#### Identifier

etd-07072004-160643

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

We construct a *G*-equivariant causal embedding of a compactly causal symmetric space *G/H* as an open dense subset of the Silov boundary *S* of the unbounded realization of a certain Hermitian symmetric space *G _{1}/K_{1}* of tube type. Then

*S*is an Euclidean space that is open and dense in the flag manifold

*G*, where

_{1}/P'*P'*denotes a certain parabolic subgroup of

*G*. The regular representation of

_{1}*G*on

*L*is thus realized on

^{2}(G/H)*L*, and we use abelian harmonic analysis in the study thereof. In particular, the holomorphic discrete series of

^{2}(S)*G/H*is being realized in function spaces on the boundary via the Euclidean Fourier transform on the boundary. Let

*P'=L*denote the Langlands decomposition of

_{1}N_{1}*P'*. The Levi factor

*L*of

_{1}*P'*then acts on the boundary

*S*, and the orbits

*O*can be characterized completely. For

*G/H*of rank one we associate to each orbit

*O*the irreducible representation

*L*:=

^{2}_{Oi}*{fεL*of

^{2}(S,dx)|supp fcO_{i}}*G*and show that the representation of

_{1}*G*on

_{1}*L*decompose as an orthogonal direct sum of these representations. We show that by restriction to

^{2}(S)*G*of the representations

*L*, we thus obtain the Plancherel decomposition of

^{2}_{Oi}*L*into series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull.

^{2}(G/H)#### Date

2004

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Johansen, Troels Roussau, "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" (2004). *LSU Doctoral Dissertations*. 1591.

http://digitalcommons.lsu.edu/gradschool_dissertations/1591

#### Committee Chair

Gestur Ólafsson