{
  "id": "2310.20399",
  "version": "v1",
  "published": "2023-10-31T12:21:12.000Z",
  "updated": "2023-10-31T12:21:12.000Z",
  "title": "Linkage for irreducible representations of ${\\rm GL}_3$ in local Base change",
  "authors": [
    "Sabyasachi Dhar"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $E$ be a finite Galois extension of a $p$-adic field $F$ with degree of extension $l$, where $l$ and $p$ are distinct primes with $p\\ne 2$. Let $\\pi_F$ be an integral cuspidal $\\overline{\\mathbb{Q}}_l$-representation of ${\\rm GL}_3(F)$ and let $\\pi_E$ be the base change lifting of $\\pi_F$ to the group ${\\rm GL}_3(E)$. Then for any ${\\rm GL}_n(E)\\rtimes{\\rm Gal}(E/F)$-stable lattice $\\mathcal{L}$ in the representation space of $\\pi_E$, the Tate cohomology group $\\widehat{H}^0({\\rm Gal}(E/F),\\mathcal{L})$ is defined as $\\overline{\\mathbb{F}}_l$-representation of ${\\rm GL}_3(F)$, and its semisimplification is denoted by $\\widehat{H}^0(\\pi_E)$. In this article, we show that the Frobenius twist of mod-$l$ reduction of $\\pi_F$ is isomorphic to the Tate cohomology group $\\widehat{H}^0({\\rm Gal}(E/F),\\mathcal{L})$. This result is related to the notion of linkage under the local base change setting. Moreover, this notion of linkage under the local base change for non-cuspidal irreducible representations of ${\\rm GL}_3$ is also investigated in this article.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-31T12:21:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "irreducible representations",
      "tate cohomology group",
      "finite galois extension",
      "representation space",
      "adic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}