On the Casselman-Jacquet functor

T. H. Chen, D. Gaitsgory, A. Yom Din

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations


We study the Casselman-Jacquet functor J, viewed as a functor from the (derived) category of (g, K)-modules to the (derived) category of (g, N )-modules, N is the negative maximal unipotent. We give a functorial definition of J as a certain right adjoint functor, and identify it as a composition of two averaging functors AvN − ! ◦ AvN∗ . We show that it is also isomorphic to the composition AvN ∗ ◦ AvN! . Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) D-modules on an algebraic stack.

Original languageEnglish (US)
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
Number of pages40
ISBN (Print)9781470442842
StatePublished - 2019
Externally publishedYes
EventConference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017 - Jerusalem, Israel
Duration: Jun 11 2017Jun 16 2017

Publication series

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


ConferenceConference on Representation Theory and Algebraic Geometry held in honor of Joseph Bernstein, 2017

Bibliographical note

Funding Information:
0.8. Acknowledgements. The second and the third authors would like to thank their teacher J. Bernstein for many illuminating discussions related to representations of real reductive groups and Harish-Chandra modules; the current paper is essentially an outcome of these conversations. The third author would like to thank Sam Raskin for very useful conversations on higher categories. The first author would like to thank the Max Planck Institute for Mathematics for support, hospitality, and a nice research environment.

Funding Information:
The research of D.G. has been supported by NSF grant DMS-1063470. The research of T.H.C. was partially supported by NSF grant DMS-1702337.

Publisher Copyright:
© 2019 American Mathematical Society.


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