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

本文考虑了严格上三角矩阵的子空间，这些子空间在李括号下与任何上三角矩阵都是稳定的。这些子空间被称为 ad-nilpotent 理想，并且有这样的子空间的 Catalan 数。每一个幂零理想$\mathrm{一世}$ 遇到一个独特的最大幂零轨道 ${\mathcal{哦}}_{\mathrm{一世}}$在所有矩阵的李代数中。论文的主要结果是在Mizuno等人研究的幂幂理想的等价关系下，等价类是幂幂理想$\mathrm{一世}$ 以至于 ${\mathcal{哦}}_{\mathrm{一世}}=\mathcal{哦}$ 对于固定幂零轨道 $\mathcal{哦}$. 我们包括该结果的两种应用，一种用于标志品种上向量束的上同调群的更高消失，另一种用于对称群的仿射 Weyl 群中的 Kazhdan-Lusztig 细胞。最后，讨论了一些组合结果。