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Geometric Eisenstein series: twisted setting. (English) Zbl 1387.11082

The author generalizes the quantum geometric Langlands correspondence (for roots of unity) from the case of split torus. It geometrizes a conjectural extension of the Langlands program to metaplectic groups proposed by Weissman. Let \(k\) be an algebraically closed field, \(G\) a simple connected group, \(X\) a smooth projective curve over \(k\), and \(\mathrm{Bun}_G\) the stack of \(G\)-torsors on \(X\). For some \(n\geq1\) let \(N=2h^\vee n\), where \({h}^\vee\) is the dual Coxeter number for \(G\), and introduce a \(\mu_N\)-gerb \(\widetilde{\mathrm{Bun}}_G\to Bun_G\) that comes from the canonical Brylinski-Deligne extension of \(G\) by \(K_2\). For the parabolic subgroup \(P\subset G\) and its Levi factor \(M\) one similarly introduces \(\widetilde{\mathrm{Bun}}_P\) and \(\widetilde{\mathrm{Bun}}_M\). For an injective character \(\zeta:\mu_N^{(k)}\to\overline{\mathbb{Q}}^*_l\) the author considers the derived category of étale \(\mathbb{Q}_l\) sheaves on which \(\mu_N^{(k)}\) acts by \(\zeta\). The Eisenstein series is defined as a functor \(\mathrm{Eis}:D_\zeta(\widetilde{\mathrm{Bun}}_G)\to\widetilde{\mathrm{Bun}}_M\) using a twisted version of \(\mathrm{Bun}_P\) intersection homology sheaf \(IC_\zeta\), and the author proves that Eis commutes with the Hecke functors with respect to the dual embedding, and that the formation of Eis is transitive for the diagram \(T\subset M\subset G\), where \(T\) is a maximal torus.
For \(G=\mathrm{SL}_2\) a partial description of the Fourier coefficients of Eis in terms of a known sheaf is given, which relates them to quantum groups. As an application, the author derives a formula for the first Whitakker coefficient of the metaplectic extension of \(\mathrm{SL}_2\), which turns out to be an \(l\)-adic analog of the space of conformal blocks of the Wess-Zumino-Witten model, and can be seen as a generalization of the central value of an abelian \(L\) function. In the case of \(X=\mathbb{P}^1\) new automorphic sheaves (“theta-sheaves”) are constructed and their Hecke property is proved.

MSC:

