The Local Langlands Conjecture for $G_2$
Forum of Mathematics, Pi, Tome 11 (2023)

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We prove the local Langlands conjecture for the exceptional group $G_2(F)$ where F is a non-archimedean local field of characteristic zero.
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Wee Teck Gan; Gordan Savin. The Local Langlands Conjecture for $G_2$. Forum of Mathematics, Pi, Tome 11 (2023). doi: 10.1017/fmp.2023.27

[A] Arthur, J., The Endoscopic Classification of Representations. Orthogonal and Symplectic Groups, American Mathematical Society Colloquium Publications, Vol. 61 (American Mathematical Society, Providence, RI, 2013), xviii+590 pp.Google Scholar

[AB] Adams, J. and Barbasch, D., ‘Reductive dual pair correspondence for complex groups’, J. Funct. Anal. 132(1) (1995), 1–42.Google Scholar | DOI

[AMS] Aubert, A. M., Moussaoui, A. and Solleveld, M., ‘Generalizations of the Springer correspondence and cuspidal Langlands parameters’, Manuscripta Math. 157(1–2) (2018), 121–192.Google Scholar | DOI

[AX] Aubert, A.M. and Xu, Y. J., ‘The explicit local Langlands correspondence for’, Preprint, 2022, arXiv, https://arxiv.org/pdf/2208.12391.pdf.Google Scholar

[BM] Barbasch, D. and Moy, A., ‘Whittaker models with an Iwahori-fixed vector’, Contemp. Math. 177 (1994), 101–105.Google Scholar | DOI

[BG] Buzzard, K. and Gee, T., ‘The conjectural connections between automorphic representations and Galois representations’, in Automorphic Forms and Galois Representations, Vol. 1, London Math. Soc. Lecture Note Ser., Vol. 414 (Cambridge Univ. Press, Cambridge, 2014), 135–187.Google Scholar | DOI

[BHV] Bekka, B., De La Harpe, P. and Valette, A., Kazhdan’s Property (T) (Cambridge Univ. Press, Cambridge, 2008).Google Scholar | DOI

[Ca] Caraiani, A., ‘Local-global compatibility and the action of monodromy on nearby cycles’, Duke Math. J. 161(12) (2012), 2311–2413.Google Scholar | DOI

[CG] Chan, P.S. and Gan, W.T., ‘The local Langlands conjecture for GSp(4) III: Stability and twisted endoscopy’, J. Number Theory 146 (2015), 69–133.Google Scholar | DOI

[CG1] Chenevier, G. and Gan, W. T., ‘is unacceptable’, Preprint, 2023, .Google Scholar | arXiv

[CG2] Chenevier, G. and Gan, W. T., ‘Triality and functoriality’, In preparation.Google Scholar

[CS] Chan, K. Y. and Savin, G., ‘Iwahori model of Bessel model spaces’, Proceedings of AMS 148(4) (2020), 1487–1497 Google Scholar | DOI

[C] Chenevier, G., ‘Subgroups of Spin(7) or SO(7) with each element conjugate to some element of and applications to automorphic forms’, Doc. Math. 24 (2019), 95–161.Google Scholar | DOI

[CKPSS] Cogdell, J. W., Kim, H., Piatetski-Shapiro, I. and Shahidi, F., ‘Functoriality for the classical groups’, Publ. Math. IHES. 99 (2004), 163–233.Google Scholar | DOI

[CZ1] Chen, R. and Zou, J. L., ‘Local Langlands correspondence for even orthogonal groups via theta lifts’, Selecta Math. (N.S.) 27(5) (2021), Paper No. 88, 71 pp.Google Scholar | DOI

[CZ2] Chen, R. and Zou, J. L., ‘Local Langlands correspondence for unitary groups via theta lifts’, Represent. Theory 25 (2021), 861–896.Google Scholar | DOI

[Co] Conway, J. B., A Course in Functional Analysis , second edn., Graduate Texts in Mathematics , Vol. 96 (Springer-Verlag, New York, 1990).Google Scholar

