{
  "id": "2111.11970",
  "version": "v4",
  "published": "2021-11-23T16:05:02.000Z",
  "updated": "2022-02-11T10:11:39.000Z",
  "title": "On tempered representations",
  "authors": [
    "David Kazhdan",
    "Alexander Yom Din"
  ],
  "comment": "Fourth version: Two main changes, 1) removed appendix B since found a proof of the bound in the literature, 2) corrected a mistake regarding the behaviour of principal series corresponding to the sign character",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $G$ be a unimodular locally compact group. We define a property of irreducible unitary $G$-representations $V$ which we call c-temperedness, and which for the trivial $V$ boils down to F{\\o}lner's condition (equivalent to the trivial $V$ being tempered, i.e. to $G$ being amenable). The property of c-temperedness is a-priori stronger than the property of temperedness. We conjecture that for semisimple groups over local fields temperedness implies c-temperedness. We check the conjecture for a special class of tempered $V$'s, as well as for all tempered $V$'s in the case of $G := SL_2 (\\mathbb{R})$. We also establish a weaker form of the conjecture, involving only $K$-finite vectors. In the $p$-adic case, we give a formula expressing the character of a tempered $V$ as an appropriately-weighted conjugation-average of a matrix coefficient of $V$, generalizing a formula of Harish-Chandra from the case when $V$ is square-integrable.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2022-02-11T10:11:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tempered representations",
      "local fields temperedness implies c-temperedness",
      "conjecture",
      "unimodular locally compact group",
      "weaker form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}