Image of a quiver variety under natural morphism We know that the natural morphism $\pi:\mathfrak{M}_{\theta}(Q,\mathbf{v},\mathbf{w})\rightarrow \mathfrak{M}_0(Q,\mathbf{v},\mathbf{w})$ between a smooth and affine quiver variety is not necessarily surjective, so denote by $\mathfrak{M}_1$ its image. 
E.g. picking the $A_1$ Dynkin quiver, standard stability condition $\theta>0,$ and dimension vectors $\mathbf{v}=4,\mathbf{w}=6,$ one gets the smooth quiver variety $\mathfrak{M}=T^*Gr(4,6)$ being cotangent bundle of Grassmanian, whereas $$\mathfrak{M}_1(A_1,4,6)=\overline{\mathcal{O}_{2211}}\subsetneq \mathfrak{M}_0(A_1,4,6)=\overline{\mathcal{O}_{222}},$$ are closures of nilpotent orbits in $\mathfrak{sl}_6.$ Also, using Nakajima reflection functors one can pass to the surjective setup. Namely, we have $\mathfrak{M}_{\theta>0}(A_1,4,6)\cong \mathfrak{M}_{\theta<0}(A_1,2,6),$ whereas both $\mathfrak{M}_0(A_1,2,6)$ and $\mathfrak{M}_1(A_1,2,6)$ are equal to $\overline{\mathcal{O}_{2211}}.$ The summary of these two examples is that: 


*
 
* $\mathfrak{M}_1$ is a stratified closed subspace of $\mathfrak{M}_0$ = closed union of strata

* $\mathfrak{M}_1$ is an afine quiver variety 

*Moreover, it is an affine quiver variety for the same quiver, though possibly with different dimension vectors 

* Using Nakajima reflection functors, we can get to the setup where $\mathfrak{M}_1$ and $\mathfrak{M}_0$ coincide.


The question is, whether any of 1-4 holds in general, or at least for ADE Dynkin quiver varieties? 
NB We assume that the complex moment parameter is equal to zero.
 A: The answer is completely known for ADE quiver varieties:
 
*

*  Holds for general quiver varieties, as $\pi$ is a projective morphism so its image is a closed Poisson subvariety of $\mathfrak{M}_0$ so it is a closure of a stratum, which is then the closed union of that strata.  
 
*  Holds for ADE quiver varieties as there $\mathfrak{M}_0(\textbf{v},\textbf{w})=\sqcup_{\textbf{v}'\leq \textbf{v}} \mathfrak{M}_0^{reg}(\textbf{v}',\textbf{w}).$ is a stratification, hence the image $\mathfrak{M}_1$ is a closure of $\mathfrak{M}_0^{reg}(\textbf{v}',\textbf{w})$ for some $\textbf{v}',$ hence $\mathfrak{M}_0(\textbf{v}',\textbf{w}).$
 
* True for ADE quiver varieties, follows from 2.  
 
*  Holds for ADE quiver varieties as for them the action of the Weyl group $w *_{\textbf{w}} \textbf{v}$ (that is involved in Nakajima reflection functors) on the vector $\textbf{v}$ can get from an arbitrary $\textbf{v}$, via some $w$, to a dominant vector $\textbf{v}'$, that is, a dimension vector for which $\textbf{w}-C\textbf{v}\geq 0$. This is not true for non-ADE quivers in general.

Still the question is, whether 2-4 are true for general quiver varieties.
