Plancherel formulae for non-symmetric polar homogeneous spaces (Q1358956)

Let \(G/H\) be a semisimple homogeneous space (\(G\) a semisimple Lie group, \(H\) a closed reductive subgroup of \(G\)). The author considers a class of such \(G/H\), namely, the so-called polar semisimple spaces. This class includes symmetric semisimple spaces. It is proved that a polar \(G/H\) has a discrete series if and only if it has a compact Cartan subspace (the notion of the Cartan subgroup for polar spaces was given by the author earlier). The method to prove it is the following. Firstly, for symmetric spaces it is a result of Flensted-Jensen, Oshima and Matsuki. Secondly, it turns out that there exist only three polar spaces \(G/H\) with \(G\) simple which are not symmetric and they are isomorphic to some real hyperbolic spaces \(G'/H'\) (symmetric spaces), i.e. \(G/H= G'/H'\), \(G\subset G'\), \(H\subset H'\), namely, to \(SO_0(4,4)/SO_0(4,3)\), \(SO_0(4,3)/SO_0(4,2)\), \(SO_0(4,3)/SO_0(3,3)\). Moreover, any irreducible representation of \(G'\) occurring in the quasi-regular representation on \(G'/H'\) remains irreducible when it is restricted to \(G\). This reduces the matter to (well-known) harmonic analysis for real hyperbolic spaces.