Analysis on a Semi-Simple Lie Group

Abstract

Let G be a connected semi-simple Lie group with finite center — then the primary objective of the present chapter is to develop machinery which, by itself, is more or less independent of representation theory but which, as will become apparent in later chapters, is of decisive importance in the Harmonic Analysis on G. The basic idea which underlies most of the discussion is that of transferring a given analytical question on G, via some device or other, to an analogous question on one of its Cartan subgroups (‘non-abelian’ → ‘abelian’...), or, at least to some lower dimensional subgroup (where, presumably, induction may be applied...). There is little to be gained by giving a detailed analysis of the main results at this point; instead we shall refer the reader to the introductory remarks which (usually) preface a given number in a given section.