A Realization of Semisimple Symmetric Spaces and Construction of Boundary Value Maps

Abstract

This chapter explains the realization of semisimple symmetric spaces and the construction of boundary value maps. A homogeneous space G/H is called a semisimple symmetric space if G is a real connected semisimple Lie group and there exists an involution of G such that H is an open subgroup of the fixed point group of the involution. The most fundamental problem on the harmonic analysis on G/H is to give an explicit decomposition of L2 (G/H) into irreducible representations of G, that is, to get a Plancherel formula for L2 (G/H). L2 (G/H) is the space of square integrable functions on G/H with respect to the invariant measure. The chapter presents a method to obtain the Plancherel formula. On the other hand, by using Flensted Jensen's duality method, one can directly study the discrete series and can get sufficient information to analyze the discrete series.