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? 



What is the significance that the Springer resolution is a moment map?
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).


