Affine Beilinson-Bernstein localization at the critical level for $\mathrm {GL}_2$

Pages 251-335 from Volume 195 (2022), Issue 1 by Sam Raskin

Abstract

We prove the rank 1 case of a conjecture of Frenkel-Gaitsgory: critical level Kac-Moody representations with regular central characters localize onto the affine Grassmannian. The method uses an analogue in local geometric Langlands of the existence of Whittaker models for most representations of $\mathrm {GL}_2$ over a non-Archimedean field.

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      zblnumber = {1251.17012},
      }
  • [fg-spherical] Go to document E. Frenkel and D. Gaitsgory, "Local geometric Langlands correspondence: the spherical case," in Algebraic Analysis and Around, Math. Soc. Japan, Tokyo, 2009, vol. 54, pp. 167-186.
    @INCOLLECTION{fg-spherical,
      author = {Frenkel, Edward and Gaitsgory, Dennis},
      title = {Local geometric {L}anglands correspondence: the spherical case},
      booktitle = {Algebraic Analysis and Around},
      series = {Adv. Stud. Pure Math.},
      volume = {54},
      pages = {167--186},
      publisher = {Math. Soc. Japan, Tokyo},
      year = {2009},
      mrclass = {22E57},
      mrnumber = {2499556},
      mrreviewer = {Peter Fiebig},
      doi = {10.2969/aspm/05410167},
      url = {https://doi.org/10.2969/aspm/05410167},
      zblnumber = {1192.17012},
      }
  • [fg-loc] Go to document E. Frenkel and D. Gaitsgory, "Localization of $\mathfrak{g}$-modules on the affine Grassmannian," Ann. of Math. (2), vol. 170, iss. 3, pp. 1339-1381, 2009.
    @ARTICLE{fg-loc,
      author = {Frenkel, Edward and Gaitsgory, Dennis},
      title = {Localization of {$\mathfrak{g}$}-modules on the affine {G}rassmannian},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {170},
      year = {2009},
      number = {3},
      pages = {1339--1381},
      issn = {0003-486X},
      mrclass = {17B67 (14F10 17B10)},
      mrnumber = {2600875},
      mrreviewer = {Arvid Siqveland},
      doi = {10.4007/annals.2009.170.1339},
      url = {https://doi.org/10.4007/annals.2009.170.1339},
      zblnumber = {1202.17009},
      }
  • [fgv] Go to document E. Frenkel, D. Gaitsgory, and K. Vilonen, "Whittaker patterns in the geometry of moduli spaces of bundles on curves," Ann. of Math. (2), vol. 153, iss. 3, pp. 699-748, 2001.
    @ARTICLE{fgv,
      author = {Frenkel, E. and Gaitsgory, D. and Vilonen, K.},
      title = {Whittaker patterns in the geometry of moduli spaces of bundles on curves},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {153},
      year = {2001},
      number = {3},
      pages = {699--748},
      issn = {0003-486X},
      mrclass = {11R39 (11F70 11G45 14D20 22E50)},
      mrnumber = {1836286},
      mrreviewer = {Igor Yu. Potemine},
      doi = {10.2307/2661366},
      url = {https://doi.org/10.2307/2661366},
      zblnumber = {1070.11050},
      }
  • [frenkel-teleman] Go to document E. Frenkel and C. Teleman, "Self-extensions of Verma modules and differential forms on opers," Compos. Math., vol. 142, iss. 2, pp. 477-500, 2006.
    @ARTICLE{frenkel-teleman,
      author = {Frenkel, Edward and Teleman, Constantin},
      title = {Self-extensions of {V}erma modules and differential forms on opers},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {142},
      year = {2006},
      number = {2},
      pages = {477--500},
      issn = {0010-437X},
      mrclass = {17B67 (17B10)},
      mrnumber = {2218907},
      mrreviewer = {Stefano Capparelli},
      doi = {10.1112/S0010437X05001958},
      url = {https://doi.org/10.1112/S0010437X05001958},
      zblnumber = {1162.17314},
      }
  • [frenkel-zhu] Go to document E. Frenkel and X. Zhu, "Any flat bundle on a punctured disc has an oper structure," Math. Res. Lett., vol. 17, iss. 1, pp. 27-37, 2010.
