Weighted Karcher means on unipotent Lie groups

A substantial theory of the Karcher mean exists in the settings of Riemannian manifolds and positive matrix and operator spaces. Here a general setting for the study of the Karcher mean on Lie groups is proposed. Local existence and uniqueness results already exist, but here, a significant global result is obtained. It is shown that a natural computable iteration scheme for the Karcher mean exists for the Lie group of upper triangular unipotent matrices and that it always converges starting from any point after finitely many steps.