{
  "abstract": "27 pages We define symplectic Dirac operators on homogeneous spaces and study their representation-theoretic role. For an invariant polarization, the symplectic Dirac operator de- composes into two symplectic Dolbeault operators. We compute their commutator as the natural symplectic analogue of the square of the classical Dirac operator. Our first main result gives a necessary and sufficient condition for this commutator to satisfy a Parthasarathy-type formula. We further prove that, whenever this condition fails, no cubic perturbation of the symplectic Dol- beault operators can yield such a formula, in contrast with Kostant’s cubic Dirac operator in the orthogonal setting. As applications, 27 pageswe establish an sl2-structure generated by the symplectic Dolbeault operators and derive Dirac-type inequalities for unitary representations of Hermitian symmetric spaces labelled by the levels of the symmetric powers of the antiholomorphic tangent space at the identity. The level-zero inequality recovers the standard Parthasarathy–Dirac in- equality, while the higher levels inequalities yield new constraints. For SU (1, n), we show that, for representations with a specific Kraljevi´c corner, the level-one inequality strengthens all basic Parthasarathy inequalities of the first kind for particular K-types, precisely those satisfying an explicit highest-weight condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09111",
  "author_total": 4,
  "authors": [
    {
      "name": "Dan Ciubotaru"
    },
    {
      "name": "Gang Liu"
    },
    {
      "name": "Salah Mehdi"
    },
    {
      "name": "Nicolas Prudhon"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6pq2rtnc245p5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09111",
  "fulltext_url": null,
  "id": "6pq2rtnc245p5",
  "issns": [],
  "landing_url": "https://hal.science/hal-05781976v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.RT"
    ],
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    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7212068310",
  "pdf_url": "https://hal.science/hal-05781976/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6pq2rtnc245p5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:49Z",
      "native_id": "hal-05781976",
      "source": "hal",
      "url": "https://hal.science/hal-05781976v1"
    },
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09111",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09111"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Symplectic Dirac operators on homogeneous spaces",
  "title_text": "Symplectic Dirac operators on homogeneous spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/6pq2rtnc245p5",
  "venue": "HAL",
  "versions": []
}