The Plancherel formula for homogeneous spaces with polynomial spectrum

Orbital Symmetric Spaces and Finite Multiplicity

“…In [19] it is shown that the latter equals q and examine [7] again, we see that our argument is reduced to showing…”

“…Thus GÂH will be orbital exactly when U = ÂS is a singleton (measure-theoretically) and R * ÂS * is orbital. We will prove (below in Lemma 5.1 see also [19,Thm. 5.3]) that R * ÂS * is symmetric and so the outline of our inductive scheme begins to emerge.…”

“…We shall construct the scheme by stirring the Mackey Machine output into the analysis of { in situations of successively greater generality. At each stage we add some new ingredients into the mixture (mainly from [1] [6] [7] [13] and [19]) that allow the structure to take shape. The stages are marked by the weaker and weaker assumptions of: (a) abelian unipotent radical, (b) no Mackey obstructions, (c) no central twist, and (d) full generality.…”

“…(See [19, for a discussion of how the symmetry of GÂH simplifies the computation.) The spaces R * ÂS * are symmetric (see [l9, Thm.…”

“…in [11] [13,96] [14] [16,94] and [19]). One interesting category is that of abelian symmetric spaces namely GÂH, where G = H _ V is a semidirect product of a normal real vector subgroup V by any Lie group H acting on V satisfying the condition that V ÂH is countably separated.…”