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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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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 PDF
Represent. Theory 19 (2015), 299-332 Request permission


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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Additional 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