{
  "id": "2005.07899",
  "version": "v1",
  "published": "2020-05-16T08:13:48.000Z",
  "updated": "2020-05-16T08:13:48.000Z",
  "title": "Endomorphism algebras and Hecke algebras for reductive p-adic groups",
  "authors": [
    "Maarten Solleveld"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let G be a reductive p-adic group and let Rep(G)^s be a Bernstein block in the category of smooth complex G-representations. We investigate the structure of Rep(G)^s, by analysing the algebra of G-endomorphisms of a progenerator \\Pi of that category. We show that Rep(G)^s is \"almost\" Morita equivalent with a (twisted) affine Hecke algebra. This statement is made precise in several ways, most importantly with a family of (twisted) graded algebras. It entails that, as far as finite length representations are concerned, Rep(G)^s and End_G (\\Pi)-Mod can be treated as the module category of a twisted affine Hecke algebra. We draw two consequences. Firstly, we show that the equivalence of categories between Rep(G)^s and End_G (\\Pi)-Mod preserves temperedness of finite length representations. Secondly, we provide a classification of the irreducible representations in Rep(G)^s, in terms of the complex torus and the finite group canonically associated to Rep(G)^s. This proves a version of the ABPS conjecture. Our methods are independent of the existence of types, and apply in complete generality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-05-16T08:13:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "20G25",
      "20C08"
    ],
    "keywords": [
      "reductive p-adic group",
      "endomorphism algebras",
      "finite length representations",
      "smooth complex g-representations",
      "twisted affine hecke algebra"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}