Normalizers of ad-nilpotent ideals

Let $\text{be}$ be a Borel subalgebra of a complex simple Lie algebra $\text{g}$. An ideal of $\text{be}$ is called ad-nilpotent, if it is contained in $[\text{be},\text{be}]$. We give several descriptions of the normalizer of an ad-nilpotent ideal: using the weight of an ideal, or the affine Weyl group, or a relationship with dominant regions of the Shi arrangement. We also give a description of those ideals whose normalizer is equal to $\text{be}$. For sl(n) and sp(2n), explicit enumerative results are obtained, which demonstrate a connection with some famous integer sequences.