rt.representation theory - Is the real Jacquet module of a Harish-Chandra module still a Harish-Chandra module?

As I understand it, the Jacquet module for $(mathfrak g, K)$-modules is defined so as to again be a $mathfrak g$-module, and in fact it is a Harish-Chandra module, not for $({mathfrak g},K)$, but rather for $(mathfrak g,N)$ (where $N$ is the unipotent radical of the parabolic with respect to which we compute the Jacquet module). (I am probably assuming that the original $(mathfrak g, K)$-module has an infinitesimal character here.)

I am using the definitions of this paper, in particular the discussion of section 2. This in turn refers to Ch. 4 of Wallach's book. So probably this latter reference will cover things in detail.

Added: I may have misunderstood the question (due in part to a confusion on my part about definitions; see the comments below), but perhaps the following remark is helpful:

If one takes the Jacquet module (say in the sense of the above referenced paper,
which is also the sense of Wallach), say for a Borel, then it is a category {mathcal O}-like
object: it is a direct sum of weight spaces for a maximal Cartan in ${mathfrak g},$
and any given weight appears only finitely many times. (See e.g. Lemma 2.3 and Prop. 2.4 in the above referenced paper; no doubt this is also in Wallach in some form; actually
these results are for the geometric Jacquet functor of that paper rather than for Wallach's
Jacquet module, but I think they should apply just as well to Wallach's.

Maybe they also apply with Casselman's definition; if so, doesn't this give the desired admissibility?