How should I think about the Grothendieck-Springer alteration?

Question

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 $\widetilde{\mathfrak{g}}$.

Answer 1

For Question 1 I agree with dhy that Namikawa's work is the relevant place to start, and from there the booming field of symplectic representation theory, which from one perspective is all about generalizations of the Grothendieck-Springer resolution (cf. a host of papers by among others Braden-Licata-Proudfoot-Webster, Losev, Braverman-Maulik-Okounkov, McGerty-Nevins, Braverman-Finkelberg-Nakajima, Bezrukavnikov, etc etc). For a context from physics this fits into the theory of moduli spaces of vacua of 3d N=4 gauge theories.

For question 2, I find the most convincing reason the Grothendieck-Springer resolution appears everywhere in representation theory to be that it's just a geometric reflection of parabolic induction/restriction, perhaps the most fundamental relation in the theory of reductive groups. Given $B\subset G$ a Borel and $B\twoheadrightarrow T$ the quotient torus, we get a correspondence $$pt/G \leftarrow pt/B \rightarrow pt/T$$ and (suitable) push-pull along this gives parabolic induction/restriction. Now pass to (shifted) cotangent bundles $$\mathfrak g^*/G \leftarrow \mathfrak b^*/B =\widetilde{\mathfrak g}/G\rightarrow \mathfrak t^*/T$$ and you find on the left the equivariant Grothendieck-Springer map, while the map to $\mathfrak t^*$ is the lift of the characteristic polynomial map. (See e.g. Safronov's https://arxiv.org/abs/1411.2962 or https://arxiv.org/abs/1709.07698 for more on this perspective.) Whether or not you care about (shifted) cotangent bundles the point is it's just a form of the fundamental parabolic induction correspondence. This is the "reason" it relates to Beilinson-Bernstein, representations of Hecke and Weyl groups (``Springer theory") and pretty much any context in which reductive groups appear.

Answer 2

The answer to 1) is that this is a special case of a broader phenomenon for symplectic resolutions (though I think some features are specific to the Grothendieck-Springer case.) For instance you have such a deformation for quiver varieties. I'm not sure what general results have been proven, but I think you can find some statements in papers of Namikawa, e.g. https://arxiv.org/abs/0902.2832. In any case, there are people on this website who are much better suited for answering this question - hopefully one of them can elaborate.

For 2), the Grothendieck-Springer resolution also naturally arises in the context of Beilinson-Bernstein. Namely, note that the usual versions of Beilinson-Bernstein involve fixing an integral Harish-Chandra central character and (at an imprecise level) relate the category of $U\mathfrak{g}$ modules at that central character and a category of $\mathcal{D}$-modules at the corresponding dominant twist. These two sides roughly come from quantizing the nilpotent cone and the cotangent bundle of the flag variety (this is the "semiclassical shadow" you mention.)

One the other hand, you can consider all possible twists at once. On the $\mathcal{D}$-module side, you have a $\mathfrak{h}^*$'s worth of twists, but on the universal enveloping algebra side, you only have a $\mathfrak{h}^*/W$'s worth of twists. That's because this situation corresponds exactly to quantizing the Grothendieck-Springer resolution, with its natural Poisson structure (instead of a symplectic structure). More explicitly, you have a sheaf $\tilde{\mathcal{D}},$ with center $\operatorname{Sym}(\mathfrak{h})$, on $G/B$ such that if you take the quotient with central character $\lambda$, you get the sheaf of $\lambda$-twisted differential operators. Taking associated graded transforms $\tilde{\mathcal{D}}$ into the symmetric algebra associated to $\tilde{\mathfrak{g}}$, viewed as a vector bundle on $G/B.$ On the other side, taking associated graded of $U\mathfrak{g}$ recovers functions on $\mathfrak{g}$. Therefore, if you consider the big (i.e. simulatneously w.r.t. all twists) localization functor $M\mapsto M\otimes_{U\mathfrak{g}}\tilde{\mathcal{D}},$ this quantizes pullback along the Grothendieck-Springer resolution. Because of the $|W|$-to-$1$ nature of the map, the geometry here is well-suited for the study of intertwining functors (which compare localization at weights in the same Weyl group orbit), and is used e.g. in Beilinson-Ginzburg https://arxiv.org/abs/alg-geom/9709022.

BTW, you can explicitly see this "semiclassical limit" in the characteristic $p$ setting, where quantum is much closer to classical. For this see the sequence of papers starting with Bezrukavnikov-Mirkovic-Rumynin, https://arxiv.org/abs/math/0205144. Again, there the difference between Grothendieck-Springer and Springer comes from what versions of central character conditions you want to impose. When you want to study all central characters at once (or even if you just care about the formal neighborhood of one central character, i.e. requiring the Harish-Chandra center to act via a generalized central character instead of strict equality), you need Grothendieck-Springer.

Let me mention one last setting where the difference between Springer and Grothendieck-Springer appears, when you want to relate equivariant coherent (derived) categories of Springer-like gadgets and categories of perverse sheaves on affine flag varieties/grassmannians (you can think of these theorems as cases of geometric Langlands on $P^1$ with points of tame ramification, composed with the long intertwining functor.) The decategorified version is a a theorem of Kazhdan-Lusztig computing equivariant K-theory of the Steinberg variety to be the affine hecke algebra (see chapter 7 of Chriss-Ginzburg.)

There are many similar such theorems in papers of Bezrukavnikov and others - let me take a specific such theorem which appears in https://arxiv.org/abs/1209.0403v4. Recall that Beilinson & Bernstein relate category O to perverse sheaves on the flag variety constant along the Schubert stratification. There are various versions of this latter category, e.g. I can take perverse sheaves on $G/B$ equivariant with respect to either $N$ or $B$,. Now let me move to the affine setting, so that $G$ and $B$ get replaced by the loop group $G(K)$ and the Iwahori $I.$ I can take either $I$- or $I_0$- (the unipotent radical of $I$) equivariant perverse sheaves on $G(K)/I$, the affine flag variety, and these two categories (or rather their derived versions) I will denote by $D_{II}$ and $D_{I_0I}$. Now Bezrukavnikov's theorem matches $D_{II}$ and $D_{I_0I}$ up with the derived categories of $G_L$-equivariant sheaves on $\tilde{\mathcal{N}}_L x_{\mathfrak{g}_L}\tilde{\mathcal{N}}_L$ and $\tilde{\mathcal{N}}_L x_{\mathfrak{g}_L}\tilde{\mathfrak{g}}_L$, respectively. So here which one to use between Springer and Grothendieck-Springer depends on whether you are considering $I-$ or $I_0$-equivariant sheaves.

Answer 3

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.