Classification of reductive homogeneous spaces satisfying strict inequality for Benoist-Kobayashi's $ρ$ functions
Kazushi Maeda
TL;DR
This work completes a precise classification of complex reductive pairs $(\mathfrak{g},\mathfrak{h})$ with $\mathfrak{g}$ complex reductive and $\mathfrak{h}$ complex semisimple that satisfy the equality-allowing Benoist–Kobayashi $\rho$-inequality $\rho_{\mathfrak{h}} \leq \rho_{\mathfrak{q}}$ while failing strict inequality everywhere, i.e., where $\rho_{\mathfrak{h}} \nless \rho_{\mathfrak{q}}$ on the maximal split abelian subspace. The authors reduce to the complex simple, classical-type setting via Dynkin's maximal-subalgebra classification, and perform explicit weight-theoretic computations to identify 11 finite families, together with witness vectors in the positive Weyl chamber that realize equality. This advances understanding of when $L^2(G/H)$ contains square-integrable or unitary-subrepresentation components and clarifies the role of $\rho$-inequalities in the structure of reductive homogeneous spaces. The results provide explicit, testable criteria for when $L^2(G/H)$ contains discrete or square-integrable contributions and offer concrete tables and vectors for further representation-theoretic and geometric applications.
Abstract
Let $G$ be a real reductive Lie group and $H$ a reductive subgroup of $G$. Benoist-Kobayashi studied when $L^2(G/H)$ is a tempered representation of $G$. They introduced the functions $ρ$ on Lie algebras and gave a necessary and sufficient condition for the temperedness of $L^2(G/H)$ in terms of an inequality on $ρ$. In a joint work with Y. Oshima, we considered when $L^2(G/H)$ is equivalent to a unitary subrepresentation of $L^2(G)$ and gave a sufficient condition for this in terms of a strict inequality of $ρ$. In this paper, we will classify the pairs $(\mathfrak{g}, \mathfrak{h})$ with $\mathfrak{g}$ complex reductive and $\mathfrak{h}$ complex semisimple which satisfy that strict inequality of $ρ$.