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Let $G$ be an algebraic group acting transitively on closed projective varieties $X$ and $Y$. Let $X \xrightarrow{f} Y$ be a bijective continuous map, that is $G-$equivariant. Does it follow that $X \to Y$ is an isomorphism of algebraic varieties?

It seems roughly intuitive because if the derivative of this map is zero somewhere then it should be zero everywhere by the equivariance.

If there is a transversality proof that is easier for $G$-manifolds and diffeomorphisms, or if it easier to prove it for complex algebraic varieties and biholomorphisms, i'd be all for it.

My motivation: There are two structures as a projective variety, that one can put on $GL(n)/B$ where $B$ is the Borel subgroup(which I define as a maximal connnected solvable subgroup of $GL(n)$).

Structure one: One can use Chevalley's theorem to define what $G/B$ means as a quasiprojective variety: its an orbit in $P(W)$ for some $G$ module $W$, which is therefore open in its closure. Then one can show that its closed $P(W)$ because $B$ is solvable, and orbits of minimal dimension are closed. Therefore $GL/B$ is a closed orbit in $P(W)$.

Structure two: There is a bijection of $G-$sets between $GL(n)/B$ and $Flag(\mathbb{C}^n)$. There is a closed embedding of this into a product of grassmanians, and grassmanians are closed subvarieties of projective space by plucker. So $GL(n)/B$ can be regarded as a closed subspace of $P(\wedge^1 \mathbb{C^n} \otimes ...\wedge^{n-1} \mathbb{C^n})$.

I want to know 'Why are these $G$-varieties isomorphic as projective varieties?' and the question above would allow me to answer this.

What I have got from professors: A professor at tea told me that I need to use the correspondence between maps from projective space and line bundles over the domain. Brainstorming: Maybe I need to use something about $G$ equivariant line bundles over $G/B$ since these will correspond to $G$-equivariant maps from $G/B$.. Borel-weil-bott theorem anyone?

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  • $\begingroup$ The answer is most likely yes for the reason you stated--as long as your are in char $0$ and algebraically closed (which is probably all you care about). You really need to show that $f$ is an open embedding or, equivalently, that it's etale and radiciel. Your morphism is generically smooth, and thus (since it's $G$-equivariant) actually smooth, but since it's a bijection, it must then be etale. You want to then see that it's universally injective. Which means that you want to show that $f^{-1}(\text{Spec}(k))$ is just $\text{Spec}(k)$. But, by assumption it's a single (physical) point and $\endgroup$ Sep 22, 2016 at 6:49
  • $\begingroup$ reduced, thus actually just $\text{Spec}(k)$. Is this the sort of thing that makes you happy? (PS: I really just answered the first part of your question--I didn't read the motivation). $\endgroup$ Sep 22, 2016 at 6:50
  • $\begingroup$ Let me look up what an etale morphism is and then i'll get back to you