On the asymptotic support of Plancherel measures for homogeneous spaces

TL;DR

This paper explores the relationship between the support of the Plancherel measure for homogeneous spaces and the moment map image, revealing that most representations are quantizations of specific coadjoint orbits and linking the existence of discrete series to elliptic elements.

Contribution

It establishes a connection between the support of $L^2(G/H)$ and the moment map image, identifying the dominant coadjoint orbits contributing to the representation spectrum.

Findings

Most representations are quantizations of coadjoint orbits related to the Levi subgroup.

The union of these coadjoint orbits asymptotically matches the moment map image.

Discrete series exist if the moment map image contains elliptic elements.

Abstract

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.