{
  "id": "2409.13914",
  "version": "v1",
  "published": "2024-09-20T21:44:01.000Z",
  "updated": "2024-09-20T21:44:01.000Z",
  "title": "Hikita conjecture for classical Lie algebras",
  "authors": [
    "Do Kien Hoang"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.CO"
  ],
  "abstract": "Let $G$ be $Sp_{2n}$, $SO_{2n}$ or $SO_{2n+1}$ and let $G^\\vee$ be its Langlands dual group. Barbash 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 $S_{e^\\vee}$ and 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 $\\tilde{\\mathbb{O}}_e^T$ 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-20T21:44:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "classical lie algebras",
      "hikita conjecture predicts",
      "sends nilpotent orbits",
      "langlands dual group",
      "special nilpotent orbits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}