Associated variety, Kostant-Sekiguchi correspondence, and locally free U(n)-action on Harish-Chandra modules

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Let \mathfrak{g} be a complex semisimple Lie algebra with symmetric decomposition \mathfrak{g}=\mathfrak{f}+\mathfrak{p}. For each irreducible Harish-Chandra (\mathfrak{g}, \mathfrak{f})-module X, we construct a family of nilpotent Lie subalgebras \mathfrak{n}(\mathcal{O}) of \mathfrak{g} whose universal enveloping algebras U(\mathfrak{n}(\mathcal{O})) act on X locally freely. The Lie subalgebras \mathfrak{n}(\mathcal{O}) are parametrized by the nilpotent orbits \mathcal{O} in the associated variety of X, and they are obtained by making use of the Cayley tranformation of \mathfrak{s}\mathfrak{l}<SUB>2</SUB>-triples (Kostant-Sekiguchi correspondence). As a consequence, it is shown that an irreducible Harish-Chandra module has the possible maximal Gelfand-Kirillov dimension if and only if it admits locally free U(\mathfrak{n}<SUB>m</SUB>)-action for \mathfrak{n}<SUB>m</SUB>=\mathfrak{n}(\mathcal{O}<SUB>max</SUB>) attached to a principal nilpotent orbit \mathcal{O}<SUB>max</SUB> in \mathfrak{p}$.

収録刊行物

  • Journal of the Mathematical Society of Japan  

    Journal of the Mathematical Society of Japan 51(1), 129-149, 1999-01 

    The Mathematical Society of Japan

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各種コード

  • NII論文ID(NAID)
    10002151444
  • NII書誌ID(NCID)
    AA0070177X
  • 本文言語コード
    ENG
  • 資料種別
    ART
  • ISSN
    00255645
  • NDL 記事登録ID
    4643293
  • NDL 雑誌分類
    ZM31(科学技術--数学)
  • NDL 請求記号
    Z53-A209
  • データ提供元
    CJP書誌  NDL  J-STAGE 
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