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Uniqueness of Ginzburg-Rallis models: The Archimedean case


Authors: Dihua Jiang, Binyong Sun and Chen-Bo Zhu
Journal: Trans. Amer. Math. Soc. 363 (2011), 2763-2802
MSC (2000): Primary 22E46; Secondary 11F70
DOI: https://doi.org/10.1090/S0002-9947-2010-05285-7
Published electronically: December 21, 2010
MathSciNet review: 2763736
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Abstract: In this paper we prove the uniqueness of Ginzburg-Rallis models in the Archimedean case. As a key ingredient, we introduce a new descent argument based on two geometric notions attached to submanifolds, which we call metrical properness and unipotent $ \chi$-incompatibility.

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  • [AG1] A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, Int. Math. Res. Notices 2008, no. 5, Art. ID mm157, 37pp. MR 2418286 (2010g:46124)
  • [AG2] A. Aizenbud and D. Gourevitch, Generalized Harish-Chandra descent, Gelfand pairs, and an Archimedean analog of Jacquet-Rallis's theorem, Duke Math. J. 149, no. 3 (2009), 509-567. MR 2553879
  • [AG3] A. Aizenbud and D. Gourevitch, Multiplicity one theorem for $ (\mathrm{GL}_{n+1}(\mathbb{R}),\mathrm{GL}_n(\mathbb{R}))$, Selecta Mathematica 15, no. 2 (2009), 271-294. MR 2529937
  • [AGJ09] A. Aizenbud, D. Gourevitch and H. Jacquet, Uniqueness of Shalika functionals: The Archimedean case, Pacific J. Math. 243 (2009), no. 2, 201-212.
  • [AGRS] A. Aizenbud, D. Gourevitch, S. Rallis and G. Schiffmann, Multiplicity One Theorems, Annals Math. 172 (2010), no. 2, 1407-1434. MR 2680495
  • [AGS07] A. Aizenbud, D. Gourevitch and E. Sayag, $ (\mathrm{GL}(n+1,F),\mathrm{GL}(n,F))$ is a Gelfand pair for any local field $ F$, Compositio Math. 144 (2008), no. 6, 1504-1524. MR 2474319 (2009k:22022)
  • [BR07] E. M. Baruch and S. Rallis, On the uniqueness of Fourier Jacobi models for representations of $ U(n,1)$, Representation Theory 11 (2007), 1-15. MR 2276364 (2007m:22011)
  • [C89] W. Casselman, Canonical extensions of Harish-Chandra modules to representations of $ G$, Canad. Jour. Math. 41 (1989), 385-438. MR 1013462 (90j:22013)
  • [CHM00] W. Casselman, H. Hecht, and D. Mili $ \breve{c}i\acute{c}$, Bruhat filtrations and Whittaker vectors for real groups, The Mathematical Legacy of Harish-Chandra, Proc. Symp. Pure Math., vol. 68, Amer. Math. Soc., Providence, RI, 2000. MR 1767896 (2002b:22023)
  • [C91] F. du Cloux, Sur les représentations différentiables des groupes de Lie algébriques, Ann. Sci. Ecole Norm. Sup. 24 (1991), no. 3, 257-318. MR 1100992 (92j:22026)
  • [GGP09] W. Gan, B. Gross and D. Prasad, Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups. arXiv:0909.2999
  • [GJ] D. Ginzburg and D. Jiang, Notes on Periods of Eisenstein series on $ E_6$, in preparation.
  • [GJR04] D. Ginzburg, D. Jiang and S. Rallis, On the nonvanishing of the central value of the Rankin-Selberg $ L$-functions, Jour. Amer. Math. Soc. 17 (2004), no. 3, 679-722. MR 2053953 (2005g:11078)
  • [GJR05] D. Ginzburg, D. Jiang and S. Rallis, On the nonvanishing of the central value of the Rankin-Selberg $ L$-functions. II, Automorphic representations, $ L$-functions and applications: progress and prospects, 157-191, Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005. MR 2192823 (2006m:11072)
  • [GJR09] D. Ginzburg, D. Jiang and S. Rallis, Models for certain residual representations of unitary groups, Automorphic forms and $ L$-functions I. Global aspects, 125-146, Contemp. Math., 488, Amer. Math. Soc., Providence, RI, 2009. MR 2522030
  • [GR00] D. Ginzburg and S. Rallis, The exterior cube $ L$-function for $ {\rm GL}(6)$, Compositio Math. 123 (2000), no. 3, 243-272. MR 1795291 (2001j:11025)
  • [GN09] H. Glöckner and K.-H. Neeb, Infinite-dimensional Lie groups: General Theory and Main Examples, Graduate Texts in Mathematics, Springer-Verlag, book to appear.
