On the Casselman-Jacquet functor

Abstract

We study the Casselman-Jacquet functor $J$, viewed as a functor from the (derived) category of $(\mathfrak{g},K)$-modules to the (derived) category of $(\mathfrak{g},N^{-})$-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 ${\text{Av}}_{!}^{N^{-}}\circ {\text{Av}}_{\ast }^N$. We show that it is also isomorphic to the composition ${\text{Av}}_{\ast }^{N^{-}}\circ {\text{Av}}_{!}^N$. Our key tool is the pseudo-identity functor that acts on the (derived) category of (twisted) $D$-modules on an algebraic stack.

Very minor modifications, compared to previous version (to appear in Proceedings of Symposia in Pure Mathematics volume, titled "Representations of Reductive Groups")

References in corpus (2)

• Sheaves of categories and the notion of 1-affineness
• On properties of the Casselman-Jacquet functor

Cited by in corpus (1)

• Wonderful asymptotics of matrix coefficient D-modules