Group theory
Vanishing of local symplectic periods for cuspidal representations of the unitary group

Presented by: the Editorial Board

Mitra, Arnab1; Offen, Omer1
1 Department of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel
Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 15-19.

We show that, for a quadratic extension of p-adic fields, no cuspidal representation of the quasi-split unitary group admits a non-trivial linear form invariant by the symplectic subgroup. Our proof is purely local.

Nous montrons que, pour une extension quadratique de corps p-adiques, aucune représentation cuspidale du groupe unitaire semi-déployé n'admet de forme linéaire non nulle invariante par l'action du sous-groupe symplectique. Notre preuve est purement locale.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.11.009
Mitra, Arnab 1; Offen, Omer 1

1 Department of Mathematics, Technion – Israel Institute of Technology, Haifa 3200003, Israel 
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Mitra, Arnab; Offen, Omer. Vanishing of local symplectic periods for cuspidal representations of the unitary group. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 15-19. doi : 10.1016/j.crma.2016.11.009. https://www.numdam.org/articles/10.1016/j.crma.2016.11.009/

[1] Ash, A.; Ginzburg, D.; Rallis, S. Vanishing periods of cusp forms over modular symbols, Math. Ann., Volume 296 (1993) no. 4, pp. 709-723 MR 1233493 (94f:11044)

[2] Bernstein, J.N. On the support of Plancherel measure, J. Geom. Phys., Volume 5 (1988) no. 4, pp. 663-710 MR 1075727 (91k:22027)

[3] Dijols, S.; Prasad, D. Symplectic models for Unitary groups | arXiv

[4] Heumos, M.J.; Rallis, S. Symplectic–Whittaker models for Gln, Pac. J. Math., Volume 146 (1990) no. 2, pp. 247-279 MR 1078382 (91k:22036)

[5] Jantzen, C. Tempered representations for classical p-adic groups, Manuscr. Math., Volume 145 (2014) no. 3–4, pp. 319-387 (MR 3268853)

[6] Kato, S.; Takano, K. Subrepresentation theorem for p-adic symmetric spaces, Int. Math. Res. Not. (2008) no. 11, p. 40 Art. ID rnn028, MR 2428854 (2009i:22021)

[7] Lagier, N. Terme constant de fonctions sur un espace symétrique réductif p-adique, J. Funct. Anal., Volume 254 (2008) no. 4, pp. 1088-1145 MR 2381204 (2009d:22013)

[8] Mœglin, C.; Tadić, M. Construction of discrete series for classical p-adic groups, J. Amer. Math. Soc., Volume 15 (2002) no. 3, pp. 715-786 (electronic), MR 1896238 (2003g:22020)

[9] Tadić, M. On tempered and square integrable representations of classical p-adic groups, Sci. China Math., Volume 56 (2013) no. 11, pp. 2273-2313 (MR 3123571)

[10] Zhang, L. Distinguished tame supercuspidal representations of symmetric pairs (Sp4n(F),Sp2n(E)), Manuscr. Math., Volume 148 (2015) no. 1–2, pp. 213-233 (with an appendix by Dihua Jiang and the author, MR 3377755)

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