| Summary: | We construct a <i>G</i>-equivariant causal embedding of a compactly causal symmetric space <i>G/H</i> as an open dense subset of the Silov boundary <i>S</i> of the unbounded realization of a certain Hermitian symmetric space <i>G<sub>1</sub>/K<sub>1</sub></i> of tube type. Then <i>S</i> is an Euclidean space that is open and dense in the flag manifold <i>G<sub>1</sub>/P'</i>, where <i>P'</i> denotes a certain parabolic subgroup of <i>G<sub>1</sub></i>. The regular representation of <i>G</i> on <i>L<sup>2</sup>(G/H)</i> is thus realized on <i>L<sup>2</sup>(S)</i>, and we use abelian harmonic analysis in the study thereof. In particular, the holomorphic discrete series of <i>G/H</i> is being realized in function spaces on the boundary via the Euclidean Fourier transform on the boundary.
Let <i>P'=L<sub>1</sub>N<sub>1</sub></i> denote the Langlands decomposition of <i>P'</i>. The Levi factor <i>L<sub>1</sub></i> of <i>P'</i> then acts on the boundary <i>S</i>, and the orbits <i>O</i> can be characterized completely. For <i>G/H</i> of rank one we associate to each orbit <i>O</i> the irreducible representation
<i>L<sup>2</sup><sub>O<sub>i</sub></sub></i>:=<i>{fεL<sup>2</sup>(S,dx)|supp fc<font style="text-decoration: overline;">O<sub>i</sub></font>}</i>
of <i>G<sub>1</sub></i> and show that the representation of <i>G<sub>1</sub></i> on <i>L<sup>2</sup>(S)</i> decompose as an orthogonal direct sum of these representations.
We show that by restriction to <i>G</i> of the representations <i>L<sup>2</sup><sub>O<sub>i</sub></sub></i>, we thus obtain the Plancherel decomposition of <i>L<sup>2</sup>(G/H)</i> into series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull.
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