Normalizers of ad-nilpotent ideals

Let b be a Borel subalgebra of a complex simple Lie algebra g . An ideal c ⊂ b is called ad-nilpotent, if it is contained in [ b , b ] . The normalizer of c in g is a standard parabolic subalgebra of g . 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 b . For s l ( n ) and s p ( 2 n ) , explicit enumerative results are obtained, which demonstrate a connection with the Motzkin and Riordan numbers, number of directed animals and trinomial coefficients.