Symplectic periods of the continuous spectrum of $\mathrm{GL}\left(2n\right)$

Yamana, Shunsuke

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 $\mathrm{GL}\left(2n\right)$ 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 $\mathrm{GL}\left(2n\right)$ and determine when it is not identically zero.

Classification : 11F67, 11F70
Keywords: symplectic periods, intertwining periods, continuous spectrum
Mot clés : périodes symplectiques, périodes d’entrelacement, spectre continu

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

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

[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 | Zbl

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

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

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

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

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

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

[10] Offen, O. On symplectic periods of discrete spectrum of ${\mathrm{GL}}_{2n}$, Israel J. Math., Volume 154 (2006), pp. 253-298 | DOI | MR | Zbl

[11] Offen, O. Residual spectrum of ${\mathrm{GL}}_{2n}$ distinguished by the symplectic group, Duke Math. J., Volume 134 (2006) no. 2, pp. 313-357 | DOI | MR | Zbl

[12] Offen, O.; Sayag, E. On unitary representations of ${\mathrm{GL}}_{2n}$ distinguished by the symplectic group, J. Number Theory, Volume 125 (2007), pp. 344-355 | DOI | MR | Zbl

[13] Vogan, D. The unitary dual of $\mathrm{GL}\left(n\right)$ over an Archimedean field, Invent. Math., Volume 83 (1986), pp. 449-505 | DOI | MR | Zbl