Ad-nilpotent ideals and The Shi arrangement

Abstract

We extend the Shi bijection from the Borel subalgebra case to parabolic subalgebras. In the process, the $I$-deleted Shi arrangement $\texttt{Shi}(I)$ naturally emerges. This arrangement interpolates between the Coxeter arrangement $\texttt{Cox}$ and the Shi arrangement $\texttt{Shi}$, and breaks the symmetry of $\texttt{Shi}$ in a certain symmetrical way. Among other things, we determine the characteristic polynomial $\chi(\texttt{Shi}(I), t)$ of $\texttt{Shi}(I)$ explicitly for $A_{n-1}$ and $C_n$. More generally, let $\texttt{Shi}(G)$ be an arbitrary arrangement between $\texttt{Cox}$ and $\texttt{Shi}$. Armstrong and Rhoades recently gave a formula for $\chi(\texttt{Shi}(G), t)$ for $A_{n-1}$. Inspired by their result, we obtain formulae for $\chi(\texttt{Shi}(G), t)$ for $B_n$, $C_n$ and $D_n$.