Stability and Hamiltonian Reduction for Grothendieck-springer Resolutions

Let G = GLn(C) and e gln the Grothendieck-Springer resolution of its Lie algebra. Assuming that the zero pre-image μ−1(0) of the moment map μ : T ∗(e gln × Cn) → gln is a complete intersection, we compute its irreducible components. These components dominate components of the corresponding moment pre-image for T (gln×C). We then analyze GIT stability of the irreducible components of μ−1(0) for various stability conditions. Unlike the case of T (gln × Cn), in which GIT quotients—both isomorphic to the Hilbert scheme (C2)[n]—can arise from only two of the n + 1 irreducible components, every component of μ−1(0) appears as a GIT quotient of μ−1(0) in the Grothendieck-Springer setting.