    @ARTICLE{frenkel-zhu,
      author = {Frenkel, Edward and Zhu, Xinwen},
      title = {Any flat bundle on a punctured disc has an oper structure},
      journal = {Math. Res. Lett.},
      fjournal = {Mathematical Research Letters},
      volume = {17},
      year = {2010},
      number = {1},
      pages = {27--37},
      issn = {1073-2780},
      mrclass = {14D24 (17B67 22E67)},
      mrnumber = {2592725},
      mrreviewer = {Matthias Wendt},
      doi = {10.4310/MRL.2010.v17.n1.a3},
      url = {https://doi.org/10.4310/MRL.2010.v17.n1.a3},
      zblnumber = {1220.14013},
      }
  • [central] Go to document D. Gaitsgory, "Construction of central elements in the affine Hecke algebra via nearby cycles," Invent. Math., vol. 144, iss. 2, pp. 253-280, 2001.
    @ARTICLE{central,
      author = {Gaitsgory, D.},
      title = {Construction of central elements in the affine {H}ecke algebra via nearby cycles},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {144},
      year = {2001},
      number = {2},
      pages = {253--280},
      issn = {0020-9910},
      mrclass = {14L15 (14C25 14F05 14M15)},
      mrnumber = {1826370},
      mrreviewer = {Burt Totaro},
      doi = {10.1007/s002220100122},
      url = {https://doi.org/10.1007/s002220100122},
      zblnumber = {1072.14055},
      }
  • [twisted-satake] Go to document D. Gaitsgory and S. Lysenko, "Parameters and duality for the metaplectic geometric Langlands theory," Selecta Math. (N.S.), vol. 24, iss. 1, pp. 227-301, 2018.
    @ARTICLE{twisted-satake,
      author = {Gaitsgory, Dennis and Lysenko, S.},
      title = {Parameters and duality for the metaplectic geometric {L}anglands theory},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {24},
      year = {2018},
      number = {1},
      pages = {227--301},
      issn = {1022-1824},
      mrclass = {14F05 (14D24 14H60)},
      mrnumber = {3769731},
      mrreviewer = {Andrew Salch},
      doi = {10.1007/s00029-017-0360-4},
      url = {https://doi.org/10.1007/s00029-017-0360-4},
      zblnumber = {1423.14124},
      }
  • [quantum-langlands-summary] D. Gaitsgory, Quantum Langlands correspondence, 2007.
    @MISC{quantum-langlands-summary, title = {Quantum {L}anglands correspondence},
      author = {Gaitsgory, Dennis},
      arxiv = {1601.05279},
      year = {2007},
      zblnumber = {},
      }
  • [lesjours] D. Gaitsgory, Whittaker categories, 2008.
    @MISC{lesjours,
      author = {Gaitsgory, Dennis},
      title = {Whittaker categories},
      note = {formerly titled \emph{Les jours et les travaux},
      available on author's website},
      year = {2008},
      zblnumber = {},
      }
  • [dgcat] D. Gaitsgory, Generalities on DG categories, 2012.
    @MISC{dgcat,
      author = {Gaitsgory, Dennis},
      title = {Generalities on {DG} categories},
      note = {available on author's website},
      year = {2012},
      zblnumber = {},
      }
  • [km-indcoh] Go to document D. Gaitsgory, Kac-Moody representations (Day IV, Talk 3), 2014.
    @MISC{km-indcoh, title = {Kac-{M}oody representations ({D}ay {IV, T}alk 3)},
      author = {Gaitsgory, Dennis},
      year = {2014},
      url = {https://sites.google.com/site/geometriclanglands2014/notes},
      zblnumber = {},
      }
  • [shvcat] Go to document D. Gaitsgory, "Sheaves of categories and the notion of 1-affineness," in Stacks and Categories in Geometry, Topology, and Algebra, Amer. Math. Soc., Providence, RI, 2015, vol. 643, pp. 127-225.
    @INCOLLECTION{shvcat,
      author = {Gaitsgory, Dennis},
      title = {Sheaves of categories and the notion of 1-affineness},
      booktitle = {Stacks and Categories in Geometry, Topology, and Algebra},
      series = {Contemp. Math.},
      volume = {643},
      pages = {127--225},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2015},
      mrclass = {14F05 (14A20)},
      mrnumber = {3381473},
      mrreviewer = {Jon Eivind Vatne},
      doi = {10.1090/conm/643/12899},
      url = {https://doi.org/10.1090/conm/643/12899},
      zblnumber = {1346.14004},
      }
  • [whit-comp] Go to document D. Gaitsgory, "The local and global versions of the Whittaker category," Pure Appl. Math. Q., vol. 16, iss. 3, pp. 775-904, 2020.
