Journal of the Mathematical Society of Japan

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

Akihiko GYOJA and Hiroshi YAMASHITA

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Let g be acomplex semisimple Lie algebra with symmetric decomposition g=k+p. For each irreducible Harish-Chandra (g,k)-module X, we construct a family of nilpotent Lie subalgebras n(O) of g whose universal enveloping algebras U(n(O)) act on X locally freely. The Lie subalgebras n(O) are parametrized by the nilpotent orbits O in the associated variety of X, and they are obtained by making use of the Cayley tranformation of sI2-triples(Kostant-Sekiguchi correspondence). As aconsequence, 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(nm)-action for nm=n(Omax) attached to aprincipal nilpotent orbit Omax in p.

Article information

J. Math. Soc. Japan, Volume 51, Number 1 (1999), 129-149.

First available in Project Euclid: 10 June 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E47: Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) [See also 17B10]
Secondary: 17B35: Universal enveloping (super)algebras [See also 16S30]

Semisimple Lie algebras Harish-Chandra modules nilpotent orbits associated variety locally free action


GYOJA, Akihiko; YAMASHITA, Hiroshi. Associated variety, Kostant-Sekiguchi correspondence, and locally free $U(\mathfrak{n})$ -action on Harish-Chandra modules. J. Math. Soc. Japan 51 (1999), no. 1, 129--149. doi:10.2969/jmsj/05110129.

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