{
  "id": "2309.11191",
  "version": "v1",
  "published": "2023-09-20T10:25:32.000Z",
  "updated": "2023-09-20T10:25:32.000Z",
  "title": "On Harish-Chandra modules over quantizations of nilpotent orbits",
  "authors": [
    "Ivan Losev",
    "Shilin Yu"
  ],
  "comment": "80 pages",
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "Let $G$ be a semisimple algebraic group over the complex numbers and $K$ be a connected reductive group mapping to $G$ so that the Lie algebra of $K$ gets identified with a symmetric subalgebra of $\\mathfrak{g}$. So we can talk about Harish-Chandra $(\\mathfrak{g},K)$-modules, where $\\mathfrak{g}$ is the Lie algebra of $G$. The goal of this paper is to give a geometric classification of irreducible Harish-Chandra modules with full support over the filtered quantizations of the algebras of the form $\\mathbb{C}[\\mathbb{O}]$, where $\\mathbb{O}$ is a nilpotent orbit in $\\mathfrak{g}$ with codimension of the boundary at least $4$. Namely, we embed the set of isomorphism classes of irreducible Harish-Chandra modules into the set of isomorphism classes of irreducible $K$-equivariant suitably twisted local systems on $\\mathbb{O}\\cap \\mathfrak{k}^\\perp$. We show that under certain conditions, for example when $K\\subset G$ or when $\\mathfrak{g}\\cong \\mathfrak{so}_n,\\mathfrak{sp}_{2n}$, this embedding is in fact a bijection. On the other hand, for $\\mathfrak{g}=\\mathfrak{sl}_n$ and $K=\\operatorname{Spin}_n$, the embedding is not bijective and we give a description of the image. Finally, we perform a partial classification for exceptional Lie algebras.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-20T10:25:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "17B10",
      "17B35"
    ],
    "keywords": [
      "nilpotent orbit",
      "irreducible harish-chandra modules",
      "quantizations",
      "isomorphism classes",
      "equivariant suitably twisted local systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 80,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}