Geometry of second adjointness for 𝑝-adic groups

Bezrukavnikov, Roman; Kazhdan, David

doi:10.1090/ert/471

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J. Bernstein, unpublished notes, available at Drinfeld’s seminar webpage.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