On the Notion of Conductor in the Local Geometric Langlands Correspondence

Kamgarpour, Masoud

Canadian journal of mathematics, Tome 69 (2017) no. 1, pp. 107-129

Résumé

Under the local Langlands correspondence, the conductor of an irreducible representation of $\text{G}{{\text{l}}_{n}}\left( F \right)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Détail

DOI : 10.4153/CJM-2016-016-1

Mots-clés : 17B67, 17B69, 22E50, 20G25, local geometric Langlands, connections, cyclic vectors, opers, conductors, Segal-Sugawara operators, Chervov-Molev operators, critical level, smooth representations, affine Kac-Moody algebra, categorical representations

Bibliographie

[1] [1] Beilinson, A. and Drinfeld, V., Quantization of Hitchin's Integrable system and Heche eigensheaves. 1997. http://www.math.uchicago.edu/-mitya/langlands/hitchin/BD-hitchin.pdf Google Scholar

[2] [2] Beilinson, A.,Opers. arxiv:math/0501398. Google Scholar

[3] [3] Casselman, W., On some results ofAtkin and Lehner. Math. Ann. 201(1973), 301–314. Google Scholar | DOI

[4] [4] Chen, T.-H. and Kamgarpour, M., Preservation of depth in local geometric Langlands correspondence. arxiv:1404.0598 Google Scholar

[5] [5] Chervov, A. V. and Molev, A. I., On higher-order Sugawara operators. Int. Math. Res. Not. IMRN 2009, no. 9, 1612–1635. http://dx.doi.Org/10.1093/imrn/rnn168 Google Scholar

[6] [6] Chervov, A. V. and Talalaev, D., Quantum spectral curves, quantum integrable systems and the geometric Langlands correspondence. arxiv:hep-th/0604128 Google Scholar

[7] [7] Deligne, P., Équations différentielles á points singuliers réguliers. Lecture Notes in Mathematics, 163, Springer-Verlag, Berlin, 1970. Google Scholar

[8] [8] Drinfeld, V. G. and Sokolov, V. V., Lie algebras and equations of Korteweg-de Vries type. Current problems in mathematics, 24, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984, pp. 81–180 Google Scholar

[9] [9] Feigin, B. and Frenkel, E., Affine Kac-Moody algebras at the critical level and Gelfand-Dikii algebras. In: Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., 16, World Sci. Publ., River Edge, NJ, 1992, pp. 197–215. Google Scholar

[10] [10] Frenkel, E., Affine Kac-Moody algebras, integrable systems, and their deformations. 2002. https://math.berkeley.edu/~frenkel/lecture.pdf Google Scholar

[11] [11] Frenkel, E., Langlands correspondence for loop groups. Cambridge Studies in Advanced Mathematics, 103, Cambridge University Press, Cambridge, 2007. Google Scholar

[12] [12] Frenkel, E. and Ben-Zvi, D., Vertex algebras and algebraic curves. Mathematical Surveys and Mono¬graphs, 88, American Mathematical Society, Providence, RI, 2004. Google Scholar

[13] [13] Frenkel, E. and Gaitsgory, D., Local geometric Langlands correspondence and affine Kac-Moody algebras. In: Algebraic geometry and number theory, Progr. Math., 253, Birkhäuser Boston, Boston, MA, 2006, pp. 69–260. http://dx.doi.Org/10.1090/surv/088 Google Scholar

[14] [14] Frenkel, E. and Zhu, X., Any flat bundle on a punctured disc has an oper structure. Math. Res. Lett. 17(2010), no. 1, 27–37. Google Scholar | DOI

[15] [15] Gaitsgory, D. and Barlev, J., Notes on Opers – seminar on quantization of the Hitchin system. 2009. http://www.math,harvard.edu/~gaitsgde/grad_2009/SeminarNotes/March2-9(Opers).pdf. Google Scholar

[16] [16] Gross, B. and Reeder, M., Arithmetic invariants of discrete Langlands parameters. Duke Math. J. 154(2010), no. 3, 431–508. Google Scholar | DOI

[17] [17] Jacquet, H., Piatetski-Shapiro, I., and Shalika, J., Conducteur des représentations du groupe linéaire. Math. Ann. 256(1981), no. 2, 199–214. Google Scholar | DOI

[18] [18] Kac, V. G., Infinite-dimensional Lie algebras. Third ed., Cambridge University Press, Cambridge,1990. Google Scholar | DOI

[19] [19] Kamgarpour, M., Compatibility of Feigin-Frenkel Isomorphism and Harish-Chandra Isomorphism for jet algebras. Transactions of AMS, 2014. Google Scholar

[20] [20] Kamgarpour, M. and Schedler, T., Geometrization of principal series representations of reductive groups. Annales de linstitut Fourier, 2015. http://dx.doi.Org/10.58O2/aif.2988 Google Scholar

[21] [21] Katz, N. M., On the calculation of some differential Galois groups. Invent. Math. 87(1987), no. 1, 13–61. http://dx.doi.Org/10.1007/BF01389152 Google Scholar

[22] [22] K. S.|Kedlaya, p-adic differential equations. Cambridge Studies in Advanced Mathematics, 125,Cambridge University Press, Cambridge, 2010. http://dx.doi.Org/10.1017/CBO9780511750922 Google Scholar

[23] [23] Komatsu, H., On the index of ordinary differential operators. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18(1971), 379–398. Google Scholar

[24] [24] Luu, M., Local Langlands duality and a duality of conformai field theories. 2015. arxiv:1506.00663 Google Scholar

[25] [25] Malgrange, B., Sur les points singuliers des équations différentielles. Enseignement Math.(2) 20(1974), 147–176. Google Scholar

[26] [26] Tsai, P.-Y., On new forms for split special odd orthogonal groups. Ph. D. thesis, Harvard University, 2013. http://math.harvard.edu/~pytsai/FixedVectors.pdf Google Scholar

[27] [27] Wedhorn, T., The local Langlands correspondence for GL(n) over p-adic fields. In: School on Auto-morphic Forms on GL(n), ICTP Lect. Notes, 21, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2008, pp. 237–320. Google Scholar