Representations of reductive p-adic groups: Localization of Hecke algebras and applications

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Abstract

Let F be a non-Archimedean local field and G be the group of F-polnts of a connected reductive group defined over F. Let M be an F-Levi subgroup of G and P = MN be a parabolic subgroup with Levi decomposition P = MN. Jacquet. or truncated, restriction gives a functor from the category of smooth representations of G to that of M. The main result describes this functor in terms of homomorphisms and localizations of Hecke algebras attached to certain compact open subgroups of G and M. This leads to new and straightforward proofs of some fundamental results. Thr first computes the smooth dual of a Jacquet module of a smooth representation of G. generalizing the corresponding result for admissible representations due to Harish-Chandra and Casselman. The second identifies the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.
Original languageEnglish
Pages (from-to)364 - 386
Number of pages23
JournalJOURNAL OF THE LONDON MATHEMATICAL SOCIETY-SECOND SERIES
Volume63
Issue number2
DOIs
Publication statusPublished - Apr 2001

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