Schwartz space of parabolic basic affine space and asymptotic Hecke algebras

Alexander Braverman, David Kazhdan

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Let F be a local non-archimedean field and G be the group of F-points of a split connected reductive group over F . In arXiv:1704.03019 we define an algebra J (G) of functions on G which contains the Hecke algebra H(G) and is contained in the Harish-Chandra Schwartz algebra C(G). We consider J (G) as an algebraic analog the algebra C(G). Given a parabolic subgroup P of G with a Levi subgroup M and the unipotent radical UP we write XP:= G/UP . Let Sc (XP ) be the space of locally constant functions on XP with compact support and Scusp,c (XP ) ⊂ Sc (XP ) be subspace of functions whose right shifts span a cuspidal representation of M. In this paper we study two versions of the Schwartz space of XP . The first is S(XP ):= J (Sc (XP )) and the 2nd is the space spanned by functions of the form ΦQ,P (φ) where Q is another parabolic with the same Levi subgroup, φ ∈ Sc (XQ ) and ΦQ,P is a normalized intertwining operator from L2 (XQ ) to L2 (XP ). We formulate a series of conjectures about these spaces; for example, we conjecture that S (XP ) ⊂ S(XP ) and that this embedding is an isomorphism on the M-cuspidal part. We give a proof of some of our conjectures (cf. Theorem 1.9).

Original languageEnglish
Title of host publicationRepresentations of Reductive Groups - Conference in honor of Joseph Bernstein Representation Theory and Algebraic Geometry, 2017
EditorsAvraham Aizenbud, Dmitry Gourevitch, Erez M. Lapid, David Kazhdan
PublisherAmerican Mathematical Society
Pages31-43
Number of pages13
ISBN (Print)9781470442842
DOIs
StatePublished - 2019
EventConference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017 - Jerusalem, Israel
Duration: 11 Jun 201716 Jun 2017

Publication series

NameProceedings of Symposia in Pure Mathematics
Volume101
ISSN (Print)0082-0717
ISSN (Electronic)2324-707X

Conference

ConferenceConference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017
Country/TerritoryIsrael
CityJerusalem
Period11/06/1716/06/17

Bibliographical note

Publisher Copyright:
© 2019 American Mathematical Society.

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