[FS] Fargues, L. and Scholze, P., ‘Geometrization of the local Langlands correspondence’, Preprint, 2021, .Google Scholar | arXiv

[G] Gan, W. T., ‘Exceptional Howe correspondences over finite fields’, Compositio Math. 118(3) (1999), 323–344.Google Scholar | DOI

[GI] Gan, W. T. and Ichino, A., ‘The Shimura–Waldspurger correspondence for ’, Ann. Math. Second Series 188(3) (2018), 965–1016.Google Scholar | DOI

[GQT] Gan, W. T., Qiu, Y. N. and Takeda, S., ‘The regularized Siegel–Weil formula (the second term identity) and the Rallis inner product formula’, Invent. Math. 198(3) (2014), 739–831.Google Scholar | DOI

[GS20] Gan, W. T. and Savin, G., ‘An exceptional Siegel–Weil formula and poles of the Spin -function of ’, Compos. Math. 156(6) (2020), 1231–1261.Google Scholar | DOI

[GS23] Gan, W. T. and Savin, G., ‘Howe duality and dichotomy for exceptional theta correspondence’, Invent. Math. 232(1) (2023), 1–78.Google Scholar | DOI

[GS24] Gan, W. T. and Savin, G., ‘A theory of -factors for ’, Preprint, 2023, .Google Scholar | arXiv

[GT] Gan, W. T. and Takeda, S., ‘The local Langlands conjecture for ’, Ann. of Math. (2) 173(3) (2011), 1841–1882.Google Scholar | DOI

[GTW] Gan, W. T. and Tantono, W., ‘The local Langlands conjecture for , II: The case of inner forms’, Amer. J. Math. 136(3) (2014), 761–805.Google Scholar | DOI

[GeSh] Gelbart, S. and Shahidi, F., Analytic Properties of Automorphic -functions, Perspectives in Mathematics, Vol. 6 (Academic Press, Boston, 1988).Google Scholar

[Gr95] Griess, R. L. Jr., ‘Basic conjugacy theorems for ’, Invent. Math. 121(2) (1995) 257–277.Google Scholar | DOI

[GrS1] Gross, B. H. and Savin, G., ‘The dual pair , Canad. Math. Bull. 40(3) (1997), 376–384.Google Scholar | DOI

[GrS2] Gross, B. H. and Savin, G., ‘Motives with Galois group of type G2: An exceptional theta-correspondence’, Compos. Math. 114 (1998), 153–217.Google Scholar | DOI

[HS] Hanzer, M. and Savin, G., ‘Eisenstein series arising from Jordan algebras’, Canad. J. Math. 72(1) (2020), 183–201.Google Scholar | DOI

[HKT] Harris, M., Khare, C. and Thorne, J., ‘A local Langlands parameterization for generic supercuspidal representations of -adic ’, Annales de l’ ENS 56(1) (2023), 257–286.Google Scholar

[HO] Heiermann, V. and Opdam, E., ‘On the tempered -functions conjecture’, Amer. J. Math. 135(3) (2013), 777–799.Google Scholar | DOI

[He] Henniart, G., ‘Une preuve simple des conjectures de Langlands pour sur un corps p-adique’, Invent. Math. 139(2) (2000), 439–455.Google Scholar | DOI

[HL] Hundley, J. and Liu, B. Y., ‘On automorphic descent from to ’, J. European Math. Society, To appear.Google Scholar

[HPS] Huang, J. S., Pandžić, P. and Savin, G., ‘New dual pair correspondences’, Duke Math. J. 82 (1996), 447–471.Google Scholar | DOI

[HT] Harris, M. and Taylor, R., The Geometry and Cohomology of Some Simple Shimura Varieties, Annals of Mathematics Studies, Vol. 151 (Princeton University Press, Princeton, NJ, 2001), viii+276 pp. With an appendix by Vladimir G. Berkovich.Google Scholar

