# Quiver variety analogue of Grothendieck-Springer resolution

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?

• 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.) – dhy Jan 25 '18 at 15:41
• 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. – 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 $$n,n-1,n-2,\ldots,1$$.
• 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 $T^* Fl(a,b,c,d;\ n)$. – Allen Knutson Apr 25 at 11:42