# 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

Abstract (Summary)

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:={f?L2(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.
Bibliographical Information:

Advisor:Lawrence Smolinsky; William M. Cready; Frank Neubrander; Jerome W. Hoffman; Mark Davidson; Gestur Ã“lafsson

School:Louisiana State University in Shreveport

School Location:USA - Louisiana

Source Type:Master's Thesis

Keywords:mathematics

ISBN:

Date of Publication:07/08/2004