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

Abstract

Let $\mathfrak{g}$ be acomplex semisimple Lie algebra with symmetric decomposition $g=\mathfrak{k}+\mathfrak{p}$. For each irreducible Harish-Chandra $(g,\mathfrak{k})$-module $X$, we construct a family of nilpotent Lie subalgebras $\mathfrak{n}\left(\mathcal{O}\right)$ of $\mathfrak{g}$ whose universal enveloping algebras $U\left(\mathfrak{n}\right(\mathcal{O}\left)\right)$ act on $X$ locally freely. The Lie subalgebras $\mathfrak{n}\left(\mathcal{O}\right)$ 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 $s{I}_2$-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 $U\left({\mathfrak{n}}_m\right)$-action for ${\mathfrak{n}}_m=\mathfrak{n}\left({\mathcal{O}}_{\max }\right)$ attached to aprincipal nilpotent orbit ${\mathcal{O}}_{\max }$ in $\mathfrak{p}$.