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Journal of the London Mathematical Society 2001 63(2):364-386; doi:10.1017/S0024610700001885
© 2001 by London Mathematical Society
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© The London Mathematical Society

Representations of Reductive p-Adic Groups: Localization of Hecke Algebras and Applications

Colin J. Bushnell

Department of Mathematics, King's College Strand, London WC2R 2LS, bushnell{at}mth.kcl.ac.uk

Received 16 June 1999. Revision received 15 May 2000.

Let F be a non-Archimedean local field and G be the group of F-points 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. The 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the co-adjoint of the Jacquet functor relative to P as the induction functor relative to the M-opposite of P, an unpublished result of J.-N. Bernstein.


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