{
  "id": "2304.06622",
  "version": "v1",
  "published": "2023-04-13T15:45:37.000Z",
  "updated": "2023-04-13T15:45:37.000Z",
  "title": "Character sheaves on tori over local fields",
  "authors": [
    "Tanmay Deshpande",
    "Saniya Wagh"
  ],
  "comment": "32 pages",
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "Let $\\breve{K}$ be a complete discrete valuation field with an algebraically closed residue field ${k}$ and ring of integers $\\breve{{O}}$. Let $T$ be a torus defined over $\\breve{K}$. Let $L^+T$ denote the connected commutative pro-algebraic group over ${k}$ obtained by applying the Greenberg functor to the connected N\\'eron model of $T$ over $\\breve{{O}}$. Following the work of Serre for the multiplicative group, we first compute the fundamental group $\\pi_1(L^+T)$. We then study multiplicative local systems (or character sheaves) on $L^+T$ and establish a local Langlands correspondence for them. Namely, we construct a canonical isomorphism of abelian groups between the group of multiplicative local systems on $L^+T$ and inertial local Langlands parameters for $T$. Finally, we relate our results to the classical local Langlands correspondence for tori over local fields due to Langlands, via the sheaf-function correspondence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-13T15:45:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local fields",
      "character sheaves",
      "inertial local langlands parameters",
      "complete discrete valuation field",
      "classical local langlands correspondence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 32,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}