Let be acomplex semisimple Lie algebra with symmetric decomposition . For each irreducible Harish-Chandra -module , we construct a family of nilpotent Lie subalgebras of whose universal enveloping algebras act on locally freely. The Lie subalgebras are parametrized by the nilpotent orbits in the associated variety of , and they are obtained by making use of the Cayley tranformation of -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 -action for attached to aprincipal nilpotent orbit in .
Akihiko GYOJA. Hiroshi YAMASHITA. "Associated variety, Kostant-Sekiguchi correspondence, and locally free -action on Harish-Chandra modules." J. Math. Soc. Japan 51 (1) 129 - 149, January, 1999. https://doi.org/10.2969/jmsj/05110129