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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© Toshiyuki Kobayashi