Kloosterman sheaves for reductive groups. (English) Zbl 1272.14012

Fix a prime \(p\), a positive integer \(n\), a finite field extension \(\mathbb F_q/\mathbb F_p\) of the field with \(p\) elements, and an element \(a\in\mathbb F_q^\times\). The Kloosterman sums \[ \text{Kl}_n(a; q) = (-1)^{n-1} \sum_{x_i\in\mathbb F_q, \prod x_i=a} \exp\left(\frac{2\pi i}p \text{Tr}_{\mathbb F_q/\mathbb F_p}(x_1+\cdots +x_n) \right). \] are of great importance in number theory and have been intensely studied since their introduction by H. Kloosterman in 1926 [Proc. Lond. Math. Soc. (2) 25, 143–173 (1926; JFM 52.0170.02)]. For instance, they occur in the Fourier coefficients of modular forms. A powerful method to study them, and to prove results such as the Weil bound \(| \text{Kl}_n(a; q) | \le nq^{(n-1)/2}\), is a sheaf-theoretic interpretation (“sheaf-function dictionary”) of Kloosterman sums. This permits to use the tool box of algebraic geometry, and also adds considerable insight from a conceptual point of view. The key is Deligne’s definition of the Kloosterman sheaf \(\text{Kl}_n = R\pi_! \sigma^* \text{AS}_\psi[n-1]\), where \(\mathbb G_a\) and \(\mathbb G_m\) are the additive and multiplicative group over \(\mathbb F_p\), \(\sigma: \mathbb G_m^n\rightarrow\mathbb G_a\) is the sum, \(\pi: \mathbb G_m^n\rightarrow \mathbb G_m\) the product, \(\text{AS}_\psi\) is the Artin-Schreier local system attached to a nontrivial character \(\psi: \mathbb F_p\rightarrow \mathbb Q_{\ell}(\mu_p)^\times\) (\(\ell\) a prime \(\neq p\)). After fixing an embedding \(\iota: \mathbb Q_{\ell}(\mu_p) \rightarrow\mathbb C\) such that \(\iota(\psi(x)) = \exp(2\pi i/p)\) for \(x\in\mathbb F_p\), the Grothendieck-Lefschatz trace formula shows that the Kloosterman sum above can be expressed as the trace of Frobenius \(\text{Frob}_a\) on the stalk at a geometric point over \(a\).
The Kloosterman sheaves were studied by Deligne, Katz and others. For example, Katz determined the Zariski closure of the image of the monodromy representation \(\pi_1(\mathbb G_m, \overline{\eta}) \rightarrow \mathrm{GL}_n(\mathbb Q_\ell(\mu_p))\), where \(\overline{\eta}\) is a geometric point of \(\mathbb G_m\). Depending on \(n\) and \(p\), the algebraic group arising in this way can be \(\mathrm{SL}_n\), \(\mathrm{Sp}_n\), \(\mathrm{SO}_n\) or, quite surprisingly, of Dynkin type \(G_2\). The \(G_2\) case occurs for \(n=7\), \(p=2\).
The goal of the paper at hand is to study the following question, raised by Katz: Do all semisimple groups appear as geometric monodromy groups of local systems on \(\mathbb G_m\)? Using the sheaf-function dictionary, one can translate this question to the question whether there exist exponential sums whose equidistribution laws are governed by arbitrary simple groups, in particular exceptional groups.
In the paper under review, a uniform construction of such local systems is given: For a split reductive group \(G^\vee\) a \(G^\vee\)-local system \(\text{Kl}_{G^\vee}\) on \(\mathbb G_m\) is constructed. This sheaf has similar local ramification properties as \(\text{Kl}_n\), and for \(G^\vee=GL_n\) the construction yields the sheaf \(\text{Kl}_n\) as defined by Deligne. The authors determine the Zariski closure of the global geometric monodromy, prove purity of the sheaf and deduce equidistribution laws. They also give a conjecture about the unicity of such local systems.
