{
  "id": "1101.0472",
  "version": "v1",
  "published": "2011-01-03T08:52:53.000Z",
  "updated": "2011-01-03T08:52:53.000Z",
  "title": "Support varieties of $(\\frak g, \\frak k)$-modules of finite type",
  "authors": [
    "Alexey V. Petukhov"
  ],
  "comment": "The notion of $(\\frak g, \\frak k)$-module have been introduced by I. Penkov, V. Serganova, G. Zuckerman. Main ingredients are Hilbert-Mumford Criterion, A. Beilinson- J. Bernstein localization theorem, O. Gabber theorem",
  "categories": [
    "math.RT"
  ],
  "abstract": "Let $\\frak g$ be a reductive Lie algebra over an algebraically closed field of characteristic 0 and $\\frak k$ be a reductive in $\\frak g$-subalgebra. Let $M$ be a finitely generated (possibly, infinite-dimensional) $\\frak g$-module. We say that $M$ is a $(\\frak g, \\frak k)$-module if $M$ is a direct sum of a (possibly, infinite) amount of simple finite-dimensional $\\frak k$-modules. We say that $M$ is of finite type if $M$ is a $(\\frak g, \\frak k)$-module and Hom$_\\frak k(V, M)<\\infty$ for any simple $\\frak k$-module $V$. Let $X$ be a variety of all Borel subalgebras of $\\frak g$. Let $M$ be a finitely generated $(\\frak g, \\frak k)$-module of finite type. In this article we prove that $M$ is holonomic, i.e. $M$ is governed by some subvariety $L_M\\subset X$ and some local system $S_M$ on it. Furthermore we provide a finite list in which L$_M$ necessarily appear.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-01-03T08:52:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "finite type",
      "support varieties",
      "local system",
      "borel subalgebras",
      "direct sum"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}