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?


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.