TITLE: How should I think about the Grothendieck-Springer alteration?
QUESTION [10 upvotes]: Given a simple complex Lie algebra $\mathfrak{g}$, recall the Springer resolution of its nilpotent cone $\widetilde{\mathcal{N}}\to \mathcal{N}$. Several times I have seen someone explaining Springer theory in terms of perverse sheaves, and each time the existence of the Grothendieck-Springer alteration $\pi: \widetilde{\mathfrak{g}}\to\mathfrak{g}$ along with the diagram
$$\require{AMScd}\begin{CD}
\widetilde{\mathcal{N}} @>>> \widetilde{\mathfrak{g}} @>>>\mathfrak{t}\\
@VVV @VVV @VVV\\
\mathcal{N} @>>> \mathfrak{g}@>>>\mathfrak{t}/W
\end{CD}$$
 is magically pulled out of a hat (eg: There also exist this other thing that...), where $\mathfrak{t}$ is the universal Cartan and $W$ is the Weyl group. Then one uses the fact that $\pi$ is a small map, giving an IC sheaf $\pi_\ast\underline{\mathbb{Q}}_\widetilde{\mathfrak{g}}$ with a natural $W$-action, which in turn induces a $W$-action on the Springer sheaf by some functoriality. I find this unsatisfying because it seems like $\widetilde{\mathfrak{g}}$ is kept mysterious. 
My vague question is how should I think about the Grothendieck-Springer resolution and what is its role in modern representation theory? I know this is not a good question, so let me try to refine it by asking the two following questions. 

1) Is there a broader theoretical context to fit the above diagram into where I am given a resolution of singularities $X_0\to Y_0$ (maybe with $X$ symplectic?), and can find a smooth family $X\to T$ and a proper map of smooth varieties $X\to Y$ fitting into the diagram
  $$\require{AMScd}\begin{CD}
X_0 @>>> X \\
@VVV @VVV\\
Y_0 @>>> Y,
\end{CD}$$
  or is the Springer map special in a sense I don't understand?

Aside from applications to proving a generalized Springer correspondence, are there other examples where the existence and properties of this remarkable space are used in representation theory?

2) What are other applications of the Grothendieck-Springer resolution? 

For example, the Springer resolution can be interpreted as a moment map, and David Ben-Zvi's answer to this question shows how this may be interpreted as the semiclassical shadow to Beilinson-Bernstein localization. Is there an analogous quantization of $\widetilde{\mathfrak{g}}\to \mathfrak{g}$? EDIT: I would be particularly interested in applications which are not so closely connected with the Springer resolution.
I'll stop here, since I have probably already asked too many questions. I would greatly appreciate any references to a modern understanding of $\widetilde{\mathfrak{g}}$.

REPLY [3 votes]: For question 1, the precise statement is due to Namikawa, but perhaps best summarized in Proposition 2.7 of the paper by Braden-Proudfoot-Webster, https://arxiv.org/abs/1208.3863.  To fit with the notation of the question, let $ X_0 \rightarrow Y_0 $ be a conical symplectic resolution.  (This means that the map is a projective resolution, that $ X $ is symplectic, and we have an action of $ \mathbb C^\times $ on the pair which acts with positive weight on the symplectic form.)  
In this case, there exists a universal Poisson deformation $ X \rightarrow H^2(X_0, \mathbb C) $.  We can then define $ Y $ to be the affinization of $ X$ and we obtain the diagram
$$\require{AMScd}\begin{CD}
X_0 @>>> X @>>> H^2(X_0, \mathbb C) \\
@VVV @VVV @VVV\\
Y_0 @>>> Y @>>> H^2(X_0, \mathbb C)/W
\end{CD}$$
where $ W $ is Namikawa's Weyl group.
In the case where $ Y_0 = \mathcal N $, this recovers the Grothendieck-Springer diagram above.
There are lots of nice examples of this diagram in general.  Of course, the one I like best is when $ Y_0 $ is an affine Grassmannian slice.