Orbit Structure on the Silov Boundary of a Tube Domain and the Plancherel Decomposition of a Causally Compact Symmetric Space, with Emphasis on the Rank One Case

• Main Author: Johansen, Troels Roussau
• Other Authors: Lawrence Smolinsky
• Format: Others
• Language: en
• Published: LSU 2004
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-07072004-160643/

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>G₁/K₁</i> of tube type. Then <i>S</i> is an Euclidean space that is open and dense in the flag manifold <i>G₁/P'</i>, where <i>P'</i> denotes a certain parabolic subgroup of <i>G₁</i>. The regular representation of <i>G</i> on <i>L²(G/H)</i> is thus realized on <i>L²(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'=L₁N₁</i> denote the Langlands decomposition of <i>P'</i>. The Levi factor <i>L₁</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>L²_{O_i}</i>:=<i>{fεL²(S,dx)|supp fc<font style="text-decoration: overline;">O_i</font>}</i> of <i>G₁</i> and show that the representation of <i>G₁</i> on <i>L²(S)</i> decompose as an orthogonal direct sum of these representations. We show that by restriction to <i>G</i> of the representations <i>L²_{O_i}</i>, we thus obtain the Plancherel decomposition of <i>L²(G/H)</i> into series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull.