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 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 G1/K1 of tube type. Then S is an Euclidean space that is open and dense in the flag manifold G1/P', where P' denotes a certain parabolic subgroup of G1. The regular representation of G on L2(G/H) is thus realized on L2(S), and we use abelian harmonic analysis in the study thereof. In particular, the holomorphic discrete series of G/H is being realized in function spaces on the boundary via the Euclidean Fourier transform on the boundary.
Let P'=L1N1 denote the Langlands decomposition of P'. The Levi factor L1 of 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
L2Oi:={fεL2(S,dx)|supp fcOi}
of G1 and show that the representation of G1 on L2(S) decompose as an orthogonal direct sum of these representations.
We show that by restriction to G of the representations L2Oi, we thus obtain the Plancherel decomposition of L2(G/H) into series of unitary irreducible representations, in the sense of Delorme, van den Ban, and Schlichtkrull.
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