{
  "id": "2309.13215",
  "version": "v1",
  "published": "2023-09-22T23:33:15.000Z",
  "updated": "2023-09-22T23:33:15.000Z",
  "title": "Unitary representations of real groups and localization theory for Hodge modules",
  "authors": [
    "Dougal Davis",
    "Kari Vilonen"
  ],
  "comment": "71 pages",
  "categories": [
    "math.RT",
    "math.AG"
  ],
  "abstract": "We prove a conjecture of Schmid and the second named author that the unitarity of a representation of a real reductive Lie group with real infinitesimal character can be read off from a canonical filtration, the Hodge filtration. Our proof rests on two main ingredients. The first is a wall crossing theory for mixed Hodge modules: the key result is that, in certain natural families, the Hodge filtration varies semi-continuously with jumps controlled by extension functors. The second ingredient is a Hodge-theoretic refinement of Beilinson-Bernstein localization: we show that the Hodge filtration of a mixed Hodge module on the flag variety satisfies the usual cohomology vanishing and global generation properties enjoyed by the underlying $\\mathcal{D}$-module. As byproducts of our work, we obtain a version of Saito's Kodaira vanishing for twisted mixed Hodge modules, a calculation of the Hodge filtration on a certain object in category $\\mathcal{O}$, and a host of new vanishing results for, for example, homogeneous vector bundles on flag varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-09-22T23:33:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "unitary representations",
      "localization theory",
      "real groups",
      "mixed hodge module",
      "hodge filtration varies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}