Doctor of Philosophy (PhD)



Document Type



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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Committee Chair

Gestur Ólafsson