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 $\mathrm{supp}{L}^2(G\mathrm{/}H)$ be the set of irreducible unitary representations of $G$ contributing to the decomposition of $L^2(G\mathrm{/}H)$, namely the support of the Plancherel measure. In this paper, we will relate $\mathrm{supp}{L}^2(G\mathrm{/}H)$ with the image of moment map from the cotangent bundle $T^{\ast }(G\mathrm{/}H)\to {\mathfrak{g}}^{\ast }$. For the homogeneous space $X=G\mathrm{/}H$, we attach a complex Levi subgroup $L_X$ of the complexification of $G$ and we show that in some sense “most” of representations in $\mathrm{supp}{L}^2(G\mathrm{/}H)$ are obtained as quantizations of coadjoint orbits $\mathcal{O}$ such that $\mathcal{O}\simeq G\mathrm{/}L$ 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 $L^2(G\mathrm{/}H)$ has a discrete series if the moment map image contains a nonempty subset of elliptic elements.