This is intended as a general question regarding whether or not a notion of left-invariant vector fields makes sense on a coset space $G/H$ (where, of course, $G$ is a Lie group and $H$ is some subgroup), and how one can determine them; however, for the sake of clarity (hopefully...), I'll illustrate what I'm getting at with an example.
$\fbox{$\textbf{Example:}$}$
$\quad$ Consider the Lie group $W:=\{w(p,x,\theta) \ | \ p,x,\theta\in\mathbb{R}\}$ with the group operation given by
$$ w(p',x',\theta')w(p,x,\theta)=w(p'+p,x'+x,\theta'+\theta+p'x-x'p), $$
called the Heisenberg-Weyl group. This can also be thought of as the unique Lie group obtained by taking the Heisenberg algebra defined by the single commutator (all others being zero) $[X,P]=2iI$ (Note: The factor of 2 here is to compensate for some factors of $1/2$ I missed below -- it's easier to fix it here. Besides, some authors discussing this algebra prefer to use the "$\hbar=2$" units anyways, so... ;) ), and exponentiating it to get
$$ w(p,x,\theta)=\exp\big[i\big(pX-xP+\theta I\big)\big]. $$
One can check that, by applying the Baker-Campbell-Hausdorff formula to a product of the above exponentials will reproduce the group operation defined above.
$\quad$ To find, for example, a representation $D(X)$ of the Lie algebra generator $X$ as a (left-invariant) vector field on $W$, one can consider a function $f\in C^{\infty}(W)$ and compute (setting $f(w(p,x,\theta))\equiv f(p,x,\theta)$, for simplicity)
\begin{align*} -i\frac{d}{dt}\bigg|_{t=0}f(w(-t,0,0)w(p,x,\theta)) &= -i\frac{d}{dt}\bigg|_{t=0}f(p-t,x,\theta-tx) \\[1em] &= -i\bigg[\frac{d(p-t)}{dt}\partial_p+\frac{d(x)}{dt}\partial_x+\frac{d(\theta-tx)}{dt}\partial_\theta\bigg]f(p-t,x,\theta-tx)\bigg|_{t=0} \\[1em] &= i\bigg[\partial_p+x\partial_\theta\bigg]f(p,x,\theta); \end{align*}
hence, we obtain the vector field $D(X)=i(\partial_p+x\partial_\theta)$. Similarly, one has (the standard expressions) $D(P)=-i(\partial_x-p\partial_\theta)$ (keeping in mind that one uses $id/dt$ in this case, due to the $-$ in the exponential for the coefficient of $P$) and $D(I)=i\partial_\theta$.
$\quad$ Now, consider the coset space $\widetilde{W}:=W/H$, where $H=\{(0,0,\theta) \ | \ \theta\in\mathbb{R}\}$. In a similar vein to the above, we can write the elements of this coset space as
$$ \widetilde{w}(p,x)=\exp\big[i\big(pX-xP\big)\big]. $$
The idea here is that we are setting $\widetilde{w}(p,x)=e^{-i\theta}w(p,x,\theta)$.
$\quad$ Even though this clearly doesn't give one a group structure, due to an extra "phase factor" (rewriting things in terms of the elements of $W$, once again using BCH, and collecting the appropriate factors in the $\theta$-direction to go back to $\widetilde{W}$):
$$ \widetilde{w}(p',x')\widetilde{w}(p,x)=\widetilde{w}(p'+p,x'+x)\exp\big[i\big(x'p-p'x\big)\big], $$
for the purposes of physics, this isn't an issue -- quantum states are only unique up to such a phase factor -- and this simply says that a representation of $W$ induces a projective representation on $\widetilde{W}$. Not only that, but they also talk about representing the generators $X$ and $P$ as (left-invariant) vector fields on $\widetilde{W}$, and this finally leads to the question at hand. Trying to repeat the above process in a naive way, one finds:
\begin{align*} -i\frac{d}{dt}\bigg|_{t=0}f(\widetilde{w}(-t,0)\widetilde{w}(p,x)) &= -i\frac{d}{dt}\bigg|_{t=0}f(\widetilde{w}(p-t,x)e^{itx}) \\[1em] \text{"} &= \text{"} -i\frac{d}{dt}\bigg|_{t=0}\bigg[f(p-t,x)e^{itx}\bigg] \\[1em] &= -i\bigg[\frac{d(p-t)}{dt}\partial_p+\frac{d(x)}{dt}\partial_x\bigg]f(p-t,x)e^{itx}\bigg|_{t=0} \\[1em] &\hspace{1.1in} -i^2xf(p-t,x)e^{itx}\bigg|_{t=0} \\[1em] &= (i\partial_p+x)f(p,x,\theta); \end{align*}
thus we arrive at a representation $\widetilde{D}(X)$ of $X$ as a differential operator on $C^{\infty}(\widetilde{W})$ given by $\widetilde{D}(X)=i\partial_p+x$. Similarly, $\widetilde{D}(P)=-i\partial_x+p$ (still keeping in mind that one uses $id/dt$ for this term).
Question: The above successfully reproduces standard expressions in the literature for both the group $W$ and the coset space $W/H$, but is there actually any sense to be made of the "$=$" above? Is it as simple as saying that what we mean by $C^{\infty}(W/H)$ is the collection of functions $f$ satisfying $f(\widetilde{w}\exp(\text{whatever}))=f(\widetilde{w})\exp(\text{whatever})$?
Or, more generally, on an arbitrary coset space $G/H$ can one make sense of left-invariant vector fields in any way remotely resembling this? (Or, if not arbitrary coset spaces, in what cases can you do something along these lines?)
After more thought, I'm fairly certain that I've already answered my own question. Instead of thinking of the operators as acting on something like $C^{\infty}(G)$, as I did in the first part of the example, for a coset space $G/H$ one works with
$$ C^{\infty}_e(G/H):=\{f\in C^{\infty}(G) \ | \ f(ge^{ia^jh_j})=f(g)e^{ia^jh_j}\}, $$
(where the $h_j$ are the generators for the Lie algebra of $H$ and the $a^j$ are parameters) and simply apply exactly the above procedure to get a suitable notion of left-invariant vector fields on $G/H$, which correctly reproduces the standard expressions given in the literature for the example considered, but can be applied more generally.