    @ARTICLE{whit-comp,
      author = {Gaitsgory, Dennis},
      title = {The local and global versions of the {W}hittaker category},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {16},
      year = {2020},
      number = {3},
      pages = {775--904},
      issn = {1558-8599},
      mrclass = {14D24 (18D99 18F20 20G25)},
      mrnumber = {4176538},
      doi = {10.4310/PAMQ.2020.v16.n3.a14},
      url = {https://doi.org/10.4310/PAMQ.2020.v16.n3.a14},
      zblnumber = {1460.14033},
      }
  • [crystals] Go to document D. Gaitsgory and N. Rozenblyum, "Crystals and D-modules," Pure Appl. Math. Q., vol. 10, iss. 1, pp. 57-154, 2014.
    @ARTICLE{crystals,
      author = {Gaitsgory, Dennis and Rozenblyum, Nick},
      title = {Crystals and {D}-modules},
      journal = {Pure Appl. Math. Q.},
      fjournal = {Pure and Applied Mathematics Quarterly},
      volume = {10},
      year = {2014},
      number = {1},
      pages = {57--154},
      issn = {1558-8599},
      mrclass = {14A20 (14F05 14F10)},
      mrnumber = {3264953},
      mrreviewer = {Andrei D. Halanay},
      doi = {10.4310/PAMQ.2014.v10.n1.a2},
      url = {https://doi.org/10.4310/PAMQ.2014.v10.n1.a2},
      zblnumber = {1327.14013},
      }
  • [indschemes] Go to document D. Gaitsgory and N. Rozenblyum, "DG indschemes," in Perspectives in representation theory, Amer. Math. Soc., Providence, RI, 2014, vol. 610, pp. 139-251.
    @INCOLLECTION{indschemes,
      author = {Gaitsgory, Dennis and Rozenblyum, Nick},
      title = {D{G} indschemes},
      booktitle = {Perspectives in representation theory},
      series = {Contemp. Math.},
      volume = {610},
      pages = {139--251},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {2014},
      mrclass = {14F05 (18F20)},
      mrnumber = {3220628},
      mrreviewer = {Jon Eivind Vatne},
      doi = {10.1090/conm/610/12080},
      url = {https://doi.org/10.1090/conm/610/12080},
      zblnumber = {1316.14006},
      }
  • [grbook] Go to document D. Gaitsgory and N. Rozenblyum, A Study in Derived Algebraic Geometry. Vol. I. Correspondences and Duality, Amer. Math. Soc., Providence, RI, 2017, vol. 221.
    @BOOK{grbook,
      author = {Gaitsgory, Dennis and Rozenblyum, Nick},
      TITLE = {A Study in Derived Algebraic Geometry. {V}ol. {I}. {C}orrespondences and Duality},
      SERIES = {Math. Surv. Monogr.},
      VOLUME = {221},
      PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2017},
      PAGES = {xl+533pp},
      ISBN = {978-1-4704-3569-1},
      MRCLASS = {14F05 (18D05 18G55)},
      MRNUMBER = {3701352},
      MRREVIEWER = {Adrian Langer},
      DOI = {10.1090/surv/221.1},
      URL = {https://doi.org/10.1090/surv/221.1},
      ZBLNUMBER = {1408.14001},
      }
  • [MR3701353] Go to document D. Gaitsgory and N. Rozenblyum, A Study in Derived Algebraic Geometry. Vol. II. Deformations, Lie Theory and Formal Geometry, Amer. Math. Soc., Providence, RI, 2017, vol. 221.
    @book {MR3701353,
      author = {Gaitsgory, Dennis and Rozenblyum, Nick},
      TITLE = {A Study in Derived Algebraic Geometry. {V}ol. {II}. {D}eformations, {L}ie Theory and Formal Geometry},
      SERIES = {Math. Surv. Monogr.},
      VOLUME = {221},
      PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2017},
      PAGES = {xxxv+436},
      ISBN = {978-1-4704-3570-7},
      MRCLASS = {14F05 (18D05 18G55)},
      MRNUMBER = {3701353},
      MRREVIEWER = {Adrian Langer},
      DOI = {10.1090/surv/221.2},
      URL = {https://doi.org/10.1090/surv/221.2},
      ZBLNUMBER = {1409.14003},
      }
  • [ginzburg-whittaker-hecke] V. Ginzburg, "Nil-Hecke algebras and Whittaker $\mathcal D$-modules," in Lie Groups, Geometry, and Representation Theory, Birkhäuser/Springer, Cham, 2018, vol. 326, pp. 137-184.
