{
  "id": "2409.05079",
  "version": "v1",
  "published": "2024-09-08T12:47:31.000Z",
  "updated": "2024-09-08T12:47:31.000Z",
  "title": "Resolutions for Locally Analytic Representations",
  "authors": [
    "Shishir Agrawal",
    "Matthias Strauch"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "The purpose of this paper is to study resolutions of locally analytic representations of a $p$-adic reductive group $G$. Given a locally analytic representation $V$ of $G$, we modify the Schneider-Stuhler complex (originally defined for smooth representations) so as to give an `analytic' variant ${\\mathcal S}^A_\\bullet(V)$. The representations in this complex are built out of spaces of analytic vectors $A_\\sigma(V)$ for compact open subgroups $U_\\sigma$, indexed by facets $\\sigma$ of the Bruhat-Tits building of $G$. These analytic representations (of compact open subgroups of $G$) are then resolved using the Chevalley-Eilenberg complex from the theory of Lie algebras. This gives rise to a resolution ${\\mathcal S}^{\\rm CE}_{q,\\bullet}(V) \\rightarrow {\\mathcal S}^A_q(V)$ for each representation ${\\mathcal S}^A_q(V)$ in the analytic Schneider-Stuhler complex. In a last step we show that the family of representations ${\\mathcal S}^{\\rm CE}_{q,j}(V)$ can be given the structure of a Wall complex. The associated total complex ${\\mathcal S}^{\\rm CE}_\\bullet(V)$ has then the same homology as that of ${\\mathcal S}^A_\\bullet(V)$. If the latter is a resolution of $V$, then one can use ${\\mathcal S}^{\\rm CE}_\\bullet(V)$ to find a complex which computes the extension group $\\underline{Ext}^n_G(V,W)$, provided $V$ and $W$ satisfy certain conditions which are satisfied when both are admissible locally analytic representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-09-08T12:47:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "20G05",
      "11F85"
    ],
    "keywords": [
      "compact open subgroups",
      "analytic schneider-stuhler complex",
      "associated total complex",
      "study resolutions",
      "wall complex"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}