The Structure of Parabolic Subgroups

Kenneth D. Johnson

Abstract: Suppose $G$ is a real connected simple noncompact Lie group with (using standard notation) Iwasawa decomposition $G=KAN$. If $M=Z(A)\cap K$, the group $B=MAN$ is a minimal parabolic subgroup of $G$. Since $A$ is a vector group and $N$ is a simply connected nilpotent group, the topological structure of $B$ is determined by the structure of $M$. When $G$ is a linear group the structure of $M$ is well known. However, if $G$ is not a linear group there is very little available information about $M$. Our purpose here is to give a description of the group $M$ for any connected, simply connected, nonlinear simple group $G$.

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