Y. Benoist and T. Kobayashi. Tempered homogeneous spaces IV. Journal of the Institute of Mathematics of Jussieu (2022), 1-28, arXiv: 2009.10391. DOI: 10.1017/S1474748022000287. Published on line, First View: 07 June, 2022.

Let $G$ be a complex semisimple Lie group and $H$ a complex closed connected subgroup. Let $\mathfrak{g}$ and $\mathfrak{h}$ be their Lie algebras. We prove that the regular representation of $G$ in $L^2(G/H)$ is tempered if and only if the orthogonal of $\mathfrak{h}$ in $\mathfrak{g}$ contains regular elements by showing simultaneously the equivalence to other striking conditions such as $\mathfrak{h}$ has a solvable limit algebra.

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