A transitivity result for ad-nilpotent ideals in type A

The paper considers subspaces of the strictly upper triangular matrices, which are stable under Lie bracket with any matrix. These called ad-nilpotent ideals and there Catalan number such subspaces. Each ideal I meets a unique largest nilpotent orbit OI in algebra all matrices. main result is that an equivalence relation on studied by Mizuno others, classes OI=O for fixed O. We include two applications result, one to higher vanishing cohomology groups vector bundles flag variety another Kazhdan–Lusztig cells affine Weyl group symmetric group. Finally, some combinatorial results discussed.