{
  "id": "2303.00913",
  "version": "v2",
  "published": "2023-03-02T02:17:57.000Z",
  "updated": "2023-05-25T04:33:23.000Z",
  "title": "Schwartz spaces, local L-factors and perverse sheaves",
  "authors": [
    "Roman Bezrukavnikov",
    "Alexander Braverman",
    "Michael Finkelberg",
    "David Kazhdan"
  ],
  "comment": "v2: references updated, section 4.2 added",
  "categories": [
    "math.RT",
    "math.AG",
    "math.NT"
  ],
  "abstract": "We propose a new conjectural way to calculate the local $L$-factor $L=L_\\chi(\\pi,\\rho,s)$ where $\\pi$ is a representation of a $p$-adic group $G$, $\\rho$ is an algebraic representation of the dual group $G^{\\vee}$ and $\\chi$ is an algebraic character of $G$ satisfying a positivity condition. A method going back to Godement and Jacquet yields a description of $L$ using as an input a certain space ${\\mathcal S}_\\rho$ of functions on $G$ depending on $\\rho$. A (partly conjectural) description of ${\\mathcal S}_\\rho$ involving trace of Frobenius functions associated to perverse sheaves on the loop space of a semigroup containing $G$ was developed %by Bouthier, Ngo and Sakellaridis, partly based on an earlier work of Braverman and Kazhdan. Here we propose a different, more general conjectural description of ${\\mathcal S}_\\rho$: it also refers to trace of Frobenius functions but instead of the loop space of a semi-group we work with the ramified global Grassmannian fibering over the configuration space of points on a global curve defined by Beilinson-Drinfeld and Gaitsgory (a relation between two approaches is discussed in the appendix). Our main result asserts validity of our conjectures where $\\pi$ is generated by an Iwahori fixed vector: we show that in this case it is compatible with the standard formula for $L$ involving local Langlands correspondence which is known for such representations $\\pi$. The proof is based on properties of the coherent realization of the affine Hecke category.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-05-25T04:33:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "perverse sheaves",
      "local l-factors",
      "schwartz spaces",
      "frobenius functions",
      "loop space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}