{
  "id": "2106.00437",
  "version": "v1",
  "published": "2021-06-01T12:33:45.000Z",
  "updated": "2021-06-01T12:33:45.000Z",
  "title": "Homological duality for covering groups of reductive $p$-adic groups",
  "authors": [
    "Dragos Fratila",
    "Dipendra Prasad"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.RA"
  ],
  "abstract": "In this largely expository paper we extend properties of the homological duality functor $RHom_{\\mathcal H}(-,{\\mathcal H})$ where ${\\mathcal H}$ is the Hecke algebra of a reductive $p$-adic group, to the case where it is the Hecke algebra of a finite central extension of a reductive $p$-adic group. The most important properties being that $RHom_{\\mathcal H}(-,{\\mathcal H})$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider--Stuhler duality for Ext groups (a Serre functor like property). Along the way we also study Grothendieck--Serre duality with respect to the Bernstein center and provide a proof of the folklore result that on admissible modules this functor is nothing but the contragredient duality. We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block. In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche, on all blocks with trivial stabilizer in the relative Weyl group.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-01T12:33:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "22E50",
      "22E45"
    ],
    "keywords": [
      "adic group",
      "homological duality",
      "covering groups",
      "hecke algebra",
      "finite central extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}