Ad-nilpotent ideals of minimal dimension

We prove the conjecture in [5] and [10] about the lower bounds of ad-nilpotent ideals with the same associated orbit. More precisely, for each nilpotent orbit of classical type, we construct explicit ad-nilpotent ideals of minimal dimension.For type An−1An−1, an explicit dimension formula for minimal ideals is provided based on partitions of n.