The Iwasawa Decomposition
Let us begin this topic with an example. Let G = GL(n, ℂ). It is the complexification of K= U(n), which is a maximal compact subgroup. Let T be the maximal torus of K consisting of diagonal matrices whose eigenvalues have absolute value 1. The complexification T ℂ of T can be factored as TA, where A is the group of diagonal matrices whose eigenvalues are positive real numbers. Let B be the group of upper triangular matrices in G,and let B 0 be the subgroup of elements of B whose diagonal entries are positive real numbers. Finally, let N be the subgroup of unipotent elements of B. Recalling that a matrix is called unipotent if its only eigenvalue is 1, the elements of N are upper triangular matrices whose diagonal entries are all equal to 1. We may factor B = TN and B 0 = AN. The subgroup N is normal in B and B 0, so these decompositions are semidirect products.