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.

We extend Gubler and Künnemann’s theory of δ-forms from algebraic varieties to good Berkovich spaces. This is based on the observation that skeletons in such spaces satisfy a tropical balancing condition. Our first theorem in this formalism is a construction of Green’s δ-forms for complete intersection cycles. Our second theorem is a formula for Artinian intersection numbers on formal models in terms of integrals of δ-forms on their generic fibers. These two results generalize known results from divisors to higher codimension. Finally, we illustrate our intersection number formula in the context of an intersection problem for Lubin–Tate spaces.