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

Let g be a complex semisimple Lie algebra with symmetric decomposition g = (iota)+p. For each irreducible Harish-Chandra (g. (iota)-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 G in the associated variety of X, and they are obtained by making use of the Cayley tranformation of siz-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(n(m))-action for n(m)= n(o (max)) attached to a principal nilpotent orbit O(max) in .