Globalization of Distinguished Supercuspidal Representations of GL(n)

Hakim, Jeffrey ; Murnaghan, Fiona

Canadian mathematical bulletin, Tome 45 (2002) no. 2, pp. 220-230

Résumé

An irreducible supercuspidal representation $\pi$ of $G\,=\,\text{GL}\left( n,\,F \right)$ , where $F$ is a nonarchimedean local field of characteristic zero, is said to be “distinguished” by a subgroup $H$ of $G$ and a quasicharacter $\text{ }\!\!\chi\!\!\text{ }$ of $H$ if $\text{Ho}{{\text{m}}_{H}}\left( \pi ,\,\text{ }\!\!\chi\!\!\text{ } \right)\,\ne \,0$ . There is a suitable global analogue of this notion for an irreducible, automorphic, cuspidal representation associated to $\text{GL}\left( n \right)$ . Under certain general hypotheses, it is shown in this paper that every distinguished, irreducible, supercuspidal representation may be realized as a local component of a distinguished, irreducible automorphic, cuspidal representation. Applications to the theory of distinguished supercuspidal representations are provided.

Détail

DOI : 10.4153/CMB-2002-025-x

Mots-clés : 22E50, 22E35, 11F70

Bibliographie

[1] [1] Bernstein, J., Deligne, P. and Kazhdan, D., Trace Paley-Weiner theorem for reductive p-adic groups. J. Anal.Math. 47 (1986), 47–1986. Google Scholar

[2] [2] Bernstein, J., Deligne, P., Kazhdan, D. and Vignéras, M.-F., Representations des groupes sur un corps local. Hermann, Paris, 1984. Google Scholar

[3] [3] Clozel, L., Invariant harmonic analysis on the Schwartz space of a reductive p-adic group. In: Harmonic Analysis on Reductive Groups, (eds.W. Barker and P. Sally), Progress in Math. 101, Birkäuser, Boston, 1991, 101–121. Google Scholar

[4] [4] Flicker, Y., On distinguished representations. J. Reine Angew. Math. 418 (1991), 418–1991. Google Scholar

[5] [5] Friedberg, S. and Jacquet, H., Linear periods. J. Reine Angew. Math. 443 (1993), 443–1993. Google Scholar

[6] [6] Gelbart, S., Lectures on the Arthur-Selberg Trace Formula. University Lecture Series 9, Amer. Math. Soc., Providence, RI, 1996. Google Scholar

[7] [7] Gelfand, I. and Kazhdan, D., Representations of GL(n, K) where K is a local field. Lie Groups and Their Representations, Wiley, New York, 1975, 95–118. Google Scholar

[8] [8] Hakim, J., Admissible distributions on p-adic symmetric spaces. J. Reine Angew. Math. 455 (1994), 455–1994. Google Scholar

[9] [9] Hakim, J., Distinguished p-adic representations. Duke Math. J. (1) 62 (1991), 62–1991. Google Scholar

[10] [10] Hakim, J. and Mao, Z., Cuspidal representations associated to (GL(n), O(n)) over finite fields and p-adic fields. J. Algebra 213 (1999), 213–1999. Google Scholar

[11] [11] Hakim, J. and Mao, Z., Supercuspidal representations of GL(n) distinguished by a unitary subgroup. Pacific J. Math. 185 (1998), 185–1998. Google Scholar

[12] [12] Hakim, J. and Murnaghan, F., Tame supercuspidal representations of GL(n) distinguished by a unitary subgroup. preprint. Google Scholar

[13] [13] Harder, G., Langlands, R. and Rapoport, M., Algebraische Zyklen auf Hilbert-Blumenthal-Flächen. J. Reine Angew. Math. 366 (1986), 366–1986. Google Scholar

[14] [14] Heumos, M. and Rallis, S., Symplectic-Whittaker models for GL . Pacific J. Math. 146 (1990), 146–1990. Google Scholar

[15] [15] Jacquet, H. and Rallis, S., Symplectic periods. J. Reine Angew. Math. 423 (1992), 423–1992. Google Scholar

[16] [16] Jacquet, H. and Rallis, S., Uniqueness of linear periods. Comp. Math. 102 (1996), 102–1996. Google Scholar

[17] [17] Jacquet, H. and Ye, Y., Distinguished representations and quadratic base change for GL(3). Trans. Amer. Math. Soc. Paris (3) 348 (1996), 348–1996. Google Scholar

[18] [18] Jacquet, H. and Ye, Y., Une remarque sur le changement de base quadratique. C. R. Acad. Sci. Paris Sér I 311 (1990), 311–1990. Google Scholar

[19] [19] Prasad, D., On the self-dual representations of a p-adic group. preprint. Google Scholar