{
  "id": "2401.09302",
  "version": "v1",
  "published": "2024-01-17T16:10:23.000Z",
  "updated": "2024-01-17T16:10:23.000Z",
  "title": "Smooth representations of involutive algebra groups over non-archimedean local fields",
  "authors": [
    "Carlos A. M. André",
    "João Dias"
  ],
  "comment": "arXiv admin note: text overlap with arXiv:1910.14639",
  "categories": [
    "math.RT"
  ],
  "abstract": "An algebra group over a field $F$ is a group of the form $G = 1+J$ where $J$ is a finite-dimensional nilpotent associative $F$-algebra. A theorem of M. Boyarchenko asserts that, in the case where $F$ is a non-archimedean local field, every irreducible smooth representation of $G$ is admissible and smoothly induced by a one-dimensional smooth representation of some algebra subgroup of $G$. If $J$ is a nilpotent algebra endowed with an involution $\\sigma:J\\to J$, then $\\sigma$ naturally defines a group automorphism of $G$, and we may consider the fixed point subgroup $C_{G}(\\sigma)$. Assuming that $F$ has characteristic different from $2$, we extend Boyarchenko's result and show that every irreducible smooth representation of $C_{G}(\\sigma)$ is admissible and smoothly induced by a one-dimensional smooth representation of a subgroup of the form $C_{H}(\\sigma)$ where $H$ is an $\\sigma$-invariant algebra subgroup of $G$. As a particular case, the result holds for maximal unipotent subgroups of the classical Chevalley groups defined over $F$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-17T16:10:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20G25",
      "22D12",
      "22D30"
    ],
    "keywords": [
      "non-archimedean local field",
      "involutive algebra groups",
      "one-dimensional smooth representation",
      "irreducible smooth representation",
      "maximal unipotent subgroups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}