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Let $\mathcal{B}$ be the flag variety and $\mathcal{N} \subset \mathfrak{g}$ is the nilpotent cone. We know that the Springer resolution $$ \mu: T^*\mathcal{B}\rightarrow \mathcal{N} $$

is the moment map, if we identify $\mathfrak{g}$ with $\mathfrak{g}^* $ by the Killing form and consider $\mathcal{N} \subset \mathfrak{g}$ as a subset of $\mathfrak{g}^*$.

As far as I know, the geometric construction of Weyl group and $U(sl_n)$ does not involve moment map or even symplectic geometry, as in the paper "Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups"

My question is: what is the consequence of the fact that the Springer resolution is a moment map?

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2 Answers 2

up vote 11 down vote accepted

One reason to emphasize the Springer resolution's role as a moment map is that it is the semiclassical shadow of Beilinson-Bernstein localization. More precisely passing to functions, the moment map description asserts that the Springer map is describing the Hamiltonian functions on the cotangent to the flag variety which generate the action of the Lie algebra. We may now quantize the cotangent bundle $T^* G/B$ to the ring of differential operators on $G/B$, and likewise quantize the dual space $g^*$ to the Lie algebra to the universal enveloping algebra $Ug$, so that the moment map describes the map from $Ug$ to global differential operators on the flag variety. What's truly significant about the Springer map (it's a birational, proper, symplectic [crepant] resolution of [rational] singularities) now translates into the Beilinson-Bernstein equivalence (for generic parameters) between $Ug$-modules and (twisted) D-modules on the flag variety, the cornerstone of geometric representation theory. There's now an entire subject (wonderfully represented in a workshop last week in Luminy) seeking to generalize all the features of this setup to other symplectic resolutions and their quantizations, viewed as the settings for "new representation theories" (the prime examples being Hilbert schemes and other quiver varieties).

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Thank you very much, David! I am trying to rephrase what you said: Since $\mu: T^*\mathcal{B}\rightarrow \mathcal{N}$ is the moment map, by considering vector fields on $\mathcal{B}$ as functions on $T^*\mathcal{B}$, we can identify the comoment map $$ \mathfrak{g}\rightarrow C^{\infty}(T^*\mathcal{B}) $$ with the infinitesimal action $$ \mathfrak{g}\rightarrow T\mathcal{B}. $$ Then we quantize the later one and get the map from $U(\mathfrak{g})$ to global differential operators on the flag variety. Now we can construct the Beilinson-Bernstein equivalence. Is that (roughly) what you mean? –  Zhaoting Wei Jul 18 '12 at 5:01
exactly! Another way to think of this is that $\mu$ is a degenerate version of the inclusion of a generic (regular semisimple) coadjoint orbit into $\mathfrak g^*$ - those orbits are affine bundles over $G/B$ (twisted cotangent bundles) and degenerate, as the eigenvalues go to zero, to the Springer resolution. –  David Ben-Zvi Jul 18 '12 at 15:24

The short answer might be that this viewpoint provides an attractive alternative way to construct the Springer resolution as a special case in a broader geometric framework, following ideas of Kostant and Souriau. I'm not at all qualified to attempt a deeper explanation of the significance of this viewpoint, but what I can do is encourage you to explore the literature beyond what Ginzburg does in his book with Chriss or in his summary lecture notes you quote from the proceedings of the 1997 conference at U. Montreal (posted on arXiv here). In expositions it is convenient for people to use the special linear case as the main example, but this is of course misleading as to the delicate complications in the general case which encourage the development of multiple approaches.

In particular, during the 1980s there was important parallel work being done on several related problems by Walter Borho, Jean-Luc Brylinski, Robert MacPherson, and others. Some detailed references occur in my attempted review of a paper by Ginzburg (as posted on MathSciNet): MR847727 (87k:17014) 17B35 Ginsburg, V. [Ginzburg, Victor], $\mathfrak{g}$-modules, Springer’s representations and bivariant Chern classes. Adv. in Math. 61 (1986), no. 1, 1–48. [Note that the first symbol in the title was originally printed upper-case but refers to a Lie algebra.]

I guess it's legal to quote my concluding reviewer's remark: "The subject matter spills over many of the conventional dividing lines between disciplines. In their abstracts prepared for the International Congress of Mathematicians in Berkeley (1986), the author and Borho deal with many of the same issues, but the author’s occurs in the section “Lie groups and representations”, while Borho’s occurs in the section “Algebra”. Both might equally well be placed in the section “Algebraic geometry”."

It's useful to look at those ICM reports (now available online at http://www.mathunion.org/ICM/) as well as Borho's incomplete lecture notes in the Canad. Math. Soc. Conf. Proc. 5 (1986), whose interesting Part II wasn't published. The more technical research papers by Borho and Brylinski in Invent. Math. 1982 and 1985 (parts I, III with a gap between) give a clearer idea of how they fit the pieces together. Then there is the 1989 B-B-M monograph Nilpotent Orbits, Primitive Ideals, and Characteristic Classes, Birkhauser series Progress in Mathematics, 78. The moral of the story seems to be that more than just the isolated construction of the Springer resolution is at stake here. I realize this doesn't directly answer your question, but may only complicate it further.

[ADDED] After taking another look at Borho's notes, my understanding is that the moment map is used initially to place the construction of the Springer resolution into an already understood classical picture. Here the flag variety is a complete variety $X$ with a natural action of the Lie algebra (viewed as vector fields). The action induces $\mu: T^*X \rightarrow \mathfrak{g^*}$ (identified with $\mathfrak{g}$ via the Killing form). In turn, some of the general theory allows one to see in the special case that the image of $\mu$ is precisely the nilpotent cone $\mathcal{N}$ (using the orthogonality of the nilradical of a Borel subalgebra to that algebra under the Killing form and taking the saturation of the nilradical under the reductive group). Similarly one sees that $\mathcal{N}$ is normal (Kostant) and that $\mu$ is a resolution of singularities.

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