Smith theory and geometric Hecke algebras

  • David TreumannEmail author


In 1960 Borel proved a “localization” result relating the rational cohomology of a topological space X to the rational cohomology of the fixed points for a torus action on X. This result and its generalizations have many applications in Lie theory. In 1934, Smith proved a similar localization result relating the mod p cohomology of X to the mod p cohomology of the fixed points for a \({\mathbb {Z}}/p\)-action on X. In this paper we study \({\mathbb {Z}}/p\)-localization for constructible sheaves and functions. We show that \({\mathbb {Z}}/p\)-localization on loop groups is related via the geometric Satake correspondence to some special homomorphisms that exist between algebraic groups defined over a field of small characteristic.



I thank Florian Herzig, Gopal Prasad, and Ting Xue for help with algebraic groups. In particular, most of the material in Sect. 3.4.5 I learned from Ting. While developing these ideas I benefited from discussions with Paul Goerss, David Nadler, and Zhiwei Yun. I thank Sam Evens for correcting some microlocal mistakes in an earlier version of this paper, and Akshay Venkatesh for improving the proof of Theorem 3.3.


  1. 1.
    Atiyah, M., Bott, R.: The moment map and equivariant cohomology. Topology 23, 1–28 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bernstein, J., Lunts, V.: Equivariant Sheaves and Functors. Springer, Berlin (1994)CrossRefzbMATHGoogle Scholar
  3. 3.
    Borel, A.: Seminar on transformation groups (AM-46). Annals of Mathematics Studies. Princeton University Press (1960)Google Scholar
  4. 4.
    Borel, A., de Siebenthal, J.: Les sous-groupes de rang maximum des groupes de Lie clos. Comment. Math. Helv. 23, 200–221 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borel, A., Tits, J.: Groupes réductifs. Inst. Hautes Études Sci. Publ. Math. 27, 55–150 (1965)CrossRefzbMATHGoogle Scholar
  6. 6.
    Braden, T.: Hyperbolic localization of intersection cohomology. Transform. Groups 8, 209–216 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bredon, G.: Fixed point sets of actions on Poincaré duality spaces. Topology 12, 159–175 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chang, T., Skjelbred, T.: Group actions on Poincaré duality spaces. Bull. Am. Math. Soc. 78, 1024–1026 (1972)CrossRefzbMATHGoogle Scholar
  9. 9.
    Goresky, M., Kottwitz, R., MacPherson, R.: Equivariant cohomology, Koszul duality, and the localization theorem. Invent. Math. 131, 25–83 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jin, X.: Holomorphic Lagrangian branes correspond to perverse sheaves. Geom. Topol. 19, 1685–1735 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kac, V.: Automorphisms of finite order of semisimple Lie algebras. Funkts. Anal. Prilozh. 3, 252–254 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kashiwara, M., Schapira, P.: Sheaves on Manifolds. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  13. 13.
    Liebeck, M., Seitz, G.: The maximal subgroups of positive dimension in exceptional algebraic groups. Memoirs of the American Mathematical Society (2004).
  14. 14.
    Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. 166, 95–143 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Nadler, D.: Perverse sheaves on real loop Grassmannians. Invent. Math. 159, 1–73 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nadler, D., Zaslow, E.: Constructible sheaves and the Fukaya category. J. Am. Math. Soc. 22, 233–286 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Quillen, D.: The spectrum of an equivariant cohomology ring. Ann. Math. 94, 549–602 (1971)Google Scholar
  18. 18.
    Schapira, P.: Operations on constructible functions. J. Pure Appl. Algebra 72, 83–93 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Shinoda, K.: The conjugacy classes of type \((\rm F_4)\) over finite fields of characteristic 2. J. Fac. Sci. Univ. Tokyo 21, 133–159 (1974)Google Scholar
  20. 20.
    Smith, P.: A theorem on fixed points for periodic transformations. Ann. Math. 35(3), 572–578 (1934)Google Scholar
  21. 21.
    Sopkina, E.: Classification of all connected subgroup schemes of a reductive group containing a split maximal torus. J. K-Theory 3, 103–122 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Spaltenstein, N.: Nilpotent classes in Lie algebras of type \(\rm F_4\) over fields of characteristic 2. J. Fac. Sci. Univ. Tokyo 30, 517–524 (1984)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Steinberg, R.: Representations of algebraic groups. Nagoya Math. J. 22, 33–56 (1963)Google Scholar

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Authors and Affiliations

  1. 1.Boston CollegeChestnut HillUSA

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