Roughly speaking, the construction is based on work of B. H. Gross and M. Reeder [Duke Math. J. 154, No. 3, 431–508 (2010; Zbl 1207.11111)] who construct an automorphic representation which should correspond to the (yet to be defined) Kloosterman sheaf. Writing down explicitly a Hecke eigenfunction for this automorphic representation, the authors are able to find a Hecke eigensheaf in terms of which the desired Kloosterman sheaf can be defined.
For more details, we refer to the introduction of the paper.


14D24 Geometric Langlands program (algebro-geometric aspects)
11L05 Gauss and Kloosterman sums; generalizations
11T23 Exponential sums
22E57 Geometric Langlands program: representation-theoretic aspects
Full Text: DOI arXiv


[1] S. Arkhipov and R. Bezrukavnikov, ”Perverse sheaves on affine flags and Langlands dual group,” Israel J. Math., vol. 170, pp. 135-183, 2009. · Zbl 1214.14011
[2] A. Beuilinson and J. Bernstein, ”A proof of Jantzen conjectures,” in I. M. Gel\('\)fand Seminar, Providence, RI: Amer. Math. Soc., 1993, vol. 16, pp. 1-50. · Zbl 0790.22007
[3] A. A. Beuilinson, J. Bernstein, and P. Deligne, ”Faisceaux pervers,” in Analysis and Topology on Singular Spaces, I, Paris: Soc. Math. France, 1982, vol. 100, pp. 5-171. · Zbl 0536.14011
[4] R. Bezrukavnikov, A. Braverman, and I. Mirkovic, ”Some results about geometric Whittaker model,” Adv. Math., vol. 186, iss. 1, pp. 143-152, 2004. · Zbl 1071.20039
[5] R. Bezrukavnikov, ”On tensor categories attached to cells in affine Weyl groups,” in Representation Theory of Algebraic Groups and Quantum Groups, Tokyo: Math. Soc. Japan, 2004, vol. 40, pp. 69-90. · Zbl 1078.20044
[6] N. Bourbaki, Éléments de Mathématique. Fasc. XXXIV. Groupes et Algèbres de Lie. Chapitre IV: Groupes de Coxeter et Systèmes de Tits. Chapitre V: Groupes Engendrés par des Réflexions. Chapitre VI: Systèmes de Racines, Paris: Hermann, 1968, vol. 1337. · Zbl 0186.33001
[7] F. Bruhat and J. Tits, ”Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée,” Inst. Hautes Études Sci. Publ. Math., vol. 60, pp. 5-184, 1984. · Zbl 0597.14041
[8] R. W. Carter, Finite Groups of Lie Type. Conjugacy Classes and Complex Characters, New York: A Wiley-Interscience Publication, John Wiley & Sons, 1985. · Zbl 0567.20023
[9] C. Chevalley, ”Sur certains groupes simples,” Tôhoku Math. J., vol. 7, pp. 14-66, 1955. · Zbl 0066.01503
[10] P. Deligne, Cohomologie Étale, New York: Springer-Verlag, 1977, vol. 569. · Zbl 0349.14008
[11] P. Deligne, ”La conjecture de Weil. II,” Inst. Hautes Études Sci. Publ. Math., vol. 52, pp. 137-252, 1980. · Zbl 0456.14014
[12] P. Deligne and J. S. Milne, ”Tannakian Categories,” in Hodge Cycles, Motives, and Shimura Varieties, New York: Springer-Verlag, 1982, vol. 900, pp. 101-228. · Zbl 0465.00010
[13] M. Demazure and A. Grothendieck, Schémas en Groupes. III: Structure des Schémas en Groupes Réductifs, New York: Springer-Verlag, 1962/1964, vol. 153. · Zbl 0212.52810
[14] G. Faltings, ”Algebraic loop groups and moduli spaces of bundles,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 5, iss. 1, pp. 41-68, 2003. · Zbl 1020.14002
[15] E. Frenkel and B. Gross, ”A rigid irregular connection on the projective line,” Ann. of Math., vol. 170, iss. 3, pp. 1469-1512, 2009. · Zbl 1209.14017
[16] D. Gaitsgory, ”Construction of central elements in the affine Hecke algebra via nearby cycles,” Invent. Math., vol. 144, iss. 2, pp. 253-280, 2001. · Zbl 1072.14055
[17] D. Gaitsgory, ”On de Jong’s conjecture,” Israel J. Math., vol. 157, pp. 155-191, 2007. · Zbl 1123.11020
[18] V. A. Ginzburg, Perverse sheaves on a Loop group and Langlands’ duality. · Zbl 0736.22009
[19] U. Görtz and T. J. Haines, ”The Jordan-Hölder series for nearby cycles on some Shimura varieties and affine flag varieties,” J. Reine Angew. Math., vol. 609, pp. 161-213, 2007. · Zbl 1157.14013
[20] B. H. Gross and M. Reeder, ”Arithmetic invariants of discrete Langlands parameters,” Duke Math. J., vol. 154, iss. 3, pp. 431-508, 2010. · Zbl 1207.11111
[21] B. H. Gross, ”Irreducible cuspidal representations with prescribed local behavior,” Amer. J. Math., vol. 133, iss. 5, pp. 1231-1258, 2011. · Zbl 1228.22017
[22] B. H. Gross, Letter to Deligne and Katz.
[23] T. Haines and M. Rapoport, ”Appendix to: Twisted loop groups and their affine flag varieties, by G. Pappas and M. Rapoport,” Adv. Math., vol. 219, iss. 1, pp. 118-198, 2008. · Zbl 1159.22010
[24] G. Harder, ”Halbeinfache Gruppenschemata über vollständigen Kurven,” Invent. Math., vol. 6, pp. 107-149, 1968. · Zbl 0186.25902
[25] J. Heinloth, ”Uniformization of \(\mathcalG\)-bundles,” Math. Ann., vol. 347, iss. 3, pp. 499-528, 2010. · Zbl 1193.14014
[26] N. M. Katz, Gauss Sums, Kloosterman Sums, and Monodromy Groups, Princeton, NJ: Princeton Univ. Press, 1988, vol. 116. · Zbl 0675.14004
[27] N. M. Katz, Exponential Sums and Differential Equations, Princeton, NJ: Princeton Univ. Press, 1990, vol. 124. · Zbl 0731.14008
[28] N. M. Katz, Rigid Local Systems, Princeton, NJ: Princeton Univ. Press, 1996, vol. 139. · Zbl 0864.14013
[29] B. Kostant, ”The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group,” Amer. J. Math., vol. 81, pp. 973-1032, 1959. · Zbl 0099.25603
[30] G. Lusztig, ”Singularities, character formulas, and a \(q\)-analog of weight multiplicities,” in Analysis and Topology on Singular Spaces, II, III, Paris: Soc. Math. France, 1983, vol. 101, pp. 208-229. · Zbl 0561.22013
[31] I. Mirković and K. Vilonen, ”Geometric Langlands duality and representations of algebraic groups over commutative rings,” Ann. of Math., vol. 166, iss. 1, pp. 95-143, 2007. · Zbl 1138.22013
[32] B. C. Ngô and P. Polo, ”Résolutions de Demazure affines et formule de Casselman-Shalika géométrique,” J. Algebraic Geom., vol. 10, iss. 3, pp. 515-547, 2001. · Zbl 1041.14002
[33] G. Pappas and M. Rapoport, ”Twisted loop groups and their affine flag varieties,” Adv. Math., vol. 219, iss. 1, pp. 118-198, 2008. · Zbl 1159.22010
[34] A. Ramanathan, ”Deformations of principal bundles on the projective line,” Invent. Math., vol. 71, iss. 1, pp. 165-191, 1983. · Zbl 0492.14007
[35] M. Reeder, ”Torsion automorphisms of simple Lie algebras,” Enseign. Math., vol. 56, iss. 1-2, pp. 3-47, 2010. · Zbl 1223.17020
[36] T. A. Springer, ”Regular elements of finite reflection groups,” Invent. Math., vol. 25, pp. 159-198, 1974. · Zbl 0287.20043
[37] J-K. Yu, Smooth models associated to concave functions in Bruhat-Tits theory. · Zbl 1356.20018
[38] X. Zhu, Frenkel-Gross’ irregular connection and Heinloth-Ngô-Yun’s are the same. · Zbl 1393.14011
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