ASSOCIATED VARIETY, KOSTANT-SEKIGUCHI CORRESPONDENCE ,AND LOCALLY FREE U(n)-ACTION ON HARISH-CHANDRA MODULES

摘要

Let g be a complex semisimple Lie algebra with symmetric decomposition g = i+p.For each irreducible Harish-Chandra (g,i)-module X, we construct a family of nilpotent Lie subalgebras n( θ) of g whose universal enveloping algebras U(θ) act on X locally freely.The Lie subalgebras n(θ) are parametrized by the nilpotent orbits θ in the associated variety of X, and they are obtained by making use of the Cayle tranformation of sl_2-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(θ_max )attached to a principal nilpotent orbit θ_max in p.