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What is the standard resolution of singularities, for the nilpotent cone (of the adjoint representation) for the symplectic group? I know how to do this for the general linear group, but am having trouble finding a good reference for the symplectic group. I understand it uses Richardson orbits. (I do know what the closure ordering should be though).

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Ben gave the general answer above. If you care specifically about the symplectic group and are interested in a "flag-like" description of its flag variety, then one exists. It is given by all half-flags of isotropic subspaces (this is just like for $SL_n$, the symplectic group acts transitively and the stabilizer of the standard half-flag will be the standard Borel). With this description, it's just as straightforward computing Springer fibers and the like as it is for the $SL_n$ case, which you're presumably familiar with.

A reference for these flag-like descriptions can be found in the section of Fulton and Harris on "Homogeneous Spaces" (there's a similar description for the special orthogonal groups).

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It's the Springer resolution; this works for all semi-simple groups. The resolution is the moment map from the cotangent bundle of $G/B$ to $\mathfrak{g}^*$. Look at section 6 of Ginzburg's notes.

Not to steal Mike's thunder, but there's an even more specific description of this resolution: given a nilpotent element $X$ of the symplectic Lie algebra, the fiber in this resolution over it is the space of complete flags $V_1\subset V_2\subset \cdots \subset \mathbb{C}^{2n}$ such that $V_i$ and $V_{2n-i}$ are symplectic orthogonal (this immediately implies that all the spaces in this flag are isotropic or coisotropic) which are preserved by $X$ ($XV_i\subset V_{i-1}$).

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