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Invariant measures on homogeneous manifolds of reductive type. (English) Zbl 0881.22013
We say that the homogeneous manifold $$G/H$$ is of reductive type if $$G$$ is a real reductive linear Lie group and if $$H$$ is a connected closed subgroup which is reductive in $$G$$. Semisimple symmetric spaces (especially, Riemannian symmetric spaces and semisimple group manifolds) and semisimple orbits are of reductive type. In this paper, we give an explicit upper estimate of the invariant measure on the homogeneous manifold $$G/H$$ of reductive type. Furthermore, we also establish a comparison theorem of the measures of homogeneous submanifolds. These results are used for the construction of new discrete series representations for non-symmetric homogeneous manifolds of reductive type in a subsequent paper [to appear in J. Funct. Anal.].

##### MSC:
 22E46 Semisimple Lie groups and their representations 43A85 Harmonic analysis on homogeneous spaces 53C35 Differential geometry of symmetric spaces 22E15 General properties and structure of real Lie groups 22E30 Analysis on real and complex Lie groups
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