Journal of Lie Theory 12 (2002), No. 2, 617--618
Copyright Heldermann Verlag 2002
Remark on the Complexified Iwasawa Decomposition
Andrew R. Sinton
Dept. of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A.
Joseph A. Wolf
Dept. of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A.
[Abstract-pdf]
\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 $G_\C /Q$.
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