## 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

## 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.## References

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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: bezrukav@math.mit.edu
**David Kazhdan**- Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
- MR Author ID: 99580
- Email: kazhdan@math.huji.ac.il
**Yakov Varshavsky**- Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
- MR Author ID: 638793
- Email: vyakov@math.huji.ac.il
- 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 - © Copyright 2015 American Mathematical Society
- Journal: Represent. Theory
**19**(2015), 299-332 - MSC (2010): Primary 20G05, 20G25, 22E35, 22E50
- DOI: https://doi.org/10.1090/ert/471
- MathSciNet review: 3430373

Dedicated: Dedicated to the memory of Izrail’ Moiseevich Gel’fand