Endoscopic character identities for metaplectic groups. (English) Zbl 1479.22015
The paper under review proves the endoscopic character identities for tempered representations of metaplectic groups \(Mp(2n)\) over a local field of characteristic zero as formulated by W. W. Li in [W. T. Gan and W.-W. Li, in: Geometric aspects of the trace formula. Proceedings of the Simons symposium, Schloss Elmau, Germany, April 10–16, 2016. Cham: Springer, 183–210 (2018; Zbl 1453.11065)]. The formulation relies on the local Langlands correspondence for \(Mp(2n)\), which has been established using theta correspondence by Adams-Barbasch in the archimedean case and Gan-Savin in the \(p\)-adic case. In the archimedean case, the endosopic character identities of \(Mp(2n)\) have been proved by D. Renard in a different form [ Am. J. Math. 121, No. 6, 1215–1243 (1999; Zbl 0942.22010)] and the author gives a reinterpretation of that result. In the nonarchimedean case the author uses a local-global argument with key inputs from the W. W. Li’s simple stable trace formula of \(Mp(2n)\), Gan-Ichino’s multiplicity formula for the tempered discrete spectrum of \(Mp(2n)\) and Arthur’s stable multiplicity formula for special odd orthogonal groups.
Reviewer: Bin Xu (Beijing)
MSC:
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
| 11F70 | Representation-theoretic methods; automorphic representations over local and global fields |
References:
| [1] | J. Adams, Genuine representations of the metaplectic group and epsilon factors, Proceedings of the International Congress of Mathematicians. Vol. 1, 2 (Zürich 1994), Birkhäuser, Basel (1995), 721-731. · Zbl 0857.11023 |
| [2] | J. Adams, Lifting of characters on orthogonal and metaplectic groups, Duke Math. J. 92 (1998), no. 1, 129-178. · Zbl 0983.11025 |
| [3] | J. Adams, Discrete series and characters of the component group, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl. 1, International Press, Somerville (2011), 369-387. · Zbl 1255.11027 |
| [4] | J. Adams and D. Barbasch, Genuine representations of the metaplectic group, Compos. Math. 113 (1998), no. 1, 23-66. · Zbl 0913.11022 |
| [5] | J. Adams, D. Barbasch and D. A. Vogan, Jr., The Langlands classification and irreducible characters for real reductive groups, Progr. Math. 104, Birkhäuser, Boston 1992. · Zbl 0756.22004 |
| [6] | J. Arthur, The invariant trace formula. II. Global theory, J. Amer. Math. Soc. 1 (1988), no. 3, 501-554. · Zbl 0667.10019 |
| [7] | J. Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc. 4, American Mathematical Society, Providence (2005), 1-263. · Zbl 1152.11021 |
| [8] | J. Arthur, The endoscopic classification of representations. Orthogonal and symplectic groups, Amer. Math. Soc. Colloq. Publ. 61, American Mathematical Society, Providence 2013. · Zbl 1310.22014 |
| [9] | A. Borel, Automorphic L-functions, Automorphic forms, representations and L-functions (Corvallis 1977), Proc. Sympos. Pure Math. 33 Part 2, American Mathematical Society, Providence (1979), 27-61. · Zbl 0412.10017 |
| [10] | P.-S. Chan and W. T. Gan, The local Langlands conjecture for \rm GSp(4) III: Stability and twisted endoscopy, J. Number Theory 146 (2015), 69-133. · Zbl 1366.11074 |
| [11] | K. Choiy and D. Goldberg, Invariance of R-groups between p-adic inner forms of quasi-split classical groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 1387-1410. · Zbl 1335.22018 |
| [12] | W. T. Gan, B. H. Gross and D. Prasad, Symplectic local root numbers, central critical L values, and restriction problems in the representation theory of classical groups, On the conjectures of Gross and Prasad. I, Astérisque 346, Société Mathématique de France, Paris (2012), 1-109. · Zbl 1280.22019 |
| [13] | W. T. Gan and A. Ichino, The Shimura-Waldspurger correspondence for \(\rm Mp}_{2n\), Ann. of Math. (2) 188 (2018), no. 3, 965-1016. · Zbl 1486.11076 |
| [14] | W. T. Gan and W.-W. Li, The Shimura-Waldspurger correspondence for {\rm Mp}(2n), Geometric aspects of the trace formula, Simons Symp., Springer, Cham (2018), 183-210. · Zbl 1453.11065 |
