Ad-nilpotent ideals and The Shi arrangement

TL;DR

This paper generalizes the Shi arrangement by introducing the $I$-deleted Shi arrangement, extending the Shi bijection to parabolic subalgebras, and provides explicit formulas for their characteristic polynomials across classical types.

Contribution

It extends the Shi bijection to parabolic subalgebras and derives explicit characteristic polynomial formulas for the $I$-deleted Shi arrangements in classical types.

Findings

Explicit formulas for $ exttt{Shi}(I)$ characteristic polynomial in type A and C.

Generalized formulas for $ exttt{Shi}(G)$ in types B, C, D.

Interprets arrangements as interpolations between Coxeter and Shi arrangements.

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 $D_n$.