Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective correspondence with the coadjoint orbits of $G$, that is, $\mathfrak g^*/\mathrm{Ad}(G)$, where $\mathfrak g^*=\mathrm{Hom}_\mathbb R(\mathfrak g,\mathbb R)$. Furtheremore, the direct integral decompositions of restriction and induction of representations can be described in terms of some natural operations on orbits.

Now let $G$ be a nilpotent complex nilpotent Lie group. Of course $G$ is also a real Lie group and therefore Kirillov theory as described above applies to $G$. But one can also imagine a complex variant of Kirillov theory, where one considers the dual space $\mathfrak g^*:=\mathrm{Hom}_\mathbb C(\mathfrak g,\mathbb C)$ - note that $\mathfrak g$ is a complex Lie algebra - and orbits of the $G$-action on $\mathfrak g^*$. Such orbits are of course complex manifolds and carry an invariant complex symplectic structure, etc.

My question is the following: is the complex variant of Kirillov theory valid as well? In other words, is it true that the representations of $G$ canonically correspond to the $G$-orbits in $\mathfrak g^*:=\mathrm{Hom}_\mathbb C(\mathfrak g,\mathbb C)$, and restriction and induction to/from complex Lie subgroups of $G$ can be described in terms of natural orbit operations?

It seems to me that at least one of the standard arguments of Kirillov theory (e.g., the book of Corwin-Greenleaf) goes through in the complex case as well. However, I am wondering if there is also a definitive reference in the literature?


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