I'm reading Geometric Invariant Theory by Mumford, and confuse about the Proposition 2.4 on P54.

It states that:

Let $G$ be a reductive group, act on an algebraic scheme. Then the action of $G$ on $X$ is proper if and only if for every non-trivial 1-PS $\lambda: \mathbb{G}_m\to G$, the induced action of $\mathbb{G}_m$ on $X$ is proper.

Here the action is proper means $(\sigma, p_2):G\times X\to X\times X$ is proper as morphism.

I get confused since in the proof of this proposition, the book use a theorem of Iwahori on P52, which is stated for $G$ semisimple of adjoint type... Mumford use a similar argument on P53, which also just require $G$ to be reductive....

I wonder that how to reduce these proofs to the case when $G$ is semisimple of adjoint type, so I can use Iwahori's theorem. I try to pass to the case when $G= G/C(G)$, but I can't find a natural action of $G/C(G)$ on $X$.

Thanks for any help.


1 Answer 1


I believe this is addressed on page 52, underneath the statement of Iwahori's theorem. He provides an argument for why Iwahori's result can be strengthened to G reductive by considering $G \rightarrow G'$ where $G $ is reductive and $G'$ is the associated adjoint group.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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