Remark on the Complexified Iwasawa Decomposition

A. R. Sinton ; J. A. Wolf

Journal of Lie theory, Tome 12 (2002) no. 2, pp. 617-618

Résumé

\def\C{\mathbb{C}} \def\R{\mathbb{R}} Let $G_\R$ be a real form of a complex semisimple Lie group $G_\C$\,. We identify the complexification $K_\C A_\C N_\C \subset G_\C$ of an Iwasawa decomposition $G_\R = K_\R A_\R N_\R$ as $\{g \in G_\C \mid gB \in \Omega\}$ where $B \subset G_\C$ is a Borel subgroup of $G_\C$ that contains $A_\R N_\R$ and $\Omega$ is the open $K_\C$--orbit on $G_\C /B$. This is done in the context of subsets $K_\C R_\C \subset G_\C\,$, where $R_\C$ is a parabolic subgroup of $G_\C$ defined over $\R$, and the open $K_\C$--orbits on complex flag manifolds $G_\C /Q$.