On the Casselman-Jacquet functor

Creators: Chen, T.-H.; Gaitsgory, D.; Din, A. Yom

Others:: Aizenbud, Avraham; Gourevitch, Dmitry; Kazhdan, David; Lapid, Erez M.

Abstract

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 Av^N−!∘Av^N∗. We show that it is also isomorphic to the composition Av^N−∗∘Av^N!. Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) D-modules on an algebraic stack.

Additional Information

© 2019 American Mathematical Society. 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. 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.

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