Let $G=R\cdot S$ be a connected reductive algebraic group where $S$ is normal semisimple and $R$ is central.

Suppose that $G$ acts algebraically on the protective variety $\mathbb P(V)$. I wanna understand the $S$-orbits in $\mathbb P(V)$. They are closed in $G$-orbits, but why they are flag varieties?


1 Answer 1


I think I know now the answer of my question:

Since $G$ is algebraic and acts algebraically on $\mathbb P(V)$, then $G$ can be represented as an algebraic subgroup of $PGL(V)$. The commutator $G'$ is -by chevalley's theorem an algebraic subgroup of $PGL(V)$. (i.e. $G'$ is zariski-closed in $PGL(V)$). Since $G$ is reductive, then $G'=S$ and hence $S$-orbits are open in their closures in $\mathbb P(V)$. But $S$ is normal in $G$, then all orbits are closed in the compact sets.

Therefore, an $S$-orbit is compact implies that the isotropy group is parabolic. Hence, $S$ orbits are flag manifolds.


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