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Geometry of second adjointness for $p$-adic groups

Authors: Roman Bezrukavnikov and David Kazhdan; \\ With an Appendix by Yakov Varshavsky; \\ With an Appendix by Roman Bezrukavnikov; \\ With an Appendix by David Kazhdan
Journal: Represent. Theory 19 (2015), 299-332
MSC (2010): Primary 20G05, 20G25, 22E35, 22E50
Published electronically: December 3, 2015
MathSciNet review: 3430373
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Abstract: 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

David Kazhdan
Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
MR Author ID: 99580

Yakov Varshavsky
Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
MR Author ID: 638793

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
Article copyright: © Copyright 2015 American Mathematical Society