On the asymptotic support of Plancherel measures for homogeneous spaces

Let $G$ be a real linear reductive group and let $H$ be a unimodular, locally algebraic subgroup. Let $\operatorname{supp} L^2(G/H)$ be the set of irreducible unitary representations of $G$ contributing to the decomposition of $L^2(G/H)$, namely the support of the Plancherel measure. In this paper, we will relate $\operatorname{supp} L^2(G/H)$ with the image of moment map from the cotangent bundle $T^*(G/H)\to \mathfrak{g}^*$. For the homogeneous space $X=G/H$, we attach a complex Levi subgroup $L_X$ of the complexification of $G$ and we show that in some sense "most" of representations in $\operatorname{supp} L^2(G/H)$ are obtained as quantizations of coadjoint orbits $\mathcal{O}$ such that $\mathcal{O}\simeq G/L$ and that the complexification of $L$ is conjugate to $L_X$. Moreover, the union of such coadjoint orbits $\mathcal{O}$ coincides asymptotically with the moment map image. As a corollary, we show that $L^2(G/H)$ has a discrete series if the moment map image contains a nonempty subset of elliptic elements.