Abstract:This paper lays the foundation for Plancherel theory on real spherical spaces $Z=G/H$, namely it provides the decomposition of $L^2(Z)$ into different series of representations via Bernstein morphisms. These series are parametrized by subsets of spherical roots which determine the fine geometry of $Z$ at infinity. In particular, we obtain a generalization of the Maass-Selberg relations. As a corollary we obtain a partial geometric characterization of the discrete spectrum: $L^2(Z)_{\mathrm {disc}}\neq \emptyset$ if $\mathfrak {h}^\perp$ contains elliptic elements in its interior. In case $Z$ is a real reductive group or, more generally, a symmetric space our results retrieve the Plancherel formula of Harish-Chandra (for the group) as well as that of Delorme and van den Ban-Schlichtkrull (for symmetric spaces) up to the explicit determination of the discrete series for the inducing datum.

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中文翻译：

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实球面空间的 Plancherel 理论：Bernstein 态射的构造

摘要：本文为实球面空间$Z=G/H$的Plancherel理论奠定了基础，即通过Bernstein态射将$L^2(Z)$分解为不同的表示系列。这些系列由球根的子集参数化，这些球根决定了无限远的 $Z$ 的精细几何形状。特别是，我们获得了 Maass-Selberg 关系的概括。作为推论，我们获得了离散谱的部分几何特征： $L^2(Z)_{\mathrm {disc}}\neq \emptyset$ 如果 $\mathfrak {h}^\perp$ 包含椭圆元素内部的。如果 $Z$ 是一个真正的还原组，或者更一般地说，

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