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 A n − 1 and C n . 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 A n − 1 . Inspired by their result, we obtain formulae for χ ( Shi ( G ) , t ) for B n , C n and D n .