标题： 计算卢斯蒂格-沃根双射 显示英文标题

标题： Computing the Lusztig--Vogan Bijection

摘要： 设 $G$ 是一个连通的复数半单代数群，其李代数为 $\mathfrak{g}$。卢斯蒂格-沃根双射将两个基联系起来，用于紧致导出范畴中 $G$-等变凝聚层在幂零锥 $\mathcal{N}$上的范畴，该幂零锥属于 $\mathfrak{g}$。 一个基由$\Lambda^+$索引，即$G$的主权值的集合，另一个由$\Omega$索引，即由一个幂零轨道$\mathcal{O} \subset \mathcal{N}$和一个不可约的$G$-等变向量丛$\mathcal{E} \rightarrow \mathcal{O}$组成的对$(\mathcal{O}, \mathcal{E})$的集合。 Lusztig--Vogan双射$\gamma \colon \Omega \rightarrow \Lambda^+$的存在性由Bezrukavnikov证明，Achar给出了在类型$A$中计算$\gamma$的算法。在此我们给出了在类型$A$中$\gamma$的组合描述，该描述涵盖了并大大简化了Achar的算法。 显示英文摘要

摘要： 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.