  • [GP92] B. Gross and D. Prasad, On the decomposition of a representation of $ {\rm SO}\sb n$ when restricted to $ {\rm SO}\sb {n-1}$, Canad. Jour. Math. 44 (1992), no. 5, 974-1002. MR 1186476 (93j:22031)
  • [GP94] B. Gross and D. Prasad, On irreducible representations of $ {\rm SO}\sb {2n+1}\times{\rm SO}\sb {2m}$, Canad. Jour. Math. 46 (1994), no. 5, 930-950. MR 1295124 (96c:22028)
  • [HK04] M. Harris and S. Kudla, On a conjecture of Jacquet, Contributions to automorphic forms, geometry, and number theory, 355-371, Johns Hopkins Univ. Press, Baltimore, MD, 2004. MR 2058614 (2005g:11081)
  • [JR96] H. Jacquet and S. Rallis, Uniqueness of linear periods, Compositio Math. 102 (1996), no. 1, 65-123. MR 1394521 (97k:22025)
  • [JS90] H. Jacquet and J. Shalika, Exterior square $ L$-functions, Automorphic forms, Shimura varieties, and $ L$-functions, Vol. II (Ann Arbor, MI, 1988), 143-226, Perspect. Math., 11, Academic Press, Boston, MA, 1990. MR 1044830 (91g:11050)
  • [J08] D. Jiang, Residues of Eisenstein series and related problems, Eisenstein Series and Applications, Progress in Math. 258, 187-204, 2008. MR 2402684 (2010a:11090)
  • [JSZ09] D. Jiang, B. Sun and C.-B. Zhu, Uniqueness of Bessel models: The Archimedean case, Geom. Funct. Anal. 20 (2010), 690-709.
  • [L01] H.Y. Loke, Trilinear forms of $ \mathfrak{g}\mathfrak{l} 2$, Pac. Jour. Math. 197 (2001), 119-144. MR 1810211 (2002b:22028)
  • [LZ97] S.T. Lee and C.-B. Zhu, Degenerate principal series and local theta correspondence II, Israel Jour. Math. 100 (1997), 29-59. MR 1469104 (99c:22022)
  • [N06] C. Nien, Models of representations of general linear groups over p-adic fields, Ph.D. Thesis, University of Minnesota, ProQuest LLC, Ann. Arbor, MI, 2006. MR 2709083
  • [P90] D. Prasad, Trilinear forms for representations of $ GL(2)$ and local epsilon factors, Compositio Math. 75 (1990), 1-46. MR 1059954 (91i:22023)
  • [Pr89] T. Przebinda, The oscillator duality correspondence for the pair $ (O(2,2), Sp(2,\mathbb{R}))$, Mem. Amer. Math. Soc. 403 (1989), 1-105. MR 979944 (90f:22019)
  • [S74] J.A. Shalika, Multiplicity one theorem for $ GL_n$, Ann. Math. 100 (1974), 171-193. MR 0348047 (50:545)
  • [S87] M. Shiota, Nash manifolds, Lect. Notes in Math., vol. 1269, Springer-Verlag, 1987. MR 904479 (89b:58011)
  • [SZ1] B. Sun and C.-B. Zhu, A general form of Gelfand-Kazhdan criterion. ArXiv:0903.1409
  • [SZ2] B. Sun and C.-B. Zhu, Multiplicity One Theorems: The Archimedean case. ArXiv:0903.1413
  • [T67] F. Treves, Topological vector spaces, distributions and kernels, Academic Press, New York, 1967. MR 0225131 (37:726)
  • [W88] N. Wallach, Real Reductive Groups I, Academic Press, San Diego, 1988. MR 929683 (89i:22029)
  • [W92] N. Wallach, Real Reductive Groups II, Academic Press, San Diego, 1992. MR 1170566 (93m:22018)

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Additional Information

Dihua Jiang
Affiliation: School of Mathematics, University of Minnesota, 206 Church Street, S.E., Minneapolis, Minnesota 55455
Email: dhjiang@math.umn.edu

Binyong Sun
Affiliation: Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, People’s Republic of China
Email: sun@math.ac.cn

Chen-Bo Zhu
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543
Email: matzhucb@nus.edu.sg

DOI: https://doi.org/10.1090/S0002-9947-2010-05285-7
Keywords: Ginzburg-Rallis models, tempered generalized functions, descent
Received by editor(s): April 16, 2009
Received by editor(s) in revised form: December 22, 2009
Published electronically: December 21, 2010
Additional Notes: The first author was supported in part by NSF (USA) grant DMS–0653742 and by a Distinguished Visiting Professorship at the Academy of Mathematics and System Sciences, the Chinese Academy of Sciences
The second author was supported by NUS-MOE grant R-146-000-102-112 and by NSFC grants 10801126 and 10931006
The third author was supported by NUS-MOE grant R-146-000-102-112
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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