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Local intertwining relation for metaplectic groups

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  01 October 2020

Hiroshi Ishimoto
Hiroshi Ishimoto*
Affiliation:
Kyoto University, Kyoto 606-8502, Japan ishimoto@math.kyoto-u.ac.jp
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Abstract

In an earlier paper of Wee Teck Gan and Gordan Savin, the local Langlands correspondence for metaplectic groups over a nonarchimedean local field of characteristic zero was established. In this paper, we formulate and prove a local intertwining relation for metaplectic groups assuming the local intertwining relation for non-quasi-split odd special orthogonal groups.

Keywords

metaplectic groupslocal intertwining relationlocal Langlands correspondencetheta correspondence

MSC classification

Primary: 11F27: Theta series; Weil representation; theta correspondences 11F85: $p$-adic theory, local fields
Type
Research Article
Information
Compositio Mathematica , Volume 156 , Issue 8 , August 2020 , pp. 1560 - 1594
Copyright
Copyright © The Author(s) 2020

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References

Arthur, J., Intertwining operators and residues I. Weighted characters, J. Funct. Anal. 84 (1989), 19–84.CrossRefGoogle Scholar
Arthur, J., Unipotent automorphic representations: conjectures, Astérisque 171–172 (1989), 13–71.Google Scholar
Arthur, J., The endoscopic classification of representations: orthogonal and symplectic groups, American Mathematical Society Colloquium Publications, vol. 61 (American Mathematical Society, Providence, RI, 2013).CrossRefGoogle Scholar
Atobe, H., The local theta correspondence and the local Gan–Gross–Prasad conjecture for the symplectic-metaplectic case, Math. Ann. 371 (2018), 225–295.CrossRefGoogle Scholar
Borel, A., Automorphic $L$-functions, in Automorphic forms, representations, and $L$-functions, Proceedings of Symposia in Pure Mathematics, vol. 33, part 2 (American Mathematical Society, Providence, RI, 1979), 27–62.Google Scholar
Gan, W. T., Gross, B. H. and Prasad, D., Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups, Astérisque 346 (2012), 1–109.Google Scholar
Gan, W. T. and Ichino, A., Formal degrees and local theta correspondence, Invent. Math. 195 (2014), 509–672.CrossRefGoogle Scholar
Gan, W. T. and Ichino, A., The Gross-Prasad conjecture and local theta correspondence, Invent. Math. 206 (2016), 705–799.Google Scholar
Gan, W. T. and Li, W. W., The Shimura-Waldspurger correspondence for Mp(2n), in Geometric aspects of the trace formula, SSTF 2016, Simons Symposia, eds. W. Müller, S. Shin and N. Templier (Springer, Cham, 2018), 183–210.Google Scholar
Gan, W. T. and Savin, G., Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compos. Math. 148 (2012), 1655–1694.Google Scholar
Gan, W. T. and Takeda, S., A proof of the Howe duality conjecture, J. Amer. Math. Soc. 29 (2016), 473–493.CrossRefGoogle Scholar
Hanzer, M., $R$-groups for metaplectic groups, Israel J. Math. 231 (2019), 467–488.CrossRefGoogle Scholar
Kaletha, T., The local Langlands conjectures for non-quasi-split groups, in Families of automorphic forms and the trace formula, Simons Symposia, eds. W. Müller, S. Shin and N. Templier (Springer, Cham, 2016), 217–257.CrossRefGoogle Scholar
Kaletha, T., Mínguez, A., Shin, S. W. and White, P.-J., Endoscopic classification of representations: inner forms of unitary groups, Preprint (2014), arXiv:1409.3731.Google Scholar
Kottwitz, R., Sign changes in harmonic analysis on reductive groups, Trans. Amer. Math. Soc. 278 (1983), 289–297.CrossRefGoogle Scholar
Kudla, S. S., Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), 361–401.CrossRefGoogle Scholar
Langlands, R. and Shelstad, D., On the definition of transfer factors, Math. Ann. 278 (1987), 219–271.CrossRefGoogle Scholar
Mok, C. P., Endoscopic classification of representations of quasi-split unitary groups, Mem. Amer. Math. Soc. 235 (2015),1–248.Google Scholar
Mœglin, C. and Renard, D., Sur les paquets d'Arthur des groupes classiques et unitaires non quasi-déployés, in Relative aspects in representation theory, Langlands functoriality and automorphic forms, Lecture Notes in Mathematics, vol. 2221, eds. V. Heiermann and D. Prasad (Springer, Cham, 2018), 341–361.Google Scholar
Mœglin, C., Vigneras, M.-F. and Waldspurger, J.-L., Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 1291 (Springer, Berlin, 1987).Google Scholar
Ranga Rao, R., On some explicit formulas in the theory of Weil representation, Pacific J. Math. 157 (1993), 335–371.CrossRefGoogle Scholar
Shahidi, F., On certain L-functions, Amer. J. Math. 103 (1981), 297–355.CrossRefGoogle Scholar
Szpruch, D., Uniqueness of Whittaker model for the metaplectic group, Pacific J. Math. 232 (2007), 453–469.CrossRefGoogle Scholar
Vogan, D. A. Jr., The local Langlands conjecture, Contemp. Math. 145 (1993), 305–379.Google Scholar
Waldspurger, J.-L., Démonstration d'une conjecture de Howe dans le cas $p$-adique, $p\neq 2$, in Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I, Israel Mathematical Conference Proceedings, vol. 2 (Weizmann Science Press, Jerusalem, 1990), 267–324.Google Scholar