NDLI: A partial analog of the integrability theorem for distributions on p-adic spaces and applications

Content Provider	Springer Nature Link
Author	Aizenbud, Avraham
Copyright Year	2012
Abstract	Let X be a smooth real algebraic variety. Let ξ be a distribution on it. One can define the singular support of ξ to be the singular support of the D $_{ X }$-module generated by ξ (sometimes it is also called the characteristic variety). A powerful property of the singular support is that it is a coisotropic subvariety of T*X. This is the integrability theorem (see [KKS, Mal, Gab]). This theorem turned out to be useful in representation theory of real reductive groups (see, e.g., [AG4, AS, Say]).The aim of this paper is to give an analog of this theorem to the non-Archimedean case. The theory of D-modules is not available to us so we need a different definition of the singular support. We use the notion wave front set from [Hef] and define the singular support to be its Zariski closure. Then we prove that the singular support satisfies some property that we call weakly coisotropic, which is weaker than being coisotropic but is enough for some applications. We also prove some other properties of the singular support that were trivial in the Archimedean case (using the algebraic definition) but not obvious in the non-Archimedean case.We provide two applications of those results: a non-Archimedean analog of the results of [Say] concerning Gel’fand property of nice symmetric pairs a proof of multiplicity one theorems for GL n which is uniform for all local fields. This theorem was proven for the non-Archimedean case in [AGRS] and for the Archimedean case in [AG4] and [SZ].
Starting Page	233
Ending Page	262
Page Count	30
File Format	PDF
ISSN	00212172
Journal	Israel Journal of Mathematics
Volume Number	193
Issue Number	1
e-ISSN	15658511
Language	English
Publisher	The Hebrew University Magnes Press
Publisher Date	2012-07-02
Publisher Place	Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Mathematics Algebra Group Theory and Generalizations Analysis Applications of Mathematics Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Mathematics

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