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

Takase, Koichi

doi:10.1090/S0002-9947-99-02168-6

On Siegel modular forms of half-integral weights and Jacobi forms
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by Koichi Takase

Trans. Amer. Math. Soc. 351 (1999), 735-780

DOI: https://doi.org/10.1090/S0002-9947-99-02168-6

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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.

References

Martin Eichler and Don Zagier, The theory of Jacobi forms, Progress in Mathematics, vol. 55, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 781735, DOI 10.1007/978-1-4684-9162-3
Godement,R., Notes on Jacquet-Langlands’ theory, The Institute for Advanced Study, 1970.
Henryk Hecht and Wilfried Schmid, On integrable representations of a semisimple Lie group, Math. Ann. 220 (1976), no. 2, 147–149. MR 399358, DOI 10.1007/BF01351699
Tomoyoshi Ibukiyama, On Jacobi forms and Siegel modular forms of half integral weights, Comment. Math. Univ. St. Paul. 41 (1992), no. 2, 109–124. MR 1190152
Jun-ichi Igusa, Harmonic analysis and theta-functions, Acta Math. 120 (1968), 187–222. MR 276688, DOI 10.1007/BF02394610
Anthony W. Knapp, Representation theory of semisimple groups, Princeton Mathematical Series, vol. 36, Princeton University Press, Princeton, NJ, 1986. An overview based on examples. MR 855239, DOI 10.1515/9781400883974
M. Kashiwara and M. Vergne, On the Segal-Shale-Weil representations and harmonic polynomials, Invent. Math. 44 (1978), no. 1, 1–47. MR 463359, DOI 10.1007/BF01389900
Michio Kuga, Fiber varieties over a symmetric space whose fibers are abelian varieties, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 338–346. MR 0206168
David Mumford, Families of abelian varieties, Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965) Amer. Math. Soc., Providence, R.I., 1966, pp. 347–351. MR 0206003
Atsushi Murase, $L$-functions attached to Jacobi forms of degree $n$. I. The basic identity, J. Reine Angew. Math. 401 (1989), 122–156. MR 1018057, DOI 10.1515/crll.1989.401.122
Ichirô Satake, Factors of automorphy and Fock representations, Advances in Math. 7 (1971), 83–110. MR 348043, DOI 10.1016/0001-8708(71)90043-0
I. Satake, Unitary representations of a semi-direct product of Lie groups on $\bar \partial$-cohomology spaces, Math. Ann. 190 (1970/71), 177–202. MR 296213, DOI 10.1007/BF01433209
Ichirô Satake, Theory of spherical functions on reductive algebraic groups over ${\mathfrak {p}}$-adic fields, Inst. Hautes Études Sci. Publ. Math. 18 (1963), 5–69. MR 195863
Ichirô Satake, Holomorphic imbeddings of symmetric domains into a Siegel space, Amer. J. Math. 87 (1965), 425–461. MR 196134, DOI 10.2307/2373012
Ichirô Satake, Algebraic structures of symmetric domains, Kanô Memorial Lectures, vol. 4, Iwanami Shoten, Tokyo; Princeton University Press, Princeton, N.J., 1980. MR 591460
Shintani,T., unpublished note.
Koichi Takase, A note on automorphic forms, J. Reine Angew. Math. 409 (1990), 138–171. MR 1061522, DOI 10.1515/crll.1990.409.138
Koichi Takase, On unitary representations of Jacobi groups, J. Reine Angew. Math. 430 (1992), 139–149. MR 1172911, DOI 10.1515/crll.1992.430.139
Koichi Takase, On two-fold covering group of $\textrm {Sp}(n,\mathbf R)$ and automorphic factor of weight $1/2$, Comment. Math. Univ. St. Paul. 45 (1996), no. 2, 117–145. MR 1416187
Tsuneo Tamagawa, On Selberg’s trace formula, J. Fac. Sci. Univ. Tokyo Sect. I 8 (1960), 363–386. MR 123633
André Weil, Sur certains groupes d’opérateurs unitaires, Acta Math. 111 (1964), 143–211 (French). MR 165033, DOI 10.1007/BF02391012