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 suppL2(G∕H) be the set of irreducible unitary representations of G contributing to the decomposition of L2(G∕H), namely, the support of the Plancherel measure. In this paper, we will relate suppL2(G∕H) with the image of the moment map from the cotangent bundle T∗(G∕H)→g∗. To the homogeneous space X=G∕H, we attach a complex Levi subgroup LX of the complexification of G, and we show that in some sense “most” of the representations in suppL2(G∕H) are obtained as quantizations of coadjoint orbits O such that O≃G∕L and the complexification of L is conjugate to LX. Moreover, the union of such coadjoint orbits O coincides asymptotically with the moment map image. As a corollary, we show that L2(G∕H) has a discrete series if the moment map image has a nonempty open subset of elliptic elements.