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Deligne–Lusztig duality and wonderful compactification

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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.

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Change history

  • 13 March 2019

    Papers [1–6] have received funding from ERC under Grant Agreement No. 669655.

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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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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Joseph Bernstein

  2. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts ave., Cambridge, MA, 02139, USA

    Roman Bezrukavnikov

  3. International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, 20 Myasnitskaya st., Moscow, Russia, 101000

    Roman Bezrukavnikov

  4. Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 9190401, Israel

    David Kazhdan

Authors
  1. Joseph Bernstein
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  2. Roman Bezrukavnikov
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  3. David Kazhdan
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Correspondence to Roman Bezrukavnikov.

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To Sasha Beilinson, with admiration and best wishes for his birthday.

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Bernstein, J., Bezrukavnikov, R. & Kazhdan, D. Deligne–Lusztig duality and wonderful compactification. Sel. Math. New Ser. 24, 7–20 (2018). https://doi.org/10.1007/s00029-018-0391-5

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