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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

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Geometry of second adjointness for $p$-adic groups
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by Roman Bezrukavnikov and David Kazhdan; With an Appendix by Yakov Varshavsky; With an Appendix by Roman Bezrukavnikov; With an Appendix by David Kazhdan
Represent. Theory 19 (2015), 299-332
Published electronically: December 3, 2015


We present a geometric proof of second adjointness for a reductive $p$-adic group. Our approach is based on geometry of the wonderful compactification and related varieties. Considering asymptotic behavior of a function on the group in a neighborhood of a boundary stratum of the compactification, we get a “cospecialization” map between spaces of functions on various varieties carrying a $G\times G$ action. These maps can be viewed as maps of bimodules for the Hecke algebra, and the corresponding natural transformations of endo-functors of the module category lead to the second adjointness. We also get a formula for the “cospecialization” map expressing it as a composition of the orispheric transform and inverse intertwining operator; a parallel result for $D$-modules was obtained by Bezrukavnikov, Finkelberg and Ostrik. As a byproduct we obtain a formula for the Plancherel functional restricted to a certain commutative subalgebra in the Hecke algebra generalizing a result by Opdam.
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Bibliographic Information
  • Roman Bezrukavnikov
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 — and — National Research University Higher School of Economics, International Laboratory of Representation Theory and Mathematical Physics, 20 Myasnitskaya st., Moscow 101000, Russia
  • MR Author ID: 347192
  • Email:
  • David Kazhdan
  • Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
  • MR Author ID: 99580
  • Email:
  • Yakov Varshavsky
  • Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
  • MR Author ID: 638793
  • Email:
  • Received by editor(s): April 2, 2014
  • Received by editor(s) in revised form: September 3, 2015, and October 4, 2015
  • Published electronically: December 3, 2015
  • Additional Notes: R.B. was supported by the NSF grant DMS-1102434 and a Simons Foundation fellowship
    D.K. was supported by the ERC grant 669655 and US-Israel Binational Science Foundation grant 2012365

  • Dedicated: Dedicated to the memory of Izrail’ Moiseevich Gel’fand
  • © Copyright 2015 American Mathematical Society
  • Journal: Represent. Theory 19 (2015), 299-332
  • MSC (2010): Primary 20G05, 20G25, 22E35, 22E50
  • DOI:
  • MathSciNet review: 3430373