| [15] | W. T. Gan and G. Savin, Representations of metaplectic groups I: Epsilon dichotomy and local Langlands correspondence, Compos. Math. 148 (2012), no. 6, 1655-1694. · Zbl 1325.11046 |
| [16] | D. Goldberg, Reducibility of induced representations for {\rmSp}(2n) and {\rmSO}(n), Amer. J. Math. 116 (1994), no. 5, 1101-1151. · Zbl 0851.22021 |
| [17] | B. H. Gross and D. Prasad, On the decomposition of a representation of \(\rm\text{SO}}_{n\) when restricted to \(\rm\text{SO}}_{n-1\), Canad. J. Math. 44 (1992), no. 5, 974-1002. · Zbl 0787.22018 |
| [18] | T. C. Hales, On the fundamental lemma for standard endoscopy: Reduction to unit elements, Canad. J. Math. 47 (1995), no. 5, 974-994. · Zbl 0840.22032 |
| [19] | T. K. Howard, Lifting of characters on p-adic orthogonal and metaplectic groups, Compos. Math. 146 (2010), no. 3, 795-810. · Zbl 1192.22008 |
| [20] | A. W. Knapp, Local Langlands correspondence: The Archimedean case, Motives (Seattle 1991), Proc. Sympos. Pure Math. 55, American Mathematical Society, Providence (1994), 393-410. · Zbl 0811.11071 |
| [21] | R. E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365-399. · Zbl 0577.10028 |
| [22] | S. Kudla, Notes on the local theta correspondence, 1996, http://www.math.toronto.edu/skudla/castle.pdf. |
| [23] | W.-W. Li, Transfert d’intégrales orbitales pour le groupe métaplectique, Compos. Math. 147 (2011), no. 2, 524-590. · Zbl 1216.22009 |
| [24] | W.-W. Li, La formule des traces pour les revêtements de groupes réductifs connexes. II. Analyse harmonique locale, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 5, 787-859. · Zbl 1330.11037 |
| [25] | W.-W. Li, Le lemme fondamental pondéré pour le groupe métaplectique, Canad. J. Math. 64 (2012), no. 3, 497-543. · Zbl 1273.11087 |
| [26] | W.-W. Li, La formule des traces pour les revêtements de groupes réductifs connexes III: Le développement spectral fin, Math. Ann. 356 (2013), no. 3, 1029-1064. · Zbl 1330.11038 |
| [27] | W.-W. Li, La formule des traces pour les revêtements de groupes réductifs connexes. I. Le développement géométrique fin, J. reine angew. Math. 686 (2014), 37-109. · Zbl 1295.22027 |
| [28] | W.-W. Li, La formule des traces pour les revêtements de groupes réductifs connexes. IV. Distributions invariantes, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 6, 2379-2448. · Zbl 1315.11041 |
| [29] | W.-W. Li, La formule des traces stable pour le groupe métaplectique: les termes elliptiques, Invent. Math. 202 (2015), no. 2, 743-838. · Zbl 1386.11074 |
| [30] | W.-W. Li, Spectral transfer for metaplectic groups. I. Local character relations, J. Inst. Math. Jussieu 18 (2019), no. 1, 25-123. · Zbl 1409.22014 |
| [31] | C. Luo, Spherical fundamental lemma for metaplectic groups, Canad. J. Math. 70 (2018), no. 4, 898-924. · Zbl 1402.22009 |
| [32] | C. Moeglin and D. Renard, Sur les paquets d’Arthur des groupes classiques et unitaires non quasi-déployés, Relative aspects in representation theory, Langlands functoriality and automorphic forms, Lecture Notes in Math. 2221, Springer, Cham (2018), 341-361. · Zbl 1457.11063 |
| [33] | D. Renard, Endoscopy for {\rm Mp}(2n,{\mathbf{R}}), Amer. J. Math. 121 (1999), no. 6, 1215-1243. · Zbl 0942.22010 |
| [34] | J. P. Schultz, Lifting of characters of SL(2)({F}) and {SO}(1,2)({F}) for F a nonarchimedean local field, ProQuest LLC, Ann Arbor, 1998; Thesis (Ph.D.)-University of Maryland, College Park. |
| [35] | F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for p-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273-330. · Zbl 0780.22005 |
| [36] | S. W. Shin, Automorphic Plancherel density theorem, Israel J. Math. 192 (2012), no. 1, 83-120. · Zbl 1300.22006 |
| [37] | J.-L. Waldspurger, Correspondance de Shimura, J. Math. Pures Appl. (9) 59 (1980), no. 1, 1-132. · Zbl 0412.10019 |
| [38] | J.-L. Waldspurger, Correspondances de Shimura et quaternions, Forum Math. 3 (1991), no. 3, 219-307. · Zbl 0724.11026 |
| [39] | J.-L. Waldspurger, Intégrales orbitales nilpotentes et endoscopie pour les groupes classiques non ramifiés, Astérisque 269, Société Mathématique de France, Paris 2001. · Zbl 0965.22012 |
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