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

Dedicated to Professor Takeshi Hirai on his sixtieth birthday

Abstract

Let \mathfrak{g} be a complex semisimple Lie algebra with symmetric decomposition \mathfrak{g}=\mathfrak{f}+\mathfrak{p}. For each irreducible Harish-Chandra (\mathfrak{g}, \mathfrak{f})-module X, we construct a family of nilpotent Lie subalgebras \mathfrak{n}(\mathcal{O}) of \mathfrak{g} whose universal enveloping algebras U(\mathfrak{n}(\mathcal{O})) act on X locally freely. The Lie subalgebras \mathfrak{n}(\mathcal{O}) are parametrized by the nilpotent orbits \mathcal{O} in the associated variety of X, and they are obtained by making use of the Cayley tranformation of \mathfrak{s}\mathfrak{l}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(\mathfrak{n}_m)-action for \mathfrak{n}_m=\mathfrak{n}(\mathcal{O}_max) attached to a principal nilpotent orbit \mathcal{O}_max in \mathfrak{p}$.