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Frenkel, Edward; Gross, Benedict

A rigid irregular connection on the projective line. (English) Zbl 1209.14017

Ann. Math. (2) 170, No. 3, 1469-1512 (2009).
The Langlands correspondence relates automorphic representations of a reductive group \(G\) over the ring of adèles of a global field \(F\) and \(\ell\)-adic representations of the Galois group of \(F\) with values in a dual group of \(G\). In some cases there is a unique irreducible automorphic representation with prescribed local behavior at finitely many points.
A special case of this is when \(F\) is the function field of the projective line \(\mathbb{P}^1\) over a finite field \(k\), and \(G\) is a simple group over \(k\). The problem is then to determine the corresponding family of \(\ell\)-adic representations of the Galois group of \(F\).
The authors construct in this paper a connection \(\nabla\) on the trivial \(G\)-bundle on \(\mathbb{P}^1\) for any simple complex algebraic group \(G\), which is regular outside of the points \(0\) and \(\infty\), has a regular singularity at \(0\) (with principal unipotent monodromy) and an irregular singularity at \(\infty\), with slope the reciprocal of the Coxeter number of \(G\).
They compute the de Rham cohomology of this connection with values in a representation \(V\) of \(G\) and describe the differential Galois group of \(\nabla\) as a subgroup of \(G\).
Reviewer: Alexandru Dimca (Nice)

Cited in 6 Reviews
Cited in 29 Documents

MSC:

14F40 de Rham cohomology and algebraic geometry
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
11R39 Langlands-Weil conjectures, nonabelian class field theory

Keywords:

Galois representation; irregular connection; simple algebraic group; Langlands correspondence.
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\textit{E. Frenkel} and \textit{B. Gross}, Ann. Math. (2) 170, No. 3, 1469--1512 (2009; Zbl 1209.14017)
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References:

