Symplectic periods of the continuous spectrum of GL(2n)  [ Périodes Symplectiques du Spectre Continu de GL(2n) ]
Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1561-1580.

On donne une formule pour la période symplectique d’une série d’Eisenstein pour le groupe GL(2n) et on détermine sous quelles conditions celle-ci n’est pas identiquement nulle.

We provide a formula for the symplectic period of an Eisenstein series on GL(2n) and determine when it is not identically zero.

DOI : https://doi.org/10.5802/aif.2890
Classification : 11F67,  11F70
Mots clés : périodes symplectiques, périodes d’entrelacement, spectre continu
@article{AIF_2014__64_4_1561_0,
     author = {Yamana, Shunsuke},
     title = {Symplectic periods of the continuous spectrum of $\mathrm{GL}(2n)$},
     journal = {Annales de l'Institut Fourier},
     pages = {1561--1580},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {64},
     number = {4},
     year = {2014},
     doi = {10.5802/aif.2890},
     mrnumber = {3329673},
     zbl = {06387317},
     language = {en},
     url = {www.numdam.org/item/AIF_2014__64_4_1561_0/}
}
Yamana, Shunsuke. Symplectic periods of the continuous spectrum of $\mathrm{GL}(2n)$. Annales de l'Institut Fourier, Tome 64 (2014) no. 4, pp. 1561-1580. doi : 10.5802/aif.2890. http://www.numdam.org/item/AIF_2014__64_4_1561_0/

[1] Arthur, J. On the inner product of truncated Eisenstein series, Duke Math. J., Volume 49 (1982), pp. 35-70 | Article | MR 650368 | Zbl 0518.22012

[2] Bernstein, J. P-invariant distributions on GL(N) and the classification of unitary representations of GL(N) (non-archimedean case), Lie Group Representations II (Lec. Notes in Math.) Volume 1041, Springer, 1984, pp. 50-102 | MR 748505

[3] Jacquet, H.; Lapid, E.; Rallis, S. A spectral identity for skew symmetric matrices, Contributions to Automorphic Forms, Geometry, and Number Theory, Johns Hopkins University Press, Baltimore, 2004, pp. 421-455 | MR 2058616 | Zbl 1082.11029

[4] Jacquet, H.; Lapid, E.; Rogawski, J. Periods of automorphic forms, J. Am. Math. Soc., Volume 12 (1999), pp. 173-240 | Article | MR 1625060 | Zbl 1012.11044

[5] Jacquet, H.; Piatetski-Shapiro, I.I.; Shalika, J. Rankin-Selberg Convolutions, Am. J. Math., Volume 105 (1983), pp. 367-464 | Article | MR 701565 | Zbl 0525.22018

[6] Jacquet, H.; Rallis, S. Symplectic periods, J. Reine Angew. Math., Volume 423 (1992), pp. 175-197 | MR 1142486 | Zbl 0734.11035

[7] Lapid, E.; Rogawski, J. Periods of Eisenstein series: the Galois case, Duke Math. J., Volume 120 (2003) no. 1, pp. 153-226 | Article | MR 2010737 | Zbl 1037.11033

[8] Moeglin, C.; Waldspurger, J.-L. Le spectre résiduel de GL(n), Ann. Sci. École Norm. Sup. (4), Volume 22 (1989), pp. 605-674 | Numdam | MR 1026752 | Zbl 0696.10023

[9] Moeglin, C.; Waldspurger, J.-L. Spectral Decomposition and Eisenstein Series, Cambridge Tracts in Mathematics, Volume 113, Cambridge University Press, 1995 | MR 1361168 | Zbl 0846.11032

[10] Offen, O. On symplectic periods of discrete spectrum of GL 2n , Israel J. Math., Volume 154 (2006), pp. 253-298 | Article | MR 2254544 | Zbl 1148.11025

[11] Offen, O. Residual spectrum of GL 2n distinguished by the symplectic group, Duke Math. J., Volume 134 (2006) no. 2, pp. 313-357 | Article | MR 2248833 | Zbl 1220.11072

[12] Offen, O.; Sayag, E. On unitary representations of GL 2n distinguished by the symplectic group, J. Number Theory, Volume 125 (2007), pp. 344-355 | Article | MR 2332593 | Zbl 1147.11028

[13] Vogan, D. The unitary dual of GL(n) over an Archimedean field, Invent. Math., Volume 83 (1986), pp. 449-505 | Article | MR 827363 | Zbl 0598.22008