Let $G$ be a real reductive Lie group, $P$ its parabolic subgroup with Levi decomposition $P=MN$, let $\mathfrak{n}$ be the nilpotent Lie algebra of $N$. Suppose given a smooth representation $(\pi,V)$ of $G$ with $V$ some Frechet space. We can form both $V/(\mathfrak{n}V)$ and $V/(N.V),$ where

$$
N.V=\operatorname{span}\{n.v-v|v\in V, n\in N\}
$$

(here we take the closure of $\mathfrak{n}V$ and $N.V$ in the two quotients). Now question is there any difference between these two quotients? or equivalently, what's the relation between the closure of $\mathfrak{n}V$ and that of $N.V$? On one direction we have $\mathfrak{n}V$ is contained in $N.V$, so the question amounts to ask whether there is some $V$, so that the latter contains the former properly.

A related phenomenon is that, when considering the minimal parabolic subgroup $B=TU$, if $\chi$ is a generic character of $U$, $\eta$ the derivative of $\chi$, so $\eta$ is a nondegenerate complex linear form on the Lie algebra $\mathfrak{u}$ of $U$. There are two versions of Whittaker functional in literature, defined either in terms of pair $(U,\chi)$ or $\mathfrak{u},\eta$. And I'm wondering if they are equivalent.

Edit (Victor Protsak): definition of N.V has been made explicit.