Decomposition \(K=MK*M\) for semisimple real Lie groups of \(R\)-rank one (Q1058612)

Let \(G\) be a semisimple real Lie group of \(R\)-rank one, \(G=KAN\) be its Iwasawa factorization and \(M=\operatorname{Cent}_K(A).\) The author studies the structure of maximal compact subgroup \(K\) of \(G\). Namely, if \(G^*\) is a subgroup of \(G\) corresponding to the Lie algebra of rank one \[ \mathfrak g^*=(\sum_{k=\pm 1,\pm 2} \mathfrak g^{k\alpha})\oplus \mathfrak m\oplus \mathbb{R} H_{\alpha} \] then the mapping \(M\times K^*\times M\to^{\phi}K\) (where \(K^*=K\cap G^*)\) defined by the multiplication in group \(G\) is onto. In addition the application of the decomposition in different problems of harmonic analysis is shown.