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On the uniqueness of Fourier Jacobi models for representations of 
Authors:
Ehud Moshe Baruch and Stephen Rallis
Journal:
Represent. Theory 11 (2007), 1-15
MSC (2000):
Primary 22E50; Secondary 11F70
Posted:
January 5, 2007
MathSciNet review:
2276364
Full-text PDF Free Access
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Additional Information
Abstract: We show that every irreducible unitary representation of  , has at most one Fourier Jacobi model.
- 1.
Ehud
Moshe Baruch, Ilya
Piatetski-Shapiro, and Stephen
Rallis, On the uniqueness of Fourier Jacobi models for
representations of 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
(2004k:22021)
- 2.
Ehud
Moshe Baruch and Steve
Rallis, A uniqueness theorem of Fourier Jacobi models for
representations of 𝑆𝑝(4), J. London Math. Soc. (2)
62 (2000), no. 1, 183–197. MR 1772180
(2001j:22021), http://dx.doi.org/10.1112/S0024610700001101
- 3.
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
(2000a:11066), http://dx.doi.org/10.1007/s002080050247
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Stephen
Gelbart, Examples of dual reductive pairs, Automorphic forms,
representations and 𝐿-functions (Proc. Sympos. Pure Math., Oregon
State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math.,
XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 287–296.
MR 546603
(81f:22035)
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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
(86f:11084), http://dx.doi.org/10.1007/BFb0073147
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S. Gelbart and Jonathan
D. Rogawski, 𝐿-functions and Fourier-Jacobi coefficients
for the unitary group 𝑈(3), Invent. Math. 105
(1991), no. 3, 445–472. MR 1117148
(93b:11059), http://dx.doi.org/10.1007/BF01232276
- 7.
Miki
Hirano, Fourier-Jacobi type spherical functions for
𝑃_{𝐽}-principal series representations of
𝑆𝑝(2,𝐑), J. London Math. Soc. (2)
65 (2002), no. 3, 524–546. MR 1895731
(2003c:11046), http://dx.doi.org/10.1112/S0024610701002927
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Yoshi-hiro
Ishikawa, The generalized Whittaker functions for
𝑆𝑈(2,1) and the Fourier expansion of automorphic
forms, J. Math. Sci. Univ. Tokyo 6 (1999),
no. 3, 477–526. MR 1726680
(2001b:22011)
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Yoshi-hiro
Ishikawa, The generalized Whittaker functions for the discrete
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Sūrikaisekikenkyūsho Kōkyūroku
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(2003m:22020)
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Li, Minimal representations & reductive dual pairs,
Representation theory of Lie groups (Park City, UT, 1998) IAS/Park City
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pp. 293–340. MR 1737731
(2001a:22013)
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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
(92k:11052)
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Atsushi
Murase and Takashi
Sugano, Fourier-Jacobi expansion of Eisenstein series on unitary
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(2002), no. 2, 347–404. MR 1904935
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(12,31d)
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A. Shalika, The multiplicity one theorem for
𝐺𝐿_{𝑛}, Ann. of Math. (2) 100
(1974), 171–193. MR 0348047
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Nolan
R. Wallach, Real reductive groups. I, Pure and Applied
Mathematics, vol. 132, Academic Press Inc., Boston, MA, 1988. MR 929683
(89i:22029)
- 1.
- Ehud Moshe Baruch, Ilya Piatetski-Shapiro, and Stephen Rallis, On the uniqueness of Fourier Jacobi models for representations of
 , Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, vol. 210, Amer. Math. Soc., Providence, RI, 2003, pp. 47-56. MR 2018352 (2004k:22021)
- 2.
- Ehud Moshe Baruch and Steve Rallis, A uniqueness theorem of Fourier Jacobi models for representations of
 , J. London Math. Soc. (2) 62 (2000), no. 1, 183-197. MR 1772180 (2001j:22021)
- 3.
- 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 (2000a:11066)
- 4.
- Stephen Gelbart, Examples of dual reductive pairs, Automorphic forms, representations and
 -functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 287-296. MR 0546603 (81f:22035)
- 5.
- Stephen Gelbart and Ilya Piatetski-Shapiro, Automorphic forms and
 -functions for the unitary group, Lie group representations, II (College Park, Md., 1982/1983), Springer, Berlin, 1984, pp. 141-184. MR 0748507 (86f:11084)
- 6.
- Stephen S. Gelbart and Jonathan D. Rogawski,
 -functions and Fourier-Jacobi coefficients for the unitary group  , Invent. Math. 105 (1991), no. 3, 445-472. MR 1117148 (93b:11059)
- 7.
- Miki Hirano, Fourier-Jacobi type spherical functions for
 -principal series representations of  , J. London Math. Soc. (2) 65 (2002), no. 3, 524-546. MR 1895731 (2003c:11046)
- 8.
- Yoshi-hiro Ishikawa, The generalized Whittaker functions for
 and the Fourier expansion of automorphic forms, J. Math. Sci. Univ. Tokyo 6 (1999), no. 3, 477-526. MR 1726680 (2001b:22011)
- 9.
- -, The generalized Whittaker functions for the discrete series representations of
 , Surikaisekikenkyusho Kokyuroku (1999), no. 1094, 97-109, Automorphic forms on  and  , II (Kyoto, 1998). MR 1751059
- 10.
- 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 (2003m:22020)
- 11.
- 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 (2001a:22013)
- 12.
- 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 (92k:11052)
- 13.
- -, 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 (2003f:11065)
- 14.
- L. Schwartz, Théorie des distributions. Tome I, Hermann & Cie., Paris, 1950. MR 0035918 (12,31d)
- 15.
- J. A. Shalika, The multiplicity one theorem for
 , Ann. of Math. (2) 100 (1974), 171-193. MR 0348047 (50:545)
- 16.
- Nolan R. Wallach, Real reductive groups. I, Academic Press Inc., Boston, MA, 1988. MR 0929683 (89i:22029)
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Additional 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
DOI:
http://dx.doi.org/10.1090/S1088-4165-07-00298-1
PII:
S 1088-4165(07)00298-1
Keywords:
Fourier Jacobi,
invariant distributions
Received by editor(s):
October 28, 2005
Received by editor(s) in revised form:
September 18, 2006
Posted:
January 5, 2007
Additional Notes:
Research of the second author was partially supported by the NSF
Article copyright:
© Copyright 2007 American Mathematical Society
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