Computing the Lusztig--Vogan Bijection

TL;DR

This paper provides a simplified combinatorial description of the Lusztig--Vogan bijection in type A, which relates bases of the derived category of $G$-equivariant sheaves on the nilpotent cone, improving upon previous algorithms.

Contribution

It introduces a new combinatorial method for computing the Lusztig--Vogan bijection in type A, simplifying and unifying prior algorithms.

Findings

Provides a simplified combinatorial description of the bijection.

Subsumes and improves upon Achar's algorithm.

Facilitates easier computation of the bijection in type A.

Abstract

Let $G$ be a connected complex reductive algebraic group with Lie algebra $\mathfrak{g}$. The Lusztig--Vogan bijection relates two bases for the bounded derived category of $G$-equivariant coherent sheaves on the nilpotent cone $\mathcal{N}$ of $\mathfrak{g}$. One basis is indexed by $\Lambda^+$, the set of dominant weights of $G$, and the other by $\Omega$, the set of pairs $(\mathcal{O}, \mathcal{E})$ consisting of a nilpotent orbit $\mathcal{O} \subset \mathcal{N}$ and an irreducible $G$-equivariant vector bundle $\mathcal{E} \rightarrow \mathcal{O}$. The existence of the Lusztig--Vogan bijection $\gamma \colon \Omega \rightarrow \Lambda^+$ was proven by Bezrukavnikov, and an algorithm computing $\gamma$ in type $A$ was given by Achar. Herein we present a combinatorial description of $\gamma$ in type $A$ that subsumes and dramatically simplifies Achar's algorithm.