On the asymptotic support of Plancherel measures for homogeneous spaces

Abstract

Let G be a real linear reductive group, and let H be a unimodular, locally algebraic subgroup. Let $\mathrm{supp}{L}^2(G∕H)$ 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 $\mathrm{supp}{L}^2(G∕H)$ with the image of the moment map from the cotangent bundle $T^{\ast }(G∕H)\to {\mathfrak{g}}^{\ast }$.

To 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 the representations in $\mathrm{supp}{L}^2(G∕H)$ are obtained as quantizations of coadjoint orbits $\mathcal{O}$ such that $\mathcal{O}\simeq G∕L$ and 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 has a nonempty open subset of elliptic elements.