Type of Document Dissertation Author Johansen, Troels Roussau Author's Email Address tjohan1@lsu.edu URN etd-07072004-160643 Title 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 Degree Doctor of Philosophy (Ph.D.) Department Mathematics Advisory Committee

Advisor Name Title Gestur Ólafsson Committee Chair Frank Neubrander Committee Member Jerome W. Hoffman Committee Member Lawrence Smolinsky Committee Member Mark Davidson Committee Member William M. Cready Dean's Representative Keywords

- regular representation
- plancherel decomposition
- hardy spaces
- orbits
- tube domain
Date of Defense 2004-06-28 Availability unrestricted AbstractWe construct aG-equivariant causal embedding of a compactly causal symmetric spaceG/Has an open dense subset of the Silov boundarySof the unbounded realization of a certain Hermitian symmetric spaceGof tube type. Then_{1}/K_{1}Sis an Euclidean space that is open and dense in the flag manifoldG, where_{1}/P'P'denotes a certain parabolic subgroup ofG. The regular representation of_{1}GonLis 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/His being realized in function spaces on the boundary via the Euclidean Fourier transform on the boundary.

Let

P'=Ldenote the Langlands decomposition of_{1}N_{1}P'. The Levi factorLof_{1}P'then acts on the boundaryS, and the orbitsOcan be characterized completely. ForG/Hof rank one we associate to each orbitOthe irreducible representation

L:=^{2}_{Oi}{fεL^{2}(S,dx)|supp fcO_{i}}

of

Gand show that the representation of_{1}Gon_{1}Ldecompose as an orthogonal direct sum of these representations.^{2}(S)

We show that by restriction to

Gof the representationsL, we thus obtain the Plancherel decomposition of^{2}_{Oi}Linto series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull.^{2}(G/H)Files

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