Endoscopy for affine Hecke categories

• Yau Wing Li ^[1]
  1. [1] University of Melbourne

     University of Melbourne

     Australia

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 5, 2024
• Idioma: inglés
• DOI: 10.1007/s00029-024-00974-y
• Enlaces
  • Texto completo
• Resumen

  • We show that the neutral block of the affine monodromic Hecke category for a reductive group is monoidally equivalent to the neutral block of the affine Hecke category for its endoscopic group. The semisimple complexes of both categories can be identified with the generalized Soergel bimodules via the Soergel functor. We extend this identification of semisimple complexes to the neutral blocks of the affine Hecke categories by the technical machinery developed by Bezrukavnikov and Yun.

• Referencias bibliográficas
  • 1. Arkhipov, S., Bezrukavnikov, R.: Perverse sheaves on affine flags and langlands dual group. Isr. J. Math. 170, 135–183 (2009)
  • 2. Arkhipov, S., Bezrukavnikov, R., Ginzburg, V.: Quantum groups, the loop Grassmannian, and the Springer resolution. J. Amer. Math. Soc....
  • 3. Behrend, K.A.: The Lefschetz trace formula for the moduli stack of principal bundles. University of California, Berkeley (1991)
  • 4. Beilinson, A., Bernstein, J. (1981) Localisation de g-modules, C. R. Acad. Sci. Paris Sér. I Math. 292(1),15–18
  • 5. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analysis and topology on singular spaces, I, 5–171, Astérisque, 100....
  • 6. Bezrukavnikov, R., Finkelberg, M., Ostrik, V.: On tensor categories attached to cells in affine Weyl groups, III. Israel J. Math. 170,...
  • 7. Beilinson, A., Ginzburg, V., Soergel, W.: Koszul duality patterns in representation theory. J. Amer. Math. Soc. 9(2), 473–527 (1996)
  • 8. Bezrukavnikov, R.: On two geometric realizations of an affine Hecke algebra. Publ. Math. Inst. Hautes Études Sci. 123, 1–67 (2016)
  • 9. Bezrukavnikov, R., Yun, Z.: On Koszul duality for Kac-Moody groups. Represent. Theory 17, 1–98 (2013)
  • 10. Björner, A., Brenti, F.: Combinatorics of Coxeter groups. Springer-Verlag, New York (2005)
  • 11. Bourbaki, N.: Groupes et algèbres de Lie, Ch. 4, 5 et 6, Hermann, Paris, (1968)
  • 12. Braden, T.: Hyperbolic localization of intersection cohomology. Transform. Groups 8(3), 209–216 (2003)
  • 13. Brylinski, J.L., Kashiwara, M.: Kazhdan-Lusztig conjecture and holonomic systems. Inventiones mathematicae 64(3), 387–410 (1981)
  • 14. Deligne, P.. La.: Conjecture de Weil II. Publ. Math. Inst. Hautes Études Sci. 52, 137–252 (1980)
  • 15. Deligne, P., Lusztig, G.: Representations of reductive groups over finite fields. Ann. Math. 103, 103–161 (1976)
  • 16. Faltings, G.: Algebraic loop groups and moduli spaces of bundles. J. Eur. Math. Soc. 5, 41–68 (2003)
  • 17. Finkelberg, M., Lysenko, S.: Twisted geometric Satake equivalence. J. Instit. Math. Jussieu 9(4), 719– 739 (2010)
  • 18. Gaitsgory, D.: Construction of central elements in the affine Hecke algebra via nearby cycles. Invent. Math. 144, 253–280 (2001)
  • 19. Gaitsgory, D.: On de Jong’s conjecture. Israel J. Math. 157, 155–191 (2007)
  • 20. Härterich, M.: Kazhdan-Lusztig-Basen, unzerlegbare Bimoduln und die Topologie der Fahnenmannigfaltigkeit einer Kac-Moody-Gruppe, Dissertation...
  • 21. Humphreys, J.: Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, vol. 94. American Mathematical...
  • 22. Kac, V.G.: Infinite-Dimensional Lie Algebras, 3rd edn. Cambridge University Press, Cambridge (1990)
  • 23. Kazhdan, D., Lusztig, G.: Representations of Coxeter groups and Hecke algebras. Inventiones mathematicae 53(2), 165–184 (1979)
  • 24. Mirkovi´c, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. of Math....
  • 25. Laszlo, Y., Sorger, C.: The line bundles on the moduli of parabolic G-bundles over curves and their sections. Ann. Sei. Ecole Norm. Sup....
  • 26. Lusztig, G.: Monodromic systems on affine flag manifolds. Proc. Roy. Soc. London 445, 231–246 (1994)
  • 27. Lusztig, G.: Characters of Reductive Groups over a Finite Field, (AM-107). Princeton University Press, Princeton (2016)
  • 28. Lusztig, G., Yun, Z.: Endoscopy for Hecke categories, character sheaves and representations. Forum Math. Pi 8, E12 (2020)
  • 29. Yun, Z.: Global Springer theory. Adv. Math. 228, 266–328 (2011)
  • 30. Yun, Z.: Rigidity in automorphic representations and local systems, Current Developments in Mathematics 2013, 73–168. International Press,...
  • 31. Zhu, X.: An introduction to affine Grassmannians and the geometric Satake equivalence, Geometry of Moduli Spaces and Representation Theory....