Hikita conjecture for classical Lie algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912078034370560 |
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| author | Hoang, Do Kien |
| author_facet | Hoang, Do Kien |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_13914 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hikita conjecture for classical Lie algebras Hoang, Do Kien Representation Theory Algebraic Geometry Combinatorics 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. |
| title | Hikita conjecture for classical Lie algebras |
| topic | Representation Theory Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2409.13914 |