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 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.

Date

2004

Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

Committee Chair

Gestur Ólafsson

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