A standard example of Nakajima quiver varieties are type A Springer resolutions $\widetilde{\mathcal{N}} \to \mathcal{N}$. In the theory of Springer resolutions it is often beneficial to consider the full Grothendieck-Springer resolution, i.e. to work with the commutative diagram

$$\require{AMScd}\begin{CD} \widetilde{\mathcal{N}} @>>> \widetilde{\mathfrak{g}} \\ @VVV @VVV\\ \mathcal{N} @>>> \mathfrak{g} \end{CD}$$

Is there a similar diagram where the left side is replaced by $\mathfrak{M} \to \mathfrak{M_0}$ for an arbitrary quiver variety.

To be clear, I do not expect that the analogue of the Grothendieck­-Springer resolution is necessarily a quiver variety itself, I merely ask whether to each (nice?) quiver variety one can associate a space that behaves similar to $\widetilde{\mathfrak{g}}$. I'm also intentionally vague about the word "similar", but for example a main feature of the above diagram is that it "lives over" $0 \hookrightarrow \mathfrak{h}//W \leftarrow \mathfrak{h}$.

Alternatively, why should I not expect such a thing to exist?

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    $\begingroup$ I believe the answer is that you get such a deformation by varying which moment map fiber you take in the Hamiltonian reduction construction (and maybe these are even known to be all deformations.) $\endgroup$ – dhy Jan 25 '18 at 15:41
  • $\begingroup$ As well as a deformation by the level of moment maps, we should also take the quotient by a `Weyl group', which is given by reflection functors defined in link.springer.com/article/10.1007%2Fs00208-003-0467-0. $\endgroup$ – Hiraku Nakajima Feb 12 '18 at 9:10

There are of course two moment maps to vary - the complex one and the real one. In most treatments of quiver varieties one fixes the complex level set to be zero and the real level set to a nonzero multiple of the identity, with the zero multiple giving the "quiver affine variety" $\mathfrak M_0$ (best not referred to as the "affine quiver variety" for fear of mis-association).

Very specifically, consider the $A_d$ quiver with only one framing vertex, attached to the first vertex, bearing dimension $n$. As you know, the various choices of dimension vector $(n_i)$ on the gauged vertices give $\mathfrak M=$ the various $d$-step (with possible repeats) flag varieties in $\mathbb C^n$. In this construction, one imposes the "preprojective" condition at each gauged vertex $v$, that the sum of all $2$-step paths $v\to w\to v$ is zero. It is fun to use this to derive that the invariant $X:$ frame $\to v_1\to$ frame satisfies $X^{d+1}=0$. Generically $X$ determines the point in the quiver variety, but not always.

The difference now is to only ask that these sums be multiples $\varepsilon_i$ of the identity, instead of actually zero. Then $X$ satisfies instead $X(X-\varepsilon_1)(X-\varepsilon_1-\varepsilon_2)\cdots = 0$ if I recall correctly, and for generic $(\varepsilon_i)$ the invariant $X$ fully determines the point in the quiver variety, i.e. the quiver variety is an affine variety $GL(n)\big /\prod_i GL(n_i)$. Which is to say, varying the multiples $(\varepsilon_i)$ exactly recovers the Grothendieck-Springer family for $\mathfrak{gl}_n$, in the case that the dimensions on the gauge vertices are $n,n-1,n-2,\ldots,1$.

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  • $\begingroup$ Thank you for the explanation. Do you have any insight on how to construct partial Grothendieck-Springer resolutions (that if the dimension vector is not (n,n-1,...,2,1)) in a quiver variety way? Or conversely have a conceptual reason why I shouldn't expect them to arise from quivers? $\endgroup$ – Clemens Koppensteiner Apr 24 '19 at 17:04
  • $\begingroup$ Have the dimensions on the gauge vectors merely be increasing to the framed vertex, e.g. $$ \begin{matrix} \fbox{n} \\ | \\ d &-& c &-& b &-& a \end{matrix}$$ where $a\leq b\leq c\leq d \leq n$ (without which the stability condition forces the quiver variety to be empty). This one will be $T^* Fl(a,b,c,d;\ n)$. $\endgroup$ – Allen Knutson Apr 25 '19 at 11:42

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