{
  "id": "2312.12031",
  "version": "v1",
  "published": "2023-12-19T10:32:00.000Z",
  "updated": "2023-12-19T10:32:00.000Z",
  "title": "Towards a theta correspondence in families for type II dual pairs",
  "authors": [
    "Gilbert Moss",
    "Justin Trias"
  ],
  "comment": "42 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $R$ be a commutative $\\mathbb{Z}[1/p]$-algebra, let $m \\leq n$ be positive integers, and let $G_n=\\text{GL}_n(F)$ and $G_m=\\text{GL}_m(F)$ where $F$ is a $p$-adic field. The Weil representation is the smooth $R[G_n\\times G_m]$-module $C_c^{\\infty}(\\text{Mat}_{n\\times m}(F),R)$ with the action induced by matrix multiplication. When $R=\\mathbb{C}$ or is any algebraically closed field of banal characteristic compared to $G_n$ and $G_m$, the local theta correspondence holds by the work of Howe and M\\'inguez. At the level of supercuspidal support, we interpret the theta correspondence as a morphism of varieties $\\theta_R$, which we describe as an explicit closed immersion. For arbitrary $R$, we construct a canonical ring homomorphism $\\theta^\\#_{R} : \\mathfrak{Z}_{R}(G_n)\\to \\mathfrak{Z}_{R}(G_m)$ that controls the action of the center $\\mathfrak{Z}_{R}(G_n)$ of the category of smooth $R[G_n]$-modules on the Weil representation. We use the rank filtration of the Weil representation to first obtain $\\theta_{\\mathbb{Z}[1/p]}^\\#$, then obtain $\\theta^\\#_R$ for arbitrary $R$ by proving $\\mathfrak{Z}_R(G_n)$ is compatible with scalar extension. In particular, the map $\\text{Spec}(\\mathfrak{Z}_R(G_m))\\to \\text{Spec}(\\mathfrak{Z}_R(G_n))$ induced by $\\theta_R^\\#$ recovers $\\theta_R$ in the $R=\\mathbb{C}$ case and in the banal case. We use gamma factors to prove $\\theta_R^\\#$ is surjective for any $R$. Finally, we describe $\\theta^\\#_R$ in terms of the moduli space of Langlands parameters and use this description to give an alternative proof of surjectivity in the tamely ramified case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-12-19T10:32:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F27",
      "11S23",
      "20C20",
      "22E50"
    ],
    "keywords": [
      "dual pairs",
      "weil representation",
      "local theta correspondence holds",
      "matrix multiplication",
      "langlands parameters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}