{
  "id": "2204.10118",
  "version": "v1",
  "published": "2022-04-21T14:14:09.000Z",
  "updated": "2022-04-21T14:14:09.000Z",
  "title": "Regular Functions on the K-Nilpotent Cone",
  "authors": [
    "Lucas Mason-Brown"
  ],
  "comment": "comments welcome!",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a complex reductive algebraic group with Lie algebra $\\mathfrak{g}$ and let $G_{\\mathbb{R}}$ be a real form of $G$ with maximal compact subgroup $K_{\\mathbb{R}}$. Associated to $G_{\\mathbb{R}}$ is a $K \\times \\mathbb{C}^{\\times}$-invariant subvariety $\\mathcal{N}_{\\theta}$ of the (usual) nilpotent cone $\\mathcal{N} \\subset \\mathfrak{g}^*$. In this article, we will derive a formula for the ring of regular functions $\\mathbb{C}[\\mathcal{N}_{\\theta}]$ as a representation of $K \\times \\mathbb{C}^{\\times}$. Some motivation comes from Hodge theory. In arXiv:1206.5547, Schmid and Vilonen use ideas from Saito's theory of mixed Hodge modules to define canonical good filtrations on many Harish-Chandra modules (including all standard and irreducible Harish-Chandra modules). Using these filtrations, they formulate a conjectural description of the unitary dual. If $G_{\\mathbb{R}}$ is split, and $X$ is the spherical principal series representation of infinitesimal character $0$, then conjecturally $\\mathrm{gr}(X) \\simeq \\mathbb{C}[\\mathcal{N}_{\\theta}]$ as representations of $K \\times \\mathbb{C}^{\\times}$. So a formula for $\\mathbb{C}[\\mathcal{N}_{\\theta}]$ is an essential ingredient for computing Hodge filtrations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-21T14:14:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E46"
    ],
    "keywords": [
      "regular functions",
      "k-nilpotent cone",
      "harish-chandra modules",
      "complex reductive algebraic group",
      "maximal compact subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}