Associated variety, Kostant-Sekiguchi correspondence, and locally free \(U(n)\)-action on Harish-Chandra modules (Q1298051)

Let \(\mathfrak g\) be a complex semisimple Lie group with an involution \(\theta\) and \({\mathfrak g}= {\mathfrak k}+{\mathfrak p}\) the associated symmetric decomposition of \(\mathfrak g\). Let \(X\) be an irreducible Harish-Chandra \((\mathfrak g\),\(\mathfrak k)\)-module. For each nilpotent \(K^{ad}_\mathbb{C}\)-orbit \(\mathcal O\) contained in the associated variety \({\mathcal V}(X)\subset {\mathfrak p}\) of \(X\), the authors construct a nilpotent Lie subalgebra \({\mathfrak n}({\mathcal O})\) of \(\mathfrak g\) whose universal enveloping algebra \(U({\mathfrak n}({\mathcal O}))\) acts on \(X\) locally freely. This construction is based on the Kostant-Sekiguchi correspondence between the set of nilpotent \(K^{ad}_\mathbb{C}\)-orbits in \(\mathfrak p\) and that of nilpotent \(G^{ad}\)-orbits in a \(\theta\)-stable real form \({\mathfrak g}_0\) of \(\mathfrak g\). They also deduce the following corollary: If a nilpotent \(K^{ad}_\mathbb{C}\)-orbit \(\mathcal O\) in \({\mathcal V}(X)\) is of maximal dimension, then \({\mathfrak n}({\mathcal O})\) is maximal among the Lie subalgebras whose enveloping algebras act on \(X\) locally freely. Especially, \({\mathcal V}(X)\) contains an open \(K^{ad}_\mathbb{C}\)-orbit in the totality of nilpotent elements of \(\mathfrak g\) contained in \(\mathfrak p\) if and only if the enveloping algebra of the Iwasawa maximal nilpotent subalgebra of \(\mathfrak g\) acts on \(X\) locally freely. In addition, they describe explicitly the Lie subalgebras \({\mathfrak n}({\mathcal O})\) associated to the holomorphic \(K^{ad}_\mathbb{C}\)-orbits \(\mathcal O\).