    @INCOLLECTION{ginzburg-whittaker-hecke,
      author = {Ginzburg, Victor},
      title = {Nil-{H}ecke algebras and {W}hittaker {$\mathcal D$}-modules},
      booktitle = {Lie Groups, Geometry, and Representation Theory},
      series = {Progr. Math.},
      volume = {326},
      pages = {137--184},
      publisher = {Birkhäuser/Springer, Cham},
      year = {2018},
      mrclass = {20G20 (14F10 16S32 17B20 20C08)},
      mrnumber = {3890208},
      mrreviewer = {Willem Adriaan de Graaf},
      zblnumber = {1455.20030},
      }
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    @ARTICLE{kt1,
      author = {Kashiwara, Masaki and Tanisaki, Toshiyuki},
      title = {Kazhdan-{L}usztig conjecture for affine {L}ie algebras with negative level},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {77},
      year = {1995},
      number = {1},
      pages = {21--62},
      issn = {0012-7094},
      mrclass = {17B67 (17B10)},
      mrnumber = {1317626},
      mrreviewer = {Alex Jay Feingold},
      doi = {10.1215/S0012-7094-95-07702-3},
      url = {https://doi.org/10.1215/S0012-7094-95-07702-3},
      zblnumber = {0829.17020},
      }
  • [htt] Go to document J. Lurie, Higher Topos Theory, Princeton Univ. Press, Princeton, NJ, 2009, vol. 170.
    @BOOK{htt,
      author = {Lurie, Jacob},
      title = {Higher Topos Theory},
      series = {Ann. of Math. Stud.},
      volume = {170},
      publisher = {Princeton Univ. Press, Princeton, NJ},
      year = {2009},
      pages = {xviii+925},
      isbn = {978-0-691-14049-0; 0-691-14049-9},
      mrclass = {18-02 (18B25 18E35 18G30 18G55 55U40)},
      mrnumber = {2522659},
      mrreviewer = {Mark Hovey},
      doi = {10.1515/9781400830558},
      url = {https://doi.org/10.1515/9781400830558},
      zblnumber = {1175.18001},
      }
  • [higheralgebra] J. Lurie, Higher algebra, 2012.
    @MISC{higheralgebra,
      author = {Lurie, Jacob},
      title = {Higher algebra},
      note = {available on author's website},
      year = {2012},
      }
  • [mirkovic-vilonen] Go to document I. Mirković and K. Vilonen, "Geometric Langlands duality and representations of algebraic groups over commutative rings," Ann. of Math. (2), vol. 166, iss. 1, pp. 95-143, 2007.
    @ARTICLE{mirkovic-vilonen,
      author = {Mirković,
      I. and Vilonen, K.},
      title = {Geometric {L}anglands duality and representations of algebraic groups over commutative rings},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {166},
      year = {2007},
      number = {1},
      pages = {95--143},
      issn = {0003-486X},
      mrclass = {22E55 (11R39 20G05)},
      mrnumber = {2342692},
      mrreviewer = {Peter Fiebig},
      doi = {10.4007/annals.2007.166.95},
      url = {https://doi.org/10.4007/annals.2007.166.95},
      zblnumber = {1138.22013},
      }
  • [mirkovic-vilonen-but-not-that-one] Go to document I. Mirković and K. Vilonen, "Characteristic varieties of character sheaves," Invent. Math., vol. 93, iss. 2, pp. 405-418, 1988.
    @ARTICLE{mirkovic-vilonen-but-not-that-one,
      author = {Mirković,
      I. and Vilonen, K.},
      title = {Characteristic varieties of character sheaves},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {93},
      year = {1988},
      number = {2},
      pages = {405--418},
      issn = {0020-9910},
      mrclass = {20G05 (22E47)},
      mrnumber = {0948107},
      mrreviewer = {Bhama Srinivasan},
      doi = {10.1007/BF01394339},
      url = {https://doi.org/10.1007/BF01394339},
      zblnumber = {0683.22012},
      }
  • [opers] Go to document S. Raskin, "A geometric proof of the Feigin-Frenkel theorem," Represent. Theory, vol. 16, pp. 489-512, 2012.
