Title: Hikita conjecture for classical Lie algebras Show Chinese title

Title: Hikita 猜想对于经典李代数

Abstract: Let $G$ be $Sp_{2n}$, $SO_{2n}$ or $SO_{2n+1}$ and let $G^\vee$ be its Langlands dual group. Barbasch and Vogan based on earlier work of Lusztig and Spaltenstein, define a duality map $D$ that sends nilpotent orbits $\mathbb{O}_{e^\vee} \subset \mathfrak{g}^\vee$ to special nilpotent orbits $\mathbb{O}_e\subset \mathfrak{g}$. In a work by Losev, Mason-Brown and Matvieievskyi, an upgraded version $\tilde{D}$ of this duality is considered, called the refined BVLS duality. $\tilde{D}(\mathbb{O}_{e^\vee})$ is a $G$-equivariant cover $\tilde{\mathbb{O}}_e$ of $\mathbb{O}_e$. Let $S_{{e^\vee}}$ be the nilpotent Slodowy slice of the orbit $\mathbb{O}_{e^\vee}$. The two varieties $X^\vee= S_{e^\vee}$ and $X=$ Spec$(\mathbb{C}[\tilde{\mathbb{O}}_e])$ are expected to be symplectic dual to each other. In this context, a version of the Hikita conjecture predicts an isomorphism between the cohomology ring of the Springer fiber $\mathcal{B}_{e^\vee}$ and the ring of regular functions on the scheme-theoretic fixed point $X^T$ for some torus $T$. This paper verifies the isomorphism for certain pairs $e$ and $e^\vee$. These cases are expected to cover almost all instances in which the Hikita conjecture holds when $e^\vee$ regular in a Levi $\mathfrak{l}^\vee\subset \mathfrak{g}^\vee$. Our results in these cases follow from the relations of three different types of objects: generalized coinvariant algebras, equivariant cohomology rings, and functions on scheme-theoretic intersections. We also give evidence for the Hikita conjecture when $e^\vee$ is distinguished. Show Chinese abstract

Abstract: 设$G$为$Sp_{2n}$、$SO_{2n}$或$SO_{2n+1}$，并设$G^\vee$为其朗兰兹对偶群。 巴尔巴斯和沃根基于卢斯蒂格和斯帕尔滕斯坦的早期工作，定义了一个对偶映射$D$，该映射将幂零轨道$\mathbb{O}_{e^\vee} \subset \mathfrak{g}^\vee$映射到特殊幂零轨道$\mathbb{O}_e\subset \mathfrak{g}$。在洛塞夫、梅森-布朗和马特维耶夫斯基的著作中，考虑了这个对偶的一个升级版本$\tilde{D}$，称为精炼的BVLS对偶。 $\tilde{D}(\mathbb{O}_{e^\vee})$是一个$G$-等变覆盖$\tilde{\mathbb{O}}_e$，它是$\mathbb{O}_e$的。 设$S_{{e^\vee}}$是轨道$\mathbb{O}_{e^\vee}$的幂零 Slodowy 切片。 两种类型 $X^\vee= S_{e^\vee}$ 和 $X=$ Spec$(\mathbb{C}[\tilde{\mathbb{O}}_e])$被期望为彼此的辛对偶。 在此背景下，Hikita猜想的一个版本预测了Springer纤维的上同调环 $\mathcal{B}_{e^\vee}$ 与某个环面 $T$的方案论固定点 $X^T$上的正则函数环之间的同构。 本文验证了某些对$e$和$e^\vee$的同构。 这些情况预计涵盖当$e^\vee$在一个 Levi$\mathfrak{l}^\vee\subset \mathfrak{g}^\vee$中正则时，Hikita 猜想成立的几乎所有实例。 我们在这些情况下的结果来自于三种不同类型的对象之间的关系：广义不变式代数、等变上同调环以及方案论交集上的函数。 我们还给出了当$e^\vee$为特殊时 Hikita 猜想的证据。