A remark on the geometric Jacquet functor (Q2902434)

Let \(G\) be a reductive linear algebraic group over the reals. Let \({\mathfrak g}_{\mathbb C}\) be a complexification of the Lie algebra of \(G\). Let \(\theta\) be a Cartan involution on \(G\) with \(K\) as its set of fixed points. Let \(V\) be some \(({\mathfrak g}_{\mathbb C}, K)\)-module and \(J(V)\) the Jacquet module of \(V\), see [\textit{W. Casselman}, Proc. int. Congr. Math., Helsinki 1978, Vol. 2, 557--563 (1980; Zbl 0425.22019)]. \textit{M. Emerton, D. Nadler} and \textit{K. Vilonen} gave a geometric description of \(J(V)\) in terms of some functor \(\Psi\) (see [Duke Math. J. 125, No. 2, 267--278 (2004; Zbl 1134.22300)]). If \(G=KAN\) is an Iwasawa decomposition, then \(J(V)\) can be regarded as a \(({\mathfrak g}_{\mathbb C}, N_{\mathbb C})\)-module. The authors describe an action of \(N\) on the functor \(\Psi\) in geometric terms.