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Selecta Mathematica

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

Deligne–Lusztig duality and wonderful compactification

  • Joseph Bernstein
  • Roman BezrukavnikovEmail author
  • David Kazhdan
Article

Abstract

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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Notes

Acknowledgements

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.

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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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