[ILM] Ichino, A., Lapid, E. and Mao, Z., ‘On the formal degrees of square-integrable representations of odd special orthogonal and metaplectic groups’, Duke Math. J. 166(7) (2017), 1301–1348.Google Scholar | DOI

[K1] Kaletha, T., ‘Regular supercuspidal representations’, J. Amer. Math. Soc. 32(4) (2019), 1071–1170.Google Scholar | DOI

[K2] Kaletha, T., ‘Supercuspidal L-packets’, Preprint, 2019, .Google Scholar | arXiv

[KMSW] Kaletha, T., Minguez, A., Shin, S. W. and White, P. J., ‘Endoscopic classification of representations: inner forms of unitary groups’, Preprint, 2014, .Google Scholar | arXiv

[KS] Kret, A. and Shin, S. W., ‘Galois representations for general symplectic groups’, J. Eur. Math. Soc. (JEMS) 25(1) (2023), 75–152.Google Scholar | DOI

[Li] Li, J.-S., ‘On the classification of irreducible low rank unitary representations of classical groups’, Compositio Math. 71(1) (1989), 29–48.Google Scholar

[Mi] Miličić, D., ‘On -algebras with bounded trace’, Glasnik Mat. 8 (28) (1973), 7–22.Google Scholar

[MR] Moeglin, 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 Math., Vol. 2221 (CIRM Jean-Morlet Ser., Springer, Cham, 2018), 341–361.Google Scholar | DOI

[M] Mok, C. P., ‘Endoscopic classification of representations of quasi-split unitary groups’, Mem. Amer. Math. Soc. 235(1108) (2015), vi+248 pp.Google Scholar

[Ro] Rodier, F., ‘Decomposition of principal series for reductive -adic groups and the Langlands’ classification’, in Operator Algebras and Group Representations, Vol. II (Neptun, 1980), Monogr. Stud. Math., Vol. 18 (Pitman, Boston, MA, 1984), 86–94.Google Scholar

[SV] Sakellaridis, Y. and Venkatesh, A., Periods and Harmonic Analysis on Spherical Varieties, Asterisque 396 (Math. Soc. France, Paris, 2017).Google Scholar

[SW] Savin, G. and Weissman, M., ‘Dichotomy for generic supercuspidal representations of ’, Compositio Math. (2010) 1–49.Google Scholar

[S] Shin, S. W., ‘Automorphic Plancherel density theorem’, Israel J. Math. 192(1) (2012), 83–120.Google Scholar | DOI

[SZ] Sun, B. Y. and Zhu, C. B., ‘Conservation relations for local theta correspondence’, J. Amer. Math. Soc. 28(4) (2015), 939–983.Google Scholar | DOI

[Ta1] Tadić, M., ‘On limits of characters of irreducible unitary representations’, Glasnik Mat. 23 (43) (1988), 15–25.Google Scholar

[Ta2] Tadić, M., ‘Geometry of dual spaces of reductive groups (non-Archimedean case)’, J. Analyse Math. 51 (1988), 139–181.Google Scholar | DOI

[TY] Taylor, R. L. and Yoshida, T., ‘Compatibility of local and global Langlands correspondences’, J. Amer. Math. Soc. 20(2) (2007), 467–493.Google Scholar | DOI

[Vo] Vogan, D., ‘Isolated unitary representations’, Automorphic Forms and Applications, IAS/Park City Math. Ser., Vol. 12 (Amer. Math. Soc., Providence, RI, 2007), 379–398.Google Scholar

[Xu1] Xu, B., ‘On lifting problem of –packets’, Compositio Math. 152(9) (2016), 1800–1850.Google Scholar | DOI

[Xu2] Xu, B., ‘–packets of quasisplit and ’, Math. Ann. 370(1–2) (2018), 71–189.Google Scholar | DOI

[Xu3] Xu, B., ‘Global L-packets of quasi-split and ’, American J. of Math., To appear.Google Scholar | arXiv

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