Maximal torus and the structure of complex flag manifolds

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Consider the compact Lie group $G=Tcdot S$ with finite intersection $I:=Tcap S$, where $T$ is the central torus and $S$ the maximal semisimple subgroup. Now, $S/Icong G/T$ has the structure of a complex flag manifold.



Since $S/I$ is a Lie group, Does it mean that it has the structure of a complex Lie group? I was thinking in the case where the intersection is finite then if $S$ is complex Lie group then it is abelian since it is compact!



So if $S/I$ is not a complex Lie group, what its complexification?




differential-geometry lie-groups




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  • It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
    – Mike Miller
    13 mins ago














up vote
1
down vote

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Consider the compact Lie group $G=Tcdot S$ with finite intersection $I:=Tcap S$, where $T$ is the central torus and $S$ the maximal semisimple subgroup. Now, $S/Icong G/T$ has the structure of a complex flag manifold.



Since $S/I$ is a Lie group, Does it mean that it has the structure of a complex Lie group? I was thinking in the case where the intersection is finite then if $S$ is complex Lie group then it is abelian since it is compact!



So if $S/I$ is not a complex Lie group, what its complexification?




differential-geometry lie-groups




share|cite|improve this question



















  • It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
    – Mike Miller
    13 mins ago












up vote
1
down vote

favorite









up vote
1
down vote

favorite











Consider the compact Lie group $G=Tcdot S$ with finite intersection $I:=Tcap S$, where $T$ is the central torus and $S$ the maximal semisimple subgroup. Now, $S/Icong G/T$ has the structure of a complex flag manifold.



Since $S/I$ is a Lie group, Does it mean that it has the structure of a complex Lie group? I was thinking in the case where the intersection is finite then if $S$ is complex Lie group then it is abelian since it is compact!



So if $S/I$ is not a complex Lie group, what its complexification?




differential-geometry lie-groups




share|cite|improve this question











Consider the compact Lie group $G=Tcdot S$ with finite intersection $I:=Tcap S$, where $T$ is the central torus and $S$ the maximal semisimple subgroup. Now, $S/Icong G/T$ has the structure of a complex flag manifold.



Since $S/I$ is a Lie group, Does it mean that it has the structure of a complex Lie group? I was thinking in the case where the intersection is finite then if $S$ is complex Lie group then it is abelian since it is compact!



So if $S/I$ is not a complex Lie group, what its complexification?





differential-geometry lie-groups





share|cite|improve this question










share|cite|improve this question




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  • It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
    – Mike Miller
    13 mins ago
















  • It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
    – Mike Miller
    13 mins ago















It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
– Mike Miller
13 mins ago




It is both a complex manifold and a Lie group, but as you observe $S$ is usually rather nonabelian, so the multiplication map is not holomorphic. I don't know the answer to your second question.
– Mike Miller
13 mins ago















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