GEOMETRY OF SECOND ADJOINTNESS FOR p-ADIC GROUPS
Name
Geometry of Second Adjointness.pdf
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452.81 KB
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Author(s)
Bezrukavnikov, Roman
Kazhdan, David
Date Issued
December 2015
Journal
Representation Theory
Publisher
American Mathematical Society (AMS)
Citation
Bezrukavnikov, Roman, and David Kazhdan. “GEOMETRY OF SECOND ADJOINTNESS FOR p-ADIC GROUPS.” Represent. Theory 19, no. 14 (December 3, 2015): 299–332. © 2015 American Mathematical Society
Version
Final published version
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 × 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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1090/ert/471
