Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-26T15:34:28.190Z Has data issue: false hasContentIssue false
  • English
  • Français

Rank One Reducibility for Metaplectic Groups via Theta Correspondence

Published online by Cambridge University Press:  20 November 2018

Marcela Hanzer  and
Goran Muić
Marcela Hanzer
Affiliation:
Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia email: [email protected]@math.hr
Goran Muić
Affiliation:
Department of Mathematics, University of Zagreb, 10 000 Zagreb, Croatia email: [email protected]@math.hr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We calculate reducibility for the representations of metaplectic groups induced from cuspidal representations of maximal parabolic subgroups via theta correspondence, in terms of the analogous representations of the odd orthogonal groups. We also describe the lifts of all relevant subquotients.

Keywords

22E5011F70
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 63 , Issue 3 , 01 June 2011 , pp. 591 - 615
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Aubert, A.-M. Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif p-adique. Trans. Amer. Math. Soc. 347(1995), no. 6, 2179–2189. doi:10.2307/2154931 Google Scholar
[2] Bernšteın, I. N. and Zelevinskiı, A. V., Representations of the group GL(n, F), where F is a local non-Archimedean field. (Russian) Uspehi Mat. Nauk 31(1976), no. 3(189), 5–70.Google Scholar
[3] Bernšteın, I. N. and Zelevinskiı, A. V., Induced representations of reductive p-adic groups. I. Ann. Sci. École Norm. Sup. (4) 10(1977), no. 4, 441–472.Google Scholar
[4] Bernstein, J., Draft of: Representations of p-adic groups. www.math.tau.ac.il/.bernstei/Publication list/publication texts/Bernst Lecture p-adic repr.pdfGoogle Scholar
[5] Bernstein, J., Second adjointness for representations of p-adic reductive groups, preprint.Google Scholar
[6] Bushnell, C. J., Representations of reductive p-adic groups: localization of Hecke algebras and applications. J. LondonMath. Soc. (2) 63(2001), no. 2, 364–386. doi:10.1017/S0024610700001885 Google Scholar
[7] Gelbart, S. S., Weil's representation and the spectrum of the metaplectic group. Lecture Notes in Mathematics, 530, Springer-Verlag, Berlin-New York, 1976.Google Scholar
[8] Hanzer, M. and Muić, G., Parabolic induction and Jacquet functors for metaplectic groups. J. Algebra 323(2010), no. 1, 241–260. doi:10.1016/j.jalgebra.2009.07.001 Google Scholar
[9] Kudla, S., Notes on the local theta correspondence (lectures at the european school in group theory). www.math.toronto.edu/.skudla/castle.pdf.Google Scholar
[10] Kudla, S. S., On the local theta-correspondence. Invent. Math. 83(1986), no. 2, 229–255. doi:10.1007/BF01388961 Google Scholar
[11] Kudla, S. S. and Rallis, S., On first occurrence in the local theta correspondence. In: Automorphic representations, L-functions and applications: progress and prospects. Ohio State Univ. Math. Res. Inst. Publ., 11, de Gruyter, Berlin, 2005, pp. 273–308.Google Scholar
[12] Manderscheid, D., Supercuspidal duality for the two-fold cover of SL2 and the split O3. Amer. J. Math. 107(1985), no. 6, 1305–1324 (1986). doi:10.2307/2374408 Google Scholar
[13] Moeglin, C., Normalisation des opérateurs d’entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques p-adiques. Ann. of Math. (2) 151(2000), no. 2, 817–847. doi:10.2307/121049 Google Scholar
[14] Moeglin, C., Vignéras, M.-F., and Waldspurger, J.-L., Correspondances de Howe sur un corps p-adique. Lecture Notes in Mathematics, 1291, Springer-Verlag, Berlin, 1987.Google Scholar
[15] Muić, G., Howe correspondence for discrete series representations; the case of (Sp(n),O(V)). J. Reine Angew. Math. 567(2004), 99–150.Google Scholar
[16] Muić, G., On the structure of the full lift for the Howe correspondence of (Sp(n),O(V)) for rank-one reducibilities. Canad. Math. Bull. 49(2006), no. 4, 578–591. doi: Google Scholar
[17] Muić, G., On the structure of theta lifts of discrete series for dual pairs (Sp(n),O(V)). Israel J. Math. 164(2008), 87–124. doi:10.1007/s11856-008-0022-5 Google Scholar
[18] Muić, G. and Savin, G., Complementary series for Hermitian quaternionic groups. Canad. Math. Bull. 43(2000), no. 1, 90–99. doi:10.4153/CMB-2000-014-5 Google Scholar
[19] Ranga Rao, R., On some explicit formulas in the theory of Weil representation. Pacific J. Math. 157(1993), no. 2, 335–371.Google Scholar
[20] Shahidi, F., A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math. (2) 132(1990), no. 2, 273–330. doi:10.2307/1971524 Google Scholar
[21] Shahidi, F., Twisted endoscopy and reducibility of induced representations for p-adic groups. Duke Math. J. 66(1992), no. 1, 1–41. doi:10.1215/S0012-7094-92-06601-4 Google Scholar
[22] Silberger, A. J., Special representations of reductive p-adic groups are not integrable. Ann. of Math. (2) 111(1980), no. 3, 571–587. doi:10.2307/1971110 Google Scholar
[23] Tadić, M., Structure arising from induction and Jacquet modules of representations of classical p-adic groups. J. Algebra 177(1995), no. 1, 1–33. doi:10.1006/jabr.1995.1284 Google Scholar
[24] Waldspurger, J.-L., Correspondance de Shimura. J. Math. Pures Appl. (9) 59(1980), no. 1, 1–132.Google Scholar
[25] Zelevinsky, A. V., Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. (4) 13(1980), no. 2, 165–210.Google Scholar