Nagoya Mathematical Journal

Some conjectures on endoscopic representations in odd orthogonal groups

David Ginzburg and Dihua Jiang

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Abstract

In this paper, we introduce two conjectures on characterizations of endoscopy structures of irreducible generic cuspidal automorphic representations of odd special orthogonal groups in terms of nonvanishing of certain period of automorphic forms. We discuss a relation between the two conjectures and prove that a special case of Conjecture 1 (and hence Conjecture 2) is true.

Article information

Source
Nagoya Math. J., Volume 208 (2012), 145-170.

Dates
First available in Project Euclid: 5 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.nmj/1354716559

Digital Object Identifier
doi:10.1215/00277630-1815231

Mathematical Reviews number (MathSciNet)
MR3006699

Zentralblatt MATH identifier
1294.11072

Subjects
Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 22E55: Representations of Lie and linear algebraic groups over global fields and adèle rings [See also 20G05]

Citation

Ginzburg, David; Jiang, Dihua. Some conjectures on endoscopic representations in odd orthogonal groups. Nagoya Math. J. 208 (2012), 145--170. doi:10.1215/00277630-1815231. https://projecteuclid.org/euclid.nmj/1354716559


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References

  • [A] J. Arthur, The endoscopic classification of representations: Orthogonal and symplectic groups, preprint, 2011.
  • [CKPS] J. W. Cogdell, H. Kim, I. I. Piatetski-Shapiro, and F. Shahidi, Functoriality for the classical groups, Publ. Math. Inst. Hautes Études Sci. 99 (2004), 163–233.
  • [GS] W. T. Gan and G. Savin, Real and global lifts from $\mathrm{PGL}_{3}$ to $G_{2}$, Int. Math. Res. Not. IMRN 50 (2003), 2699–2724.
  • [G1] D. Ginzburg, A construction of CAP representations for classical groups, Int. Math. Res. Not. IMRN 20 (2003), 1123–1140.
  • [G2] D. Ginzburg, Certain conjectures relating unipotent orbits to automorphic representations, Israel J. Math. 151 (2006), 323–355.
  • [G3] D. Ginzburg, Endoscopic lifting in classical groups and poles of tensor $L$-functions, Duke Math. J. 141 (2008), 447–503.
  • [GRS] D. Ginzburg, S. Rallis, and D. Soudry, The Descent Map from Automorphic Representations of $\operatorname{GL}(n)$ to Classical Groups, World Sci., Hackensack, N.J., 2011.
  • [Ich] A. Ichino, On the regularized Siegel-Weil formula, J. Reine Angew. Math. 539 (2001), 201–234.
  • [Ik] T. Ikeda, On the theory of Jacobi forms and Fourier-Jacobi coefficients of Eisenstein series, Kyoto J. Math. 34 (1994), 615–636.
  • [J] D. Jiang, On the fundamental automorphic $L$-functions of $\operatorname{SO}(2n+1)$, Int. Math. Res. Not. IMRN 2006, Art. ID 64069.
  • [JQ] D. Jiang and Y.-J. Qin, Residues of Eisenstein series and generalized Shalika models for $\operatorname{SO}(4n)$, J. Ramanujan Math. Soc. 22 (2007), 101–133.
  • [JS] D. Jiang and D. Soudry, On the genericity of cuspidal automorphic forms of $\operatorname{SO}_{2n+1}$, II, Compos. Math 143 (2007), 721–748.
  • [KR] S. S. Kudla and S. Rallis, A regularized Siegel-Weil formula: The first term identity, Ann. of Math. (2) 140 (1994), 1–80.
  • [S] D. Soudry, Rankin-Selberg integrals, the descent method, and Langlands functoriality, Int. Congr. Math. 2, Eur. Math. Soc., Zürich, 2006, 1311–1325.