  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.
