Duflo's conjecture for the branching to the Iwasawa $AN$-subgroup

The purpose of this paper is to prove Duflo’s conjecture for $(G,\pi ,AN)$ where $G$ is a simple Lie group of Hermitian type and $\pi$ is a discrete series of $G$ and $AN$ is the maximal exponential solvable subgroup for an Iwasawa decomposition $G=KAN$. This is essentially reduced from the following general theorem we prove in this paper: let $G$ be a connected semisimple Lie group . Then a strongly elliptic $G$-coadjoint orbit $\mathcal{O}$ is holomorphic if and only if $\text{p}(\mathcal{O})$ is an open $AN$-coadjoint orbit, where $$\text{p}$ : \mathfrak{g}^* \longrightarrow (\mathfrak{a}\oplus\mathfrak{n})^*$ is the natural projection.