Compact submanifold of the projective space and semismple algebraic group action

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Let $X$ be a compact submanifold of $mathbb CP^n$ such that there exists a semisimple complex linear algebraic group $G$ acts transitively on $X$. If $X$ is $G$-equivariantly embedded in $mathbb CP^n$. How to show that $X$ is a flag manifold?







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    Let $X$ be a compact submanifold of $mathbb CP^n$ such that there exists a semisimple complex linear algebraic group $G$ acts transitively on $X$. If $X$ is $G$-equivariantly embedded in $mathbb CP^n$. How to show that $X$ is a flag manifold?







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      up vote
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      down vote

      favorite









      up vote
      1
      down vote

      favorite











      Let $X$ be a compact submanifold of $mathbb CP^n$ such that there exists a semisimple complex linear algebraic group $G$ acts transitively on $X$. If $X$ is $G$-equivariantly embedded in $mathbb CP^n$. How to show that $X$ is a flag manifold?







      share|cite|improve this question












      Let $X$ be a compact submanifold of $mathbb CP^n$ such that there exists a semisimple complex linear algebraic group $G$ acts transitively on $X$. If $X$ is $G$-equivariantly embedded in $mathbb CP^n$. How to show that $X$ is a flag manifold?









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      asked Aug 12 at 0:03









      Ronald

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