedberg, S., Differential operators and theta series, Trans. Amer. Math. Soc, 287 (1985), 569–589.CrossRefGoogle Scholar
[3] Kaufhold, G., Dirichletsche Reihe mit Funktionalgleichung in der Theorie der Modulfunktion 2. Grades, Math. Ann., 137 (1959), 454–476.Google Scholar
[4] Koecher, M., Über Thetareihen indefiniter quadratischer Formen, Math. Nachr., 9 (1953), 51–85.Google Scholar
[5] Maass, H., Die Differentialgleichungen in der Théorie der Siegelschen Modulfunktionen, Math. Ann., Bd., 126 (1953), 44–68.Google Scholar
[6] Maass, H., Über eine neue Art von nichtanalitischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Functionalgleichungen, Math. Ann., 121 (1949), 141–183.CrossRefGoogle Scholar
[7] Maass, H., Dirichletsche Reihen und Modulfunktionen zweiten Grades, Acta Arith., 24 (1973), 225–238.Google Scholar
[8] Maass, H., Über die räumliche Verteilung der Punkte in Gittern mit indefiniter Metrik, Math. Ann., 138 (1959), 287–315.Google Scholar
[9] Nakajima, S., On invariant differential operators on bounded symmetric domains of type IV, Proc. Japan Acad., 58, Ser. A (1982), 235–238.Google Scholar
[10] Nakajima, S., Invariant differential operators on SO(2, q)/SO(2)×SO(q) (q≥3), Master these.Google Scholar
[11] Shimura, G., Confluent hypergeometric functions on tube domains, Math. Ann., 265 (1982), 269–302.Google Scholar
[12] Yamashita, H., On Whittaker vectors for generalized Gelfand-Graev representations of semisimple Lie groups, J. Math. Kyoto Univ., 26–2 (1986), 263–298.Google Scholar
[13] Yamashita, H., On Whittaker vectors for generalized Gelfand-Graev representations of semi-simple Lie groups, Proc. Japan Acad., 61, Ser. A (1985), 213–216.Google Scholar
[14] Yamashita, H., Finite multiplicity theorems for induced representations of semisimple Lie groups and their applications to generalized Gelfand-Graev representations, Proc. Japan Acad., 63, Ser. A (1987), 153–156.Google Scholar
[15] Yamashita, H., Multiplicity one theorems for generalized Gelfand-Graev representations of semisimple Lie groups and Whittaker models for the discrete series, Preprint.Google Scholar
[16] Yoshida, H., Siegel’s modular forms and the arithmetic of quadratic forms, Invent. Math., 60 (1980), 193–248.Google Scholar
[17] Yoshida, H., On Siegel modular forms obtained from theta series, J. reine angew. Math., 352 (1984), 184–219.Google Scholar