Representations of Semisimple Lie Groups

Let G be a Lie group and § a Banach space. A representation n of G on § is a mapping which assigns to every element x in G a bounded linear operator n(x) on § such that the following two conditions are fulfilled: (1) 7t(xy) = n(x) 7i(y) (x, y e G),n(l) = / and (2) the mapping (x, tp) ->uz(x)y) of G x § into § is continuous. (Here 1 is the unit element of G and I is the unit operator,) In particular if § is a Hilbert space and n(x) is a unitary operator for every x e G, we say that n is a unitary representation. In the study of finite-dimensional representations the success of the infinitesimal method is well known. Our object is to make this method applicable also to the infinite-dimensional case. First we discuss a few preliminary notions. Let / be a mapping of some neighborhood of the origin on the real line into the Banach space § . We say / is analytic at zero if it is possible to write it in the form