Invariant eigendistributions on a semisimple Lie group

Abstract

2. The mapping Tx.Let G be a Lie group, g its Lie algebra over R and © the universal enveloping algebra of gc.For any X e gc, let Lx and Rx, respectively, denote the endomorphisms g->Xg and g->gX (ge©) of (5.Fix xe G and defined) (cf.[4(e), p. 114]) ax(X) = Lx-ix-Rx (Xeqc).Note that Lx and RY commute and(2)for X, Yegc.Hence it follows immediately that ax is a representation of gcon®.It may, therefore, be extended (uniquely) to a representation of © which we shall again denote by ax.Let Tx denote the linear mapping of © ® (5 into © such that Tx(gy ® g2) = 0-x(gy)g2 (gugl