{
  "id": "2210.17249",
  "version": "v1",
  "published": "2022-10-31T12:15:43.000Z",
  "updated": "2022-10-31T12:15:43.000Z",
  "title": "On commutations of derivatives and integrals of $\\square$-irreducible representations for $p$-adic $\\mathrm{GL}$",
  "authors": [
    "Kei Yuen Chan"
  ],
  "comment": "29 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $G_n$ be an inner form of the general linear group over a non-Archimedean field $F$. For a $\\square$-irreducible representation $\\sigma$ of $G_n$ and an irreducible representation $\\pi$ of $G_m$, the parabolically induced modules $\\sigma \\times \\pi$ and $\\pi \\times \\sigma$ have irreducible socles. This was conjectured by Leclerc (2003) for quantum affine algebras, and is now proved by Kang-Kashiwara-Kim-Oh (2015). For $p$-adic general linear groups, it is explicated by Lapid-M\\'inguez. Denote the socles respectively by $I^L_{\\sigma}(\\pi)$ and $I^R_{\\sigma}(\\pi)$, called integrals. We denote respectively the inverse operators by $D^L_{\\sigma}$ and $D^R_{\\sigma}$, called derivatives. We study a sufficient condition arising from geometric lemma such that the commutation: \\[ D^R_{\\sigma}\\circ I^L_{\\sigma'}(\\pi) \\cong I^L_{\\sigma'}\\circ D^R_{\\sigma}(\\pi) \\] holds. In particular, we develop a dual theory for such condition. When $\\sigma$ and $\\sigma'$ are essentially square-integrable representations, we give several equivalent sufficient conditions for the commutation. Such commutation will play an important role in formulating a notion of generalized Gan-Gross-Prasad relevant pairs in the sequel.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-31T12:15:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "20C08",
      "11F70"
    ],
    "keywords": [
      "irreducible representation",
      "commutation",
      "derivatives",
      "adic general linear groups",
      "equivalent sufficient conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}