Ad-nilpotent ideals and the Shi arrangement

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the I-deleted Shi arrangement Shi(I) naturally emerges. This arrangement interpolates between the Coxeter arrangement Cox and the Shi arrangement Shi, and breaks the symmetry of Shi in a certain symmetrical way. Among other things, we determine the characteristic polynomial χ(Shi(I),t) of Shi(I) explicitly for An−1 and Cn. More generally, let Shi(G) be an arbitrary arrangement between Cox and Shi. Armstrong and Rhoades recently gave a formula for χ(Shi(G),t) for An−1. Inspired by their result, we obtain formulae for χ(Shi(G),t) for Bn, Cn and Dn.