Resolution of singularities for nilpotent cone of the symplectic group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:21:38Z http://mathoverflow.net/feeds/question/7858 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7858/resolution-of-singularities-for-nilpotent-cone-of-the-symplectic-group Resolution of singularities for nilpotent cone of the symplectic group Vinoth 2009-12-05T09:55:27Z 2009-12-05T17:38:01Z <p>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). </p> http://mathoverflow.net/questions/7858/resolution-of-singularities-for-nilpotent-cone-of-the-symplectic-group/7872#7872 Answer by Ben Webster for Resolution of singularities for nilpotent cone of the symplectic group Ben Webster 2009-12-05T15:57:25Z 2009-12-05T17:38:01Z <p>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 <a href="http://front.math.ucdavis.edu/9802.5004" rel="nofollow">Ginzburg's notes</a>.</p> <p>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}\$).</p> http://mathoverflow.net/questions/7858/resolution-of-singularities-for-nilpotent-cone-of-the-symplectic-group/7874#7874 Answer by Mike Skirvin for Resolution of singularities for nilpotent cone of the symplectic group Mike Skirvin 2009-12-05T16:21:04Z 2009-12-05T16:21:04Z <p>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.</p> <p>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).</p>