{
  "id": "2212.13436",
  "version": "v1",
  "published": "2022-12-27T10:20:48.000Z",
  "updated": "2022-12-27T10:20:48.000Z",
  "title": "Almost commuting scheme of symplectic matrices and quantum Hamiltonian reduction",
  "authors": [
    "Pallav Goyal"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.QA"
  ],
  "abstract": "Losev introduced the scheme $X$ of almost commuting elements (i.e., elements commuting upto a rank one element) of $\\mathfrak{g}=\\mathfrak{sp}(V)$ for a symplectic vector space $V$ and discussed its algebro-geometric properties. We construct a Lagrangian subscheme $X^{nil}$ of $X$ and show that it is a complete intersection of dimension $\\text{dim}(\\mathfrak{g})+\\frac{1}{2}\\text{dim}(V)$ and compute its irreducible components. We also study the quantum Hamiltonian reduction of the algebra $\\mathcal{D}(\\mathfrak{g})$ of differential operators on the Lie algebra $\\mathfrak{g}$ tensored with the Weyl algebra with respect to the action of the symplectic group, and show that it is isomorphic to the spherical subalgebra of a certain rational Cherednik algebra of Type $C$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-27T10:20:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum hamiltonian reduction",
      "symplectic matrices",
      "commuting scheme",
      "rational cherednik algebra",
      "symplectic vector space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}