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

The third version, quasi-antichains are shown to be in bijection with elements of L(Cox). arXiv admin note: text overlap with arXiv:1009.1655 by other authors

References in corpus (4)

• Abelian ideals in a Borel subalgebra of a complex simple Lie algebra
• Quantizations of modules of differential operators
• Symmetric Spaces in Supergravity
• Enumeration of ad-nilpotent ideals of parabolic subalgebras for exceptional types