## On the uniqueness of Fourier Jacobi models for representations of $U(n,1)$

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- by Ehud Moshe Baruch and Stephen Rallis
- Represent. Theory
**11**(2007), 1-15 - DOI: https://doi.org/10.1090/S1088-4165-07-00298-1
- Published electronically: January 5, 2007
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## Abstract:

We show that every irreducible unitary representation of $U(n,1)$, has at most one Fourier Jacobi model.## References

- Ehud Moshe Baruch, Ilya Piatetski-Shapiro, and Stephen Rallis,
*On the uniqueness of Fourier Jacobi models for representations of $\textrm {U}(2,1)$*, Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 47–56. MR**2018352**, DOI 10.1090/trans2/210/04 - Ehud Moshe Baruch and Steve Rallis,
*A uniqueness theorem of Fourier Jacobi models for representations of $\textrm {Sp}(4)$*, J. London Math. Soc. (2)**62**(2000), no. 1, 183–197. MR**1772180**, DOI 10.1112/S0024610700001101 - S. Böcherer, J. H. Bruinier, and W. Kohnen,
*Non-vanishing of scalar products of Fourier-Jacobi coefficients of Siegel cusp forms*, Math. Ann.**313**(1999), no. 1, 1–13. MR**1666805**, DOI 10.1007/s002080050247 - Stephen Gelbart,
*Examples of dual reductive pairs*, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 287–296. MR**546603** - Stephen Gelbart and Ilya Piatetski-Shapiro,
*Automorphic forms and $L$-functions for the unitary group*, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 141–184. MR**748507**, DOI 10.1007/BFb0073147 - Stephen S. Gelbart and Jonathan D. Rogawski,
*$L$-functions and Fourier-Jacobi coefficients for the unitary group $\textrm {U}(3)$*, Invent. Math.**105**(1991), no. 3, 445–472. MR**1117148**, DOI 10.1007/BF01232276 - Miki Hirano,
*Fourier-Jacobi type spherical functions for $P_J$-principal series representations of $\textrm {Sp}(2,\mathbf R)$*, J. London Math. Soc. (2)**65**(2002), no. 3, 524–546. MR**1895731**, DOI 10.1112/S0024610701002927 - Yoshi-hiro Ishikawa,
*The generalized Whittaker functions for $\textrm {SU}(2,1)$ and the Fourier expansion of automorphic forms*, J. Math. Sci. Univ. Tokyo**6**(1999), no. 3, 477–526. MR**1726680** - Yoshi-hiro Ishikawa,
*The generalized Whittaker functions for the discrete series representations of $\textrm {SU}(3,1)$*, Sūrikaisekikenkyūsho K\B{o}kyūroku**1094**(1999), 97–109. Automorphic forms on $\textrm {Sp}(2;\textbf {R})$ and $\textrm {SU}(2,2)$, II (Kyoto, 1998). MR**1751059** - Shin-ichi Kato, Atsushi Murase, and Takashi Sugano,
*Whittaker-Shintani functions for orthogonal groups*, Tohoku Math. J. (2)**55**(2003), no. 1, 1–64. MR**1956080** - Jian-Shu Li,
*Minimal representations & reductive dual pairs*, Representation theory of Lie groups (Park City, UT, 1998) IAS/Park City Math. Ser., vol. 8, Amer. Math. Soc., Providence, RI, 2000, pp. 293–340. MR**1737731** - Atsushi Murase and Takashi Sugano,
*Whittaker-Shintani functions on the symplectic group of Fourier-Jacobi type*, Compositio Math.**79**(1991), no. 3, 321–349. MR**1121142** - Atsushi Murase and Takashi Sugano,
*Fourier-Jacobi expansion of Eisenstein series on unitary groups of degree three*, J. Math. Sci. Univ. Tokyo**9**(2002), no. 2, 347–404. MR**1904935** - L. Schwartz,
*Théorie des distributions. Tome I*, Publ. Inst. Math. Univ. Strasbourg, vol. 9, Hermann & Cie, Paris, 1950 (French). MR**0035918** - J. A. Shalika,
*The multiplicity one theorem for $\textrm {GL}_{n}$*, Ann. of Math. (2)**100**(1974), 171–193. MR**348047**, DOI 10.2307/1971071 - Nolan R. Wallach,
*Real reductive groups. I*, Pure and Applied Mathematics, vol. 132, Academic Press, Inc., Boston, MA, 1988. MR**929683**

## Bibliographic Information

**Ehud Moshe Baruch**- Affiliation: Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel
- Email: embaruch@math.technion.ac.il
**Stephen Rallis**- Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
- Email: haar@math.ohio-state.edu
- Received by editor(s): October 28, 2005
- Received by editor(s) in revised form: September 18, 2006
- Published electronically: January 5, 2007
- Additional Notes: Research of the second author was partially supported by the NSF
- © Copyright 2007 American Mathematical Society
- Journal: Represent. Theory
**11**(2007), 1-15 - MSC (2000): Primary 22E50; Secondary 11F70
- DOI: https://doi.org/10.1090/S1088-4165-07-00298-1
- MathSciNet review: 2276364