A Method of Harmonic Analysis on Semisimple Symmetric Spaces

Abstract

This chapter discusses a method of harmonic analysis on semisimple symmetric spaces. A homogeneous space X = G/H of a connected Lie group G is called a symmetric space if there exists an involutive automorphism σ of G such that H is an open subgroup of the fixed point group of σ. It considers a case where G is a connected real linear semisimple Lie group. Then the symmetric space is called a semisimple symmetric space. Moreover, if σ is a Cartan involution, the group H is a maximal compact subgroup of G and the symmetric space is called a Riemannian symmetric space of noncompact type. T has been assumed that G is a connected real linear semisimple Lie group. Then the symmetric space X = G/H admits an invariant measure and have a unitary representation of G on L2(X). The most fundamental problem on the harmonic analysis on the semisimple symmetric space is to give an explicit decomposition of L2(X) into irreducible unitary representations of G, which is called a Plancherel formula.