Harmonic Analysis On Two Kinds Of Semisimple Symmetric Spaces

This thesis is a study on the harmonic analysis of two kinds of symmetric spaces:the Paley-Wiener type theorem on the tangent space of the rank two causal symmetric space SO(2n + 1,C)/SO(2n-1,2),explicit Plancherel formula on the line bundle over SL(n+1,R)/S(GL(1,R)x GL(n,R)).This thesis is organized as follows:Chapter 1 is devoted to the classical Paley-Wiener on the Euclidean space and its generalization to the Riemannian symmetric spaces,then we prove the Paley-Wiener theorem on the tangent space of the rank one causal symmetric space SO(1,3)/SO(1,2).Based on the result of rank one case,we use deformed shift operator to obtain Paley-Wiener type theorems on the tangent spaces of the rank two causal symmetric space SO(2n + 1,C)/SO(2n-1,2).In Chapter 2,we calculate the Casimir operator on the line bundles over the semisimple symmetric space SL(n + 1,R)/S(GL(1,R)x GL(n,R)),obtain the con-crete expressions of the x-spherical distributions and finally we write out the explicit Plancherel formula.