## Journal of the Mathematical Society of Japan

### Associated variety, Kostant-Sekiguchi correspondence, and locally free $U(\mathfrak{n})$-action on Harish-Chandra modules

#### Abstract

Let $\mathfrak{g}$ be acomplex semisimple Lie algebra with symmetric decomposition $\mathrm{g}=\mathfrak{k}+\mathfrak{p}$. For each irreducible Harish-Chandra $(\mathrm{g},\mathfrak{k})$-module $\mathrm{X}$, we construct a family of nilpotent Lie subalgebras $\mathfrak{n}(\mathscr{O})$ of $\mathfrak{g}$ whose universal enveloping algebras $U(\mathfrak{n}(\mathscr{O}))$ act on $\mathrm{X}$ locally freely. The Lie subalgebras $\mathfrak{n}(\mathscr{O})$ are parametrized by the nilpotent orbits $\mathscr{O}$ in the associated variety of $\mathrm{X}$, and they are obtained by making use of the Cayley tranformation of $\mathrm{sI}_{2}$-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 $U(\mathfrak{n}_{m})$-action for $\mathfrak{n}_{m}=\mathfrak{n}(\mathscr{O}_{\max})$ attached to aprincipal nilpotent orbit $\mathscr{O}_{\max}$ in $\mathfrak{p}$.

#### Article information

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

Dates
First available in Project Euclid: 10 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1213108354

Digital Object Identifier
doi:10.2969/jmsj/05110129

Mathematical Reviews number (MathSciNet)
MR1661024

Zentralblatt MATH identifier
0931.22008

#### Citation

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. https://projecteuclid.org/euclid.jmsj/1213108354