{
  "id": "2406.07279",
  "version": "v1",
  "published": "2024-06-11T14:06:10.000Z",
  "updated": "2024-06-11T14:06:10.000Z",
  "title": "$\\mathscr{D}$-modules on the basic affine space and large $\\mathfrak{g}$-modules",
  "authors": [
    "Masatoshi Kitagawa"
  ],
  "comment": "30 pages",
  "categories": [
    "math.RT"
  ],
  "abstract": "In this paper, we treat $\\mathscr{D}$-modules on the basic affine space $G/U$ and their global sections for a semisimple complex algebraic group $G$. Our aim is to prepare basic results about large non-irreducible modules for the branching problem and harmonic analysis of reductive Lie groups. A main tool is a formula given by Bezrukavnikov--Braverman--Positselskii. The formula is about a product of functions and their Fourier transforms on $G/U$ like Capelli's identity. Using the formula, we give a generalization of the Beilinson--Bernstein correspondence. We show that the global sections of holonomic $\\mathscr{D}$-modules are also holonomic using the formula. As a consequence, we give a large algebra action on the $\\mathfrak{u}$-cohomologies $H^i(\\mathfrak{u}; V)$ of a $\\mathfrak{g}$-module $V$ when $V$ is realized as a holonomic $\\mathscr{D}$-module. We consider affinity of the supports of the $\\mathfrak{t}$-modules $H^i(\\mathfrak{u}; V)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-11T14:06:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "basic affine space",
      "global sections",
      "semisimple complex algebraic group",
      "prepare basic results",
      "large algebra action"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}