[1] D. Arinkin, ”Fourier transform and middle convolution for irregular \(\mathcal D\)-modules,” , preprint , 2008.
[2] P. P. Boalch, ”Stokes matrices, Poisson Lie groups and Frobenius manifolds,” Invent. Math., vol. 146, iss. 3, pp. 479-506, 2001. · Zbl 1044.53060 · doi:10.1007/s002220100170
[3] A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves. · Zbl 0864.14007
[4] A. Beilinson and D. Drinfeld, ”Opers,” , preprint , 2005.
[5] S. Bloch and H. Esnault, ”Local Fourier transforms and rigidity for \(\mathcal D\)-modules,” Asian J. Math., vol. 8, iss. 4, pp. 587-605, 2004. · Zbl 1082.14506 · doi:10.4310/AJM.2004.v8.n4.a16
[6] R. W. Carter, Finite Groups of Lie Type, New York: John Wiley & Sons, 1985. · Zbl 0567.20023
[7] P. Deligne, ”Application de la formule des traces aux sommes trigonometriques,” in Cohomology Étale, New York: Springer-Verlag, 1977, vol. 569, pp. 168-239. · Zbl 0349.10031
[8] P. Deligne, Équations Différentielles à Points Singuliers Réguliers, New York: Springer-Verlag, 1970, vol. 163. · Zbl 0244.14004 · doi:10.1007/BFb0061194
[9] P. Deligne and J. Milne, ”Tannakian categories,” Lecture Notes in Math., vol. 900, pp. 101-228, 1982. · Zbl 0477.14004
[10] V. Drinfeld and V. Sokolov, ”Lie algebras and KdV type equations,” J. Soviet Math., vol. 30, pp. 1975-2036, 1985. · Zbl 0578.58040
[11] E. B. Dynkin, ”Semisimple subalgebras of semisimple Lie algebras (Russian),” Mat. Sb., vol. 30, pp. 349-462, 1952. · Zbl 0048.01701
[12] B. Feigin and E. Frenkel, ”Quantization of soliton systems and Langlands duality,” , preprint , 2007. · Zbl 1245.81062
[13] B. Feigin, E. Frenkel, and L. Rybinikov, ”Opers with irregular singularity and spectra of the shift of argument subalgebra,” , preprint , 2007.
[14] B. Feigin, E. Frenkel, and V. Toledano Laredo, ”Gaudin models with irregular singularities,” , preprint , 2006. · Zbl 1186.81065 · doi:10.1016/j.aim.2009.09.007
[15] E. Frenkel, Langlands Correspondence for Loop Groups, Cambridge: Cambridge Univ. Press, 2007. · Zbl 1133.22009
[16] E. Frenkel, ”Lectures on the Langlands program and conformal field theory,” in Frontiers in Number Theory, Physics, and Geometry. II, New York: Springer-Verlag, 2007, pp. 387-533. · Zbl 1196.11091
[17] E. Frenkel and D. Gaitsgory, ”Local geometric Langlands correspondence and affine Kac-Moody algebras,” in Algebraic geometry and number theory, Boston, MA: Birkhäuser, 2006, pp. 69-260. · Zbl 1184.17011 · doi:10.1007/978-0-8176-4532-8_3
[18] E. Frenkel and X. Zhu, ”Any flat bundle on a punctured disc has an oper structure,” , preprint , 2008. · Zbl 1220.14013 · doi:10.4310/MRL.2010.v17.n1.a3
[19] S. I. Gelfand and Y. I. Manin, Homological Algebra, New York: Springer-Verlag, 1994. · Zbl 0787.18008
[20] B. H. Gross, ”On the motive of \(G\) and the principal homomorphism \({ SL}_2\to\widehat G\),” Asian J. Math., vol. 1, iss. 1, pp. 208-213, 1997. · Zbl 0942.20031
[21] B. H. Gross, ”On the motive of a reductive group,” Invent. Math., vol. 130, iss. 2, pp. 287-313, 1997. · Zbl 0904.11014 · doi:10.1007/s002220050186
[22] B. H. Gross, ”Irreducible cuspidal representations with prescribed local behavior,” , preprint. · Zbl 1228.22017 · doi:10.1353/ajm.2011.0035
[23] B. H. Gross, ”Algebraic modular forms,” Israel J. Math., vol. 113, pp. 61-93, 1999. · Zbl 0965.11020 · doi:10.1007/BF02780173
[24] B. H. Gross and M. Reeder, ”Arithmetic invariants of discrete Langlands parameters,” , preprint. · Zbl 1207.11111 · doi:10.1215/00127094-2010-043
[25] V. G. Kac, ”Infinite-dimensional algebras, Dedekind’s \(\eta\)-function, classical Möbius function and the very strange formula,” Adv. Math., vol. 30, pp. 85-136, 1978. · Zbl 0391.17010 · doi:10.1016/0001-8708(78)90033-6
[26] V. G. Kac, Infinite-dimensional Lie Algebras, Third ed., Cambridge: Cambridge Univ. Press, 1990. · Zbl 0716.17022
[27] N. M. Katz, ”On the calculation of some differential Galois groups,” Invent. Math., vol. 87, iss. 1, pp. 13-61, 1987. · Zbl 0609.12025 · doi:10.1007/BF01389152
[28] N. M. Katz, Exponential sums and differential equations, Princeton, NJ: Princeton Univ. Press, 1990. · Zbl 0731.14008 · doi:10.1515/9781400882434
[29] B. Kostant, ”Lie group representations on polynomial rings,” Amer. J. Math., vol. 85, pp. 327-404, 1963. · Zbl 0124.26802 · doi:10.2307/2373130
[30] R. Kottwitz, ”Tamagawa numbers,” Ann. of Math., vol. 127, pp. 629-646, 1988. · Zbl 0678.22012 · doi:10.2307/2007007
[31] J. Saxl and G. M. Seitz, ”Subgroups of groups containing a principal unipotent element,” J. London Math. Soc., vol. 55, pp. 370-386, 1997. · Zbl 0955.20033 · doi:10.1112/S0024610797004808
[32] T. Springer, ”Regular elements of finite reflection groups,” Invent. Math., vol. 25, pp. 159-198, 1974. · Zbl 0287.20043 · doi:10.1007/BF01390173
[33] N. M. Katz, Rigid Local Systems, Princeton, NJ: Princeton Univ. Press, 1996. · Zbl 0864.14013 · doi:10.1515/9781400882595
[34] A. Weil, Dirichlet Series and Automorphic Forms, New York: Springer-Verlag, 1971, vol. 189. · Zbl 0218.10046 · doi:10.1007/BFb0061201
[35] E. Witten, ”Gauge theory and wild ramification,” Anal. Appl. \((\)Singapore\()\), vol. 6, iss. 4, pp. 429-501, 2008. · Zbl 1177.81101 · doi:10.1142/S0219530508001195
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