Lie algebra stability implies Lie group stability for a certain representation Suppose $G$ is a connected complex algebraic group with lie algebra $\mathfrak{g}$ and $V$ a (potentially infinite-dimensional) vector space with actions of $G$ and $\mathfrak{g}$ which are 'compatible', i.e. for all $g\in G$, $X\in\mathfrak{g}$, $v\in V$, we have
$$g\cdot X\cdot v=Ad(g)(X)\cdot g\cdot v$$
Note that I am not assuming that $V$ is a 'nice' representation of $G$, I'm only assuming it is a group action by its complex points, $G(\mathbb{C})$.  Although see below for where the action originates.
My question is: suppose that $\mathfrak{g}$ stabilizes some finite-dimensional subspace $W\subseteq V$.   Then must $G$ also stabilize $W$?
Actual motivation and setting of question: I have an action of $G$ on an algebraic variety $X$, and there is a finite-dimensional subspace $W$ of the field of rational functions on $X$, that the action of $\mathfrak{g}$ preserves, and I want to know if that implies $G$ must preserve it as well.

I see no one has answered your question, so I will try.
I think the answer is no. Just consider an any Lie group $G$ action on a vector space and have its lie algebra $\mathfrak{g}$ act trivially on it. Then the actions will be compatible. But, any subspace will be stabilized by $\mathfrak{g}$, although some may not be stabilized by $G$.
I think you need more hypothesis to get the result.