    @ARTICLE{opers,
      author = {Raskin, Sam},
      title = {A geometric proof of the {F}eigin-{F}renkel theorem},
      journal = {Represent. Theory},
      fjournal = {Representation Theory. An Electronic Journal of the Amer. Math. Soc.},
      volume = {16},
      year = {2012},
      pages = {489--512},
      mrclass = {17B65 (14L24 81R10)},
      mrnumber = {2972556},
      mrreviewer = {Volodymyr Mazorchuk},
      doi = {10.1090/S1088-4165-2012-00417-4},
      url = {https://doi.org/10.1090/S1088-4165-2012-00417-4},
      zblnumber = {1302.17035},
      }
  • [chiralcats] S. Raskin, Chiral categories, 2015.
    @MISC{chiralcats,
      author = {Raskin, Sam},
      title = {Chiral categories},
      note = {available on author's website},
      year = {2015},
      zblnumber = {07374911},
      }
  • [dmod] S. Raskin, $D$-modules on infinite dimensional varieties, 2015.
    @MISC{dmod,
      author = {Raskin, Sam},
      note = {available on author's website},
      title = {$D$-modules on infinite dimensional varieties},
      year = {2015},
      zblnumber = {},
      }
  • [cpsii] Go to document S. D. Raskin, Chiral Principal Series Categories, , 2014.
    @BOOK{cpsii,
      author = {Raskin, Sam D.},
      title = {Chiral Principal Series Categories},
      note = {Ph.D. {T}hesis, Harvard Univ.},
      year = {2014},
      pages = {340},
      isbn = {978-1321-02173-8},
      mrclass = {Thesis},
      url = {http://nrs.harvard.edu/urn-3:HUL.InstRepos:12274305},
      mrnumber = {3251357},
      zblnumber = {},
      }
  • [methods] S. Raskin, Homological methods in semi-infinite contexts, 2019.
    @MISC{methods,
      author = {Raskin, Sam},
      title = {Homological methods in semi-infinite contexts},
      note = {available on author's website},
      year = {2019},
      zblnumber = {},
      }
  • [b-fn] Go to document S. Raskin, "A generalization of the $b$-function lemma," Compos. Math., vol. 157, iss. 10, pp. 2199-2214, 2021.
    @ARTICLE{b-fn,
      author = {Raskin, Sam},
      title = {A generalization of the {$b$}-function lemma},
      journal = {Compos. Math.},
      fjournal = {Compositio Mathematica},
      volume = {157},
      year = {2021},
      number = {10},
      pages = {2199--2214},
      issn = {0010-437X},
      mrclass = {14F10 (17B55 32C38)},
      mrnumber = {4311556},
      doi = {10.1112/S0010437X21007491},
      url = {https://doi.org/10.1112/S0010437X21007491},
      zblnumber = {},
      }
  • [whit] Go to document S. Raskin, "$\mathcal W$-algebras and Whittaker categories," Selecta Math. (N.S.), vol. 27, iss. 3, p. 46, 2021.
    @ARTICLE{whit,
      author = {Raskin, Sam},
      title = {{$\mathcal W$}-algebras and {W}hittaker categories},
      journal = {Selecta Math. (N.S.)},
      fjournal = {Selecta Mathematica. New Series},
      volume = {27},
      year = {2021},
      number = {3},
      pages = {Paper No. 46, 114},
      issn = {1022-1824},
      mrclass = {17B67 (22E57)},
      mrnumber = {4273643},
      doi = {10.1007/s00029-021-00641-6},
      url = {https://doi.org/10.1007/s00029-021-00641-6},
      zblnumber = {07364461},
      }

Keywords
, , ,
Mathematical Subject Classification
Primary: ,
DOI
https://doi.org/10.4007/annals.2022.195.1.4
MR
4358416
zbMATH
7483858
Milestones
Received: 3 February 2020
Revised: 7 July 2021
Accepted: 9 September 2021
Published online: 1 January 2022

Authors

Sam Raskin

The University of Texas at Austin, Austin, TX