## 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 - DOI: https://doi.org/10.1090/ert/471
- Published electronically: December 3, 2015
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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.## References

- J. Bernstein, unpublished notes, available at Drinfeld’s seminar webpage.
- J. N. Bernstein,
*Le “centre” de Bernstein*, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32 (French). Edited by P. Deligne. MR**771671** - Roman Bezrukavnikov, Michael Finkelberg, and Victor Ostrik,
*Character $D$-modules via Drinfeld center of Harish-Chandra bimodules*, Invent. Math.**188**(2012), no. 3, 589–620. MR**2917178**, DOI 10.1007/s00222-011-0354-3 - M. Brion,
*Lectures on the geometry of flag varieties*, http://www-fourier.ujf-grenoble.fr/ mbrion/lecturesrev.pdf. - Michel Brion,
*The behaviour at infinity of the Bruhat decomposition*, Comment. Math. Helv.**73**(1998), no. 1, 137–174. MR**1610599**, DOI 10.1007/s000140050049 - Michel Brion and Shrawan Kumar,
*Frobenius splitting methods in geometry and representation theory*, Progress in Mathematics, vol. 231, Birkhäuser Boston, Inc., Boston, MA, 2005. MR**2107324**, DOI 10.1007/b137486 - Michel Brion and Patrick Polo,
*Large Schubert varieties*, Represent. Theory**4**(2000), 97–126. MR**1789463**, DOI 10.1090/S1088-4165-00-00069-8 - W. Casselman,
*Introduction to admissible representations of $p$-adic groups*, preprint available at: https://www.math.ubc.ca/\textasciitilde cass/research/publications.html. - C. De Concini and C. Procesi,
*Complete symmetric varieties*, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 1–44. MR**718125**, DOI 10.1007/BFb0063234 - I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro,
*Representation theory and automorphic functions*, Generalized Functions, vol. 6, Academic Press, Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition. MR**1071179** - S. Evens, B. F. Jones,
*On the wonderful compactification,*arxiv:0801.0456. - Xuhua He and Jesper Funch Thomsen,
*Geometry of $B\times B$-orbit closures in equivariant embeddings*, Adv. Math.**216**(2007), no. 2, 626–646. MR**2351372**, DOI 10.1016/j.aim.2007.06.001 - George Lusztig,
*Some examples of square integrable representations of semisimple $p$-adic groups*, Trans. Amer. Math. Soc.**277**(1983), no. 2, 623–653. MR**694380**, DOI 10.1090/S0002-9947-1983-0694380-4 - M. Marten, M. Thaddeus,
*Compactifications of reductive groups as moduli stacks of bundles,*arXiv:1105.4830 - Eric M. Opdam,
*A generating function for the trace of the Iwahori-Hecke algebra*, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 301–323. MR**1985730** - David Renard,
*Représentations des groupes réductifs $p$-adiques*, Cours Spécialisés [Specialized Courses], vol. 17, Société Mathématique de France, Paris, 2010 (French). MR**2567785** - Y. Sakellaridis, A. Venkatesh,
*Periods and harmonic analysis on spherical varieties,*preprint arXiv:1203.0039, 291pp. - Tonny A. Springer,
*Some results on compactifications of semisimple groups*, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1337–1348. MR**2275648** - T. A. Springer,
*Linear algebraic groups*, 2nd ed., Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR**1642713**, DOI 10.1007/978-0-8176-4840-4

## 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: 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