Y. Benoist and T. Kobayashi, Tempered homogeneous spaces III, Journal of Lie Theory 31 (2021), 833-869..

Let $G$ be a real semisimple algebraic Lie group and $H$ a real reductive algebraic subgroup. We describe the pairs $(G,H)$ for which the representation of $G$ in $L^2(G/H)$ is tempered. When $G$ and $H$ are complex Lie groups, the temperedness condition is characterized by the fact that {\it the stabilizer in $H$ of a generic point on $G/H$ is virtually abelian}.

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