{
  "id": "2410.13519",
  "version": "v1",
  "published": "2024-10-17T13:08:26.000Z",
  "updated": "2024-10-17T13:08:26.000Z",
  "title": "Schubert cells and Whittaker functionals for $\\text{GL}(n,\\mathbb{R})$ part I: Combinatorics",
  "authors": [
    "Doyon Kim"
  ],
  "comment": "33 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We give a formula for a birational map on the Schubert cell associated to each Weyl group element of $G=\\text{GL}(n)$. The map simplifies the UDL decomposition of matrices, providing structural insight into the Schubert cell decomposition of the flag variety $G/B$, where $B$ is a Borel subgroup. An application of the formula includes a new proof of the existence of Whittaker functionals for principal series representations of $\\text{GL}(n,\\mathbb{R})$ via integration by parts. In this paper, we establish combinatorial properties of the birational map and prove auxiliary results.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-17T13:08:26.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "05E14"
    ],
    "keywords": [
      "whittaker functionals",
      "combinatorics",
      "birational map",
      "schubert cell decomposition",
      "principal series representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}