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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On Siegel modular forms of
half-integral weights and Jacobi forms

Author: Koichi Takase
Journal: Trans. Amer. Math. Soc. 351 (1999), 735-780
MSC (1991): Primary 11F37, 11F27; Secondary 11F70
MathSciNet review: 1466958
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Abstract: We will establish a bijective correspondence between the space of the cuspidal Jacobi forms and the space of the half-integral weight Siegel cusp forms which is compatible with the action of the Hecke operators. This correspondence is based on a bijective correspondence between the irreducible unitary representations of a two-fold covering group of a symplectic group and a Jacobi group (that is, a semidirect product of a symplectic group and a Heisenberg group). The classical results due to Eichler-Zagier and Ibukiyama will be reconsidered from our representation theoretic point of view.

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  • [E-Z] M.Eichler and D.Zagier, Theory of Jacobi Forms, Progress in Math. 55, 1985. MR 86j:11043
  • [God] Godement,R., Notes on Jacquet-Langlands' theory, The Institute for Advanced Study, 1970.
  • [H-S] Hecht,H. and Schmid,W., On Integrable Representations of a Semisimple Lie Group, Math.Ann. 220 (1976), 147-149. MR 53:3203
  • [Ibu] Ibukiyama,T., On Jacobi forms and Siegel modular forms of half integral weights, Comment. Math. Univ. St. Pauli 41 (1992), 109-124. MR 94a:11069
  • [Igu] Igusa,J.-I., Harmonic analysis and theta-functions, Acta Math. 120 (1968), 187-222. MR 43:2428
  • [Knp] Knapp,A.W., Representation Theory of Semisimple Groups, Princeton Univ. Press, 1986. MR 87j:22022
  • [K-V] Kashiwara,M. and Vergne,M., On the Segal-Shale-Weil Representations and Harmonic Polynomials, Inv.Math. 44 (1978), 1-47. MR 57:3311
  • [Kug] Kuga,M., Fibre varieties over a symmetric space whose fibres are abelian varieties., Algebraic Groups and Discontinuous Subgroups P. S. P. M. vol. 9 (Amer. Math. Soc.) (1966), 338-346. MR 34:5990
  • [Mum] Mumford,D., Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups P.S.P.M. vol.9 (Amer.Math.Soc.) (1966), 347-351. MR 34:5828
  • [Mur] Murase,A., $L$-functions attached to Jacobi forms of degree $n$, Part I. The basic identity, J. reine angew. Math. 401 (1989), 122-156. MR 90k:11064
  • [Sat1] Satake,I., Factors of Automorphy and Fock Representations, Advances in Math. 7 (1971), 83-110. MR 50:541
  • [Sat2] Satake,I., Unitary representations of semi-direct product of Lie groups of $\bar {\partial }$-cohomology, Math. Ann. 190 (1971), 177-202. MR 45:5274
  • [Sat3] Satake,I., Theory of spherical functions on reductive algebraic groups over $p$-adic fields, Publ. Math., I.H.E.S. 18 (1963), 5-69. MR 33:4059
  • [Sat4] Satake,I., Holomorphic imbedding of symmetric domains into Siegel space, Amer. J. Math. 87 (1965), 425-461. MR 33:4326
  • [Sat5] Satake,I., Algebraic Structures of Symmetric Domains, Iwanami Shoten and Princeton Univ. Press, 1980. MR 82i:32003
  • [Shn] Shintani,T., unpublished note.
  • [Tak1] Takase,K., A note on automorphic forms, J.reine angew.Math. 409 (1990), 138-171. MR 91h:11040
  • [Tak2] Takase,K., On unitary representation of Jacobi group, J. reine angew. Math. 430 (1992), 139-149 (Correction: J. reine angew. Math. 441 (1993), 215-216). MR 93e:11064; CMP 93:15
  • [Tak3] Takase,K., On Two-fold covering group of $Sp(n,\mathbb{R})$ and automorphic factor of weight $1/2$, Comment. Math. Univ. Sancti Pauli 45 (1996), 117-145. MR 98c:11041
  • [Tam] Tamagawa,T., On Selberg's trace formula, J.Fac.Sci.Univ.Tokyo 8 (1960), 363-386. MR 23:A958
  • [Wei] Weil,A., Sur certains groupes d'opérateurs unitaires, Acta Math. 111 (1964), 143-211. MR 29:2324

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Additional Information

Koichi Takase
Affiliation: Department of Mathematics, Miyagi University of Education, Aoba-ku, Sendai 980, Japan

Received by editor(s): February 5, 1997
Article copyright: © Copyright 1999 American Mathematical Society

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