Abstract
For holomorphic modular forms on tube domains, there are two types of known Fourier expansions, i.e. the classical Fourier expansion and the Fourier-Jacobi expansion. Either of them is along a maximal parabolic subgroup. In this paper, we discuss Fourier expansion of holomorphic modular forms on tube domains of classical type along the minimal parabolic subgroup. We also relate our Fourier expansion to the two known ones in terms of Fourier coefficients and theta series appearing in these expansions.
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References
W. Baily Jr. andA. Borel, Compactification of arithmetic quotients of bounded symmetric domains.Ann. Math. 84 (1966), 442–528.
H. Braun, Hermitian modular functions.Ann. Math. 50 (1949), 827–855.
L. Corwin andF. P. Greenleaf, Intertwining operators for representations induced from uniform subgroups.Ada Math. 136 (1976), 275–301.
-,Representations of nilpotent Lie groups and their applications Part 1: Basic theory and examples. Cambridge Studies in Advanced Mathematics18, Cambridge University Press, 1990.
M. Eichler and D. Zagier,The theory of Jacobi forms. Progress in Mathematics 55, Birkhaeuser Verlag, 1985.
J. Faraut andA. Koranyi, Analysis on symmetric cones. Oxford Mathematical Monographs, Oxford Science Publications, 1994.
A. Knapp,Representation theory of semisimple groups, an overview based on examples. Princeton Mathematical Series36, Princeton University Press, 1986.
A. Krieg,Modular forms on half-spaces of quaternions. Lecture Notes in Math.1143, Springer-Verlag, 1985.
V. A. Gritsenko, Fourier-Jacobi functions of in.J. Soviet Math. 53 (1991), 243–252.
J. Igusa,Thetafunctions. Springer Verlag, 1972.
A. Murase andT. Sugano, Fourier-Jacobi expansion of Eisenstein series on unitary groups of degree 3.J. Math. Sci. Univ. Tokyo 9 (2002), 347–404.
H. Narita, Fourier expansion of holomorphic Siegel modular forms of genusn along the minimal parabolic subgroup.J. Math. Sci. Univ. Tokyo 10 (2003), 311–353. Erratum in:J. Math. Sci. Univ. Tokyo 10 (2003), 579-580.
I. I. Piatetskii-Shapiro,Automorphic functions and the geometry of classical domains. Mathematics and its Applications 8, Gordon and Breach Science Publishers, 1969.
I. Satake,Algebraic structure of symmetric domains. Publication of the Mathematical Society of Japan14, Iwanami Shoten Publishers and Princeton University Press, 1980.
C. L. Siegel, Über die analytische Theorie der Quadratischen Formen.Ann. Math. 36 (1935), 527–606.
T. Shintani, On automorphic forms on unitary groups of order 3. Unpublished manuscript.
N. Wallach,Real reductive groups I. Academic Press, 1988.
C. Ziegler, Jacobi forms of higher degree.Abh. Math. Sem. Univ. Hamburg 59 (1989), 191–224.
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R. Berndt
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Narita, H. Fourier expansion of holomorphic modular forms on classical lie groups of tube type along the minimal parabolic subgroup. Abh.Math.Semin.Univ.Hambg. 74, 253–279 (2004). https://doi.org/10.1007/BF02941540
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DOI: https://doi.org/10.1007/BF02941540