{
  "id": "1302.1031",
  "version": "v2",
  "published": "2013-02-05T13:35:44.000Z",
  "updated": "2014-06-05T11:49:56.000Z",
  "title": "Invariants and K-spectrums of local theta lifts",
  "authors": [
    "Hung Yean Loke",
    "Jia-jun Ma"
  ],
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $(G,G')$ be a type I irreducible reductive dual pair in $\\mathrm{Sp}(W_{\\mathbb{R}})$. We assume that $(G,G')$ is in the stable range where $G$ is the smaller member. Let $K$ and $K'$ be maximal compact subgroups of $G$ and $G'$ respectively. Let $\\mathfrak{g} = \\mathfrak{k} \\oplus \\mathfrak{p}$ and $\\mathfrak{g}' = \\mathfrak{k}' \\oplus \\mathfrak{p}'$ be the complexified Cartan decompositions of the Lie algebras of $G$ and $G'$ respectively. Let ${\\widetilde{K}}$ and ${\\widetilde{K}}'$ be the inverse images of $K$ and $K'$ in the metaplectic double cover $\\widetilde{\\mathrm{Sp}}(W_\\mathbb{R})$ of ${\\mathrm{Sp}}(W_\\mathbb{R})$. Let $\\rho$ be a genuine irreducible $(\\mathfrak{g},{\\widetilde{K}})$-module. Our first main result is that if $\\rho$ is unitarizable, then except for one special case, the full local theta lift $\\rho' = \\Theta(\\rho)$ is equal to the local theta lift $\\theta(\\rho)$. Thus excluding the special case, the full theta lift $\\rho'$ is an irreducible and unitarizable $(\\mathfrak{g}',{\\widetilde{K}}')$-module. Our second main result is that the associated variety and the associated cycle of $\\rho'$ are the theta lifts of the associated variety and the associated cycle of the contragredient representation $\\rho^*$ respectively. Finally we obtain some interesting $(\\mathfrak{g},{\\widetilde{K}})$-modules whose ${\\widetilde{K}}$-spectrums are isomorphic to the spaces of global sections of some vector bundles on some nilpotent $K_\\mathbb{C}$-orbits in $\\mathfrak{p}^*$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-05T11:49:56.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "special case",
      "invariants",
      "k-spectrums",
      "full local theta lift",
      "first main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1302.1031Y"
    }
  }
}