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 language | English |
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Title of host publication | Representations of Reductive Groups - Conference in honor of Joseph Bernstein Representation Theory and Algebraic Geometry, 2017 |
Editors | Avraham Aizenbud, Dmitry Gourevitch, Erez M. Lapid, David Kazhdan |
Publisher | American Mathematical Society |
Pages | 31-43 |
Number of pages | 13 |
ISBN (Print) | 9781470442842 |
DOIs | |
State | Published - 2019 |
Event | Conference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017 - Jerusalem, Israel Duration: 11 Jun 2017 → 16 Jun 2017 |
Publication series
Name | Proceedings of Symposia in Pure Mathematics |
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Volume | 101 |
ISSN (Print) | 0082-0717 |
ISSN (Electronic) | 2324-707X |
Conference
Conference | Conference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017 |
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Country/Territory | Israel |
City | Jerusalem |
Period | 11/06/17 → 16/06/17 |
Bibliographical note
Publisher Copyright:© 2019 American Mathematical Society.