Normalizers of ad-nilpotent ideals

Let bb be a Borel subalgebra of a complex simple Lie algebra gg. An ideal c⊂bc⊂b is called ad-nilpotent, if it is contained in [b,b][b,b]. The normalizer of cc in gg is a standard parabolic subalgebra of gg. We give several descriptions of the normalizer: (1) using the weight of an ideal, or (2) the affine Weyl group and integer points in a certain simplex, or (3) a relationship with dominant regions of the Shi arrangement. We also characterise the ad-nilpotent ideals whose normalizer is equal to bb. For sl(n)sl(n) and sp(2n)sp(2n), explicit enumerative results are obtained, which demonstrate a connection with the Motzkin and Riordan numbers, number of directed animals and trinomial coefficients.