{
  "id": "2302.02207",
  "version": "v1",
  "published": "2023-02-04T17:39:20.000Z",
  "updated": "2023-02-04T17:39:20.000Z",
  "title": "The Weyl law for congruence subgroups and arbitrary $K_\\infty$-types",
  "authors": [
    "Werner Mueller"
  ],
  "comment": "50 pages. arXiv admin note: text overlap with arXiv:2002.04598",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $G$ be a reductive algebraic group over $\\mathbb{Q}$ and $\\Gamma\\subset G(\\mathbb{Q})$ an arithmetic subgroup. Let $K_\\infty\\subset G(\\mathbb{R})$ be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of $G(\\mathbb{R})$ in $L^2(\\Gamma\\backslash G(\\mathbb{R}))$ of a fixed $K_\\infty$-type $\\sigma$. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type $\\sigma$ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about $L$-functions occurring in the constant terms of Eisenstein series. If $G$ satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-04T17:39:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F72",
      "58J35"
    ],
    "keywords": [
      "congruence subgroups",
      "weyl law",
      "residual spectrum",
      "counting function",
      "conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 50,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}