Selecta Mathematica

, Volume 24, Issue 1, pp 7–20 | Cite as

Deligne–Lusztig duality and wonderful compactification

  • Joseph Bernstein
  • Roman BezrukavnikovEmail author
  • David Kazhdan


We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for \(G=GL(n)\) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.

Mathematics Subject Classification

20G05 20G25 20J05 22E35 


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We thank Vladimir Drinfeld for many helpful conversations over the years. The second author is also grateful to Michael Finkelberg, Leonid Rybnikov and Jonathan Wang for motivating discussions. The impetus for writing this note came from a talk given by the second author at the Higher School for Economics (Moscow), he thanks this institution for the stimulating opportunity. Finally, we thank Victor Ginzburg for an inspiring correspondence which has motivated Sect. 3.4. J.B. acknowledges partial support by the ERC Grant 291612, R.B. was partly supported by the NSF Grant DMS-1601953 and Russian Academic Excellence Project 5-100, D.K. was supported by an EPRC Grant, their collaboration was supported by the US-Israel BSF Grant 2016363.


  1. 1.
    Alvis, D.: The duality operation in the character ring of a finite Chevalley group. Am. Math. Soc. Bull. New Ser. 1(6), 907–911 (1979)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aubert, A.-M.: Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie dun groupe réductif \(p\)-adique, Trans. Am. Math. Soc. 347 (1995), no. 6, 2179–2189, erratum in: Trans. Am. Math. Soc. 348 (1996), no. 11, 4687–4690Google Scholar
  3. 3.
    Bernstein, J.: Le “centre” de Bernstein. In: Deligne, P. (ed.) Travaux en Cours, Representations of Reductive Groups Over a Local Field, pp. 1–32. Hermann, Paris (1984)Google Scholar
  4. 4.
    Bernstein, J., Braverman, A., Gaitsgory, D.: The Cohen–Macaulay property of the category of \(({\mathfrak{g}},K)\)-modules. Selecta Math. (N.S.) 3(3), 303–314 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bernstein, J.: Second adjointness for representations of \(p\)-adic groups, preprint, available at:
  6. 6.
    Bezrukavnikov, R.: Homological properties of representations of \(p\)-adic groups related to geometry of the group at infinity, Ph.D. thesis, arxiv preprint arXiv:math/0406223
  7. 7.
    Bezrukavnikov, R., Kazhdan, D.: Geometry of second adjointness for \(p\)-adic groups, with an appendix by Y. Varshavsky, Bezrukavnikov and Kazhdan. Represent. Theory 19, 299–332 (2015)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bezrukavnikov, R., Kaledin, D.: McKay equivalence for symplectic resolutions of singularities, Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 20–42; translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 13–33Google Scholar
  9. 9.
    Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and mutations, Math. USSR-Izv. 35 (1990), no. 3, 519–541; translated from Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205Google Scholar
  10. 10.
    Curtis, C.: Truncation and duality in the character ring of a finite group of Lie type. J. Algebra 62(2), 320–332 (1980)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Deligne, P., Lusztig, G.: Duality for representations of a reductive group over a finite field. J. Algebra 74(1), 284–291 (1982)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Drinfeld, V., Wang, J.: On a strange invariant bilinear form on the space of automorphic forms. Selecta Math. (N.S.) 22(4), 1825–1880 (2016)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gaitsgory, D.: A “strange” functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles. Ann. Sci. Ec. Norm. Sup. (4) 50 (5), 1123–1162 (2017)Google Scholar
  14. 14.
    Gromov, M.: Hyperbolic Groups. In: Gersten, S.M. (ed.) “Essays in Group Theory,” Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar
  15. 15.
    Hiraga, K.: On functoriality of Zelevinski involutions. Compos. Math. 140, 1625–1656 (2004)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kato, S.: Duality for representations of a Hecke algebra. Proc. Am. Math. Soc. 119, 941–946 (1993)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Mirković, I., Riche, S.: Iwahori–Matsumoto involution and linear Koszul duality. Int. Math. Res. Not. IMRN 2013(1), 150–196 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat-Tits building. Inst. Hautes Études Sci. Publ. Math. 85, 97–191 (1997)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Zelevinsky, A.: Induced representations of reductive p-adic groups, II: on irreducible representations of \(GL(n)\). Ann. Sci. Ec. Norm. Sup. 13, 165–210 (1980)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Joseph Bernstein
    • 1
  • Roman Bezrukavnikov
    • 2
    • 3
    Email author
  • David Kazhdan
    • 4
  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  3. 3.International Laboratory of Representation Theory and Mathematical PhysicsNational Research University Higher School of EconomicsMoscowRussia
  4. 4.Einstein Institute of MathematicsThe Hebrew University of JerusalemJerusalemIsrael

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