Normalizers of ad-nilpotent ideals

Let $\be$ be a Borel subalgebra of a complex simple Lie algebra $\g$. An ideal of $\be$ is called ad-nilpotent, if it is contained in $[\be,\be]$. 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 $\be$. For sl(n) and sp(2n), explicit enumerative results are obtained, which demonstrate a connection with some famous integer sequences.