Variation of Additive Characters in the Transfer for Mp(2n)

Abstract

Let Mp(2n) be the metaplectic group of rank n over a local field F of characteristic zero. In this note, we determine the behavior of endoscopic transfer for Mp(2n) under variation of additive characters of F. The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan–Savin, which describes how the local Langlands correspondence for Mp(2n) depends on the additive characters.

Keywords

Endoscopy / metaplectic group / local Langlands correspondence / 22E50 / 11F70 / 11F72

References

[1] Adams J, Barbasch D. Reductive dual pair correspondence for complex groups. J. Funct. Anal.. 1995, 132(1): 1-42.

[2] Adams J, Barbasch D. Genuine representations of the metaplectic group. Composition Math.. 1998, 113(1): 23-66.

[3] Arthur J. The Endoscopic Classification of Representations—Orthogonal and Symplectic Groups. 2013, Providence, RI, American Mathematical Society. 61

[4] Borel A. Automorphic L-functions. Automorphic Forms, Representations and L-functions. 1977, Corvallis, Ore., Oregon State Univ.2761Part 2, Proc. Sympos. Pure Math., XXXIII, Providence, RI: American Mathematical Society, 1979

[5] Fintzen J., Kaletha T., Spice L., On certain sign characters of tori and their extensions to Bruhat-Tits groups. 2019, arXiv:1912.03286v1

[6] Gan WT, Gao F. The Langlands–Weissman program for Brylinski–Deligne extensions. Astérisque. 2018, 398: 187-275

[7] Gan WT, Gross BH, Prasad D. Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups. Astérisque. 2012, 346: 1-109

[8] Gan WT, Ichino A. The Shimura–Waldspurger correspondence for Mp(2n). Ann. of Math. (2). 2018, 188(3): 965-1016.

[9] Gan WT, Savin G. Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence. Compos. Math.. 2012, 148(6): 1655-1694.

[10] Li W-W. Transfert d’intégrales orbitales pour le groupe métaplectique. Compos. Math.. 2011, 147(2): 524-590.

[11] Li W-W. La formule des traces stable pour le groupe métaplectique: les termes elliptiques. Invent. Math.. 2015, 202(2): 743-838.

[12] Li W-W. Spectral transfer for metaplectic groups, I. Local character relations. J. Inst. Math. Jussieu. 2019, 18(1): 25-123.

[13] Li W.-W., Stable conjugacy and epipelagic L-packets for Brylinski–Deligne covers of Sp(2n). Selecta Math. (N.S.), 2020, 26(1): Paper No. 12, 123 pp.

[14] Li W.-W., Arthur packets for metaplectic groups. 2024, arXiv:2410.13606

[15] Luo C. Endoscopic character identities for metaplectic groups. J. Reine Angew. Math.. 2020, 768: 1-37.

[16] Mœglin C, Renard D. Paquets d’Arthur des groupes classiques complexes. Around Langlands Correspondences. 2017, Providence, RI, American Mathematical Society203256. 691

[17] Mœglin C, Vignéras M-F, Waldspurger J-L. Correspondances de Howe sur un corps p-adique. 1987, Berlin, Springer-Verlag1291

[18] Moore CC. Group extensions of p-adic and adelic linear groups. Inst. Hautes Études Sci. Publ. Math.. 1968, 35: 157-222.

[19] Prasad D. Generalizing the MVW involution, and the contragredient. Trans. Amer. Math. Soc.. 2019, 372(1): 615-633.

[20] Scharlau W. Quadratic and Hermitian Forms. 1985, Berlin, Springer-Verlag270

[21] Weissman MH. L-groups and parameters for covering groups. Astérisque. 2018, 398: 33-186

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