11R39 Langlands-Weil conjectures, nonabelian class field theory
11G45 Geometric class field theory
14H60 Vector bundles on curves and their moduli
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[1] Bezrukavnikov, R., Finkelberg, M., Schechtman, V.: Factorizable Sheaves and Quantum Groups. Lecture Notes in Math. 1691, Springer (1998)Zbl 0938.17016 MR 1641131 · Zbl 0938.17016
[2] Braverman, A., Finkelberg, M., Gaitsgory, D., Mirkovic, I.: Intersection cohomology of Drinfeld compactifications. Selecta Math. (N.S.) 8, 381-418 (2002); Erratum, ibid. 10, 429-430 (2004)Zbl 1063.14503 MR 2099075 · Zbl 1031.14019
[3] Braverman, A., Gaitsgory, D.: Geometric Eisenstein series. Invent. Math. 150, 287-384 (2002)Zbl 1046.11048 MR 1933587 Geometric Eisenstein series: twisted setting3251 · Zbl 1046.11048
[4] Brubaker, B., Bump, D., Friedberg, S.: Eisenstein series, crystals, and ice. Notices Amer. Math. Soc. 58, 1563-1571 (2011)Zbl 1309.11043 MR 2896085 · Zbl 1309.11043
[5] Brylinski, J.-L., Deligne, P.: Central extensions of reductive groups by K2. Publ. Math. IHES 94, 5-85 (2001)Zbl 1093.20027 MR 1896177 · Zbl 1093.20027
[6] Bump, D.: Introduction: Multiple Dirichlet series. In: Multiple Dirichlet Series, L-functions and Automorphic Forms, Progr. Math. 300, Birkh¨auser, 1-36 (2012)Zbl 06153592 MR 2952570 · Zbl 1370.11062
[7] Deligne, P.: Cohomologie ´etale. S´eminaire de G´eom´etrie Alg´ebrique du Bois-Marie SGA41, 2 Lecture Notes in Math. 569, Springer (1977)Zbl 0349.14013 MR 0463174
[8] Deligne, P.: Le d´eterminant de la cohomologie. In: Current Trends in Arithmetical Algebraic Geometry (Arcata, CA, 1985), Contemp. Math. 67, Amer. Math. Soc., 93-177 (1987) Zbl 0629.14008 MR 0902592
[9] Finkelberg, M., Lysenko, S.: Twisted geometric Satake equivalence. J. Inst. Math. Jussieu 9, 719-739 (2010)Zbl 1316.22019 MR 2684259 · Zbl 1316.22019
[10] Finkelberg,M.,Schechtman,V.:MicrolocalapproachtoLusztig’ssymmetries. arXiv:1401.5885(2014)
[11] Gaitsgory, D.: Central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144, 253-280 (2001)Zbl 1072.14055 MR 1826370 · Zbl 1072.14055
[12] Gaitsgory, D.: Twisted Whittaker model and factorizable sheaves. Selecta Math. (N.S.) 13, 617-659 (2008)Zbl 1160.17009 MR 2403306 · Zbl 1160.17009
[13] Gaitsgory, D.: Quantum Langlands correspondence.arXiv:1601.05279(2016)
[14] Gaitsgory, D., Lysenko, S.: Parameters and duality for the metaplectic geometric Langlands theory.arXiv:1608.00284(2016) · Zbl 1423.14124
[15] Gan, W. T., Gao, F.: The Langlands-Weissman program for Brylinski-Deligne extensions. arXiv:1409.4039(2014)
[16] Gao, F.: The Gindikin-Karpelevich formula and constant terms of Eisenstein series for Brylinski-Deligne extensions. PhD. thesis, Univ. of Singapore (2014)
[17] Laszlo, Y., Olsson, M.: The six operations for sheaves on Artin stacks II: adic coefficients. Publ. Math. IHES 107, 169-210 (2008)Zbl 1191.14003 MR 2434692 · Zbl 1191.14003
[18] Lysenko, S.: Twisted geometric Langlands correspondence for a torus. Int. Math. Res. Notices 2015, 8680-8723Zbl 1345.14018 MR 3417689 · Zbl 1345.14018
[19] Lysenko, S.: Geometric Whittaker models and Eisenstein series for Mp2.arXiv:1211.1596 (2013)
[20] Lysenko, S.: Geometric Waldspurger periods II.arXiv:1308.6531(2013) · Zbl 1209.14010
[21] Lysenko, S.: Moduli of metaplectic bundles on curves and theta-sheaves. Ann. Sci. ´Ecole Norm. Sup. 39, 415-466 (2006)Zbl 1111.14029 MR 2265675 · Zbl 1111.14029
[22] Lysenko, S.: Twisted geometric Satake equivalence: reductive case.arXiv:1411.6782(2014)
[23] Lysenko, S.: Twisted Whittaker models for metaplectic groups. Geom. Funct. Anal. 27, 289- 372 (2017)Zbl 06721408 MR 3626614 · Zbl 1379.22018
[24] Moeglin, C., Waldspurger, J.-L.: Spectral Decomposition and Eisenstein Series: une Paraphrase de l’ ´Ecriture. Cambridge Univ. Press (1995)Zbl 1141.11029 MR 1361168 · Zbl 0846.11032
[25] Savin, G.: On unramified representations of covering groups. J. Reine Angew. Math. 566, 111-134 (2004)Zbl 1032.22006 MR 2039325 · Zbl 1032.22006
[26] Schechtman, V.: Dualit´e de Langlands quantique. Ann. Fac. Sci. Toulouse Math. 23, 129-158 (2014)Zbl 06293506 MR 3204734 · Zbl 1373.22028
[27] Schieder, S.: The Harder-Narasimhan stratification of BunGvia Drinfeld’s compactifications. Selecta Math. (N.S.) 21, 763-831 (2015)Zbl 1341.14006 MR 3366920 3252Sergey Lysenko · Zbl 1341.14006
[28] Springer, T. A.: Reductive groups. In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, part I, Amer. Math. Soc., 1-27 (1979)Zbl 0416.20034 MR 0546587 · Zbl 0416.20034
[29] Weissman, M.: L-groups and parameters for covering groups.arXiv:1507.01042(2015)
[30] Weissman, M.: Split metaplectic groups and their L-groups. J. Reine Angew. Math. 696, 89-141 (2014)Zbl 1321.22027 MR 3276164 · Zbl 1321.22027
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