{
  "id": "2404.07463",
  "version": "v1",
  "published": "2024-04-11T03:55:45.000Z",
  "updated": "2024-04-11T03:55:45.000Z",
  "title": "Generic representations, open parameters and ABV-packets for $p$-adic groups",
  "authors": [
    "Clifton Cunningham",
    "Sarah Dijols",
    "Andrew Fiori",
    "Qing Zhang"
  ],
  "categories": [
    "math.RT",
    "math.AG",
    "math.NT"
  ],
  "abstract": "If $\\pi$ is a representation of a $p$-adic group $G(F)$, and $\\phi$ is its Langlands parameter, can we use the moduli space of Langlands parameters to find a geometric property of $\\phi$ that will detect when $\\pi$ is generic? In this paper we show that if $G$ is classical or if we assume the Kazhdan-Lusztig hypothesis for $G$, then the answer is yes, and the property is that the orbit of $\\phi$ is open. We also propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter $\\phi$ for a $p$-adic group $G(F)$, the ABV-packet $\\Pi^{\\mathrm{ABV}}_\\phi(G(F))$ contains a generic representation if and only if the local adjoint L-function $L(s,\\phi,\\mathop{\\text{Ad}})$ is regular at $s=1$, and show that this condition is equivalent to the \"open parameter\" condition above. We show that this genericity conjecture for ABV-packets follows from other standard conjectures and we verify its validity with the same conditions on $G$. We show that, in this case, the ABV-packet for $\\phi$ coincides with its $L$-packet. Finally, we prove Vogan's conjecture on $A$-packets for tempered parameters.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-11T03:55:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "32S60"
    ],
    "keywords": [
      "adic group",
      "open parameter",
      "generic representation",
      "abv-packet",
      "langlands parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}