{
  "id": "2502.08479",
  "version": "v1",
  "published": "2025-02-12T15:14:17.000Z",
  "updated": "2025-02-12T15:14:17.000Z",
  "title": "How does the restriction of representations change under translations? A story for the general linear groups and the unitary groups",
  "authors": [
    "Toshiyuki Kobayashi",
    "Birgit Speh"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "We present a new approach to symmetry breaking for pairs of real forms of $(GL(n, \\mathbb{C}), GL(n-1, \\mathbb{C}))$. While translation functors are a useful tool for studying a family of representations of a single reductive group $G$, when applied to a pair of groups $G \\supset G'$,translation functors can significantly alter the nature of symmetry breaking between the representations of $G$ and $G'$, even within the same Weyl chamber of the direct product group $G \\times G'$. We introduce the concept of \\lq\\lq{fences for the interlacing pattern}\\rq\\rq,which provides a refinement of the usual notion of \\lq\\lq{walls for Weyl chambers}\\rq\\rq. We then present a theorem that states that multiplicity is constant unless these \\lq\\lq{fences}\\rq\\rq\\ are crossed. This general theorem is illustrated with examples of both tempered and non-tempered representations. Additionally,we provide a new non-vanishing theorem of period integrals for pairs of reductive symmetric spaces,which is further strengthened through this approach.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-12T15:14:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "general linear groups",
      "unitary groups",
      "representations change",
      "translation functors",
      "restriction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}