Calabi-Yau structures on the complexifications of rank two symmeric spaces

For a (Reimannian) symmetric space $G\mathrm{/}K$ of compact type, the natural action of $G$ on its complexification $G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$ (which is an anti-Kaehler symmetric space) is one of the isometric actions called ``Hermann type action’'. Let $\psi$ be the $G$-invariant strictly plurisubharmonic $C^{\mathrm{\infty }}$-function on an open set of $G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$ arising from a $W$-invariant strictly convex $C^{\mathrm{\infty }}$-function $\rho$ on an open set of a maximal abelian subspace ${\mathfrak{a}}^d$ of ${\mathfrak{p}}^d$, where ${\mathfrak{p}}^d$ is the subspace of the Lie algebra ${\mathfrak{g}}^d$ of $G^d$ such that ${\mathfrak{g}}^d=\mathfrak{k}\oplus {\mathfrak{p}}^d$ gives the Cartan decomposition associated to the dual symmetric space $G^d\mathrm{/}K$ of $G\mathrm{/}K$ and $W$ is the Weyl group assocaited to ${\mathfrak{a}}^d$. In this paper, we first give a new proof of a known relation between the complex Hessian of $\psi$ and the Hessian of $\rho$. This new proof is performed from the viewpoint of the orbit geometry of the Hermann type action $G\curvearrowright G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$. In more detail, it is performed by using the explicit descriptions of the shape operators of the orbits of the isotropy action $K\curvearrowright G^d\mathrm{/}K$ and the Hermann type action $G\curvearrowright G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$. Next we prove that there exists a $C^{\mathrm{\infty }}$-Calabi-Yau structure on the whole of the complexification $G^{\mathbb{C}}\mathrm{/}{K}^{\mathbb{C}}$ in the case where $G\mathrm{/}K$ is of rank two on the basis of this relation. In the future, the above new proof will be useful to investigate the existence of invariant Calabi-Yau structure on an anti-Kaehler manifold equipped with a certain kind of complex hyperpolar action in more general, where we note that Hermann type actions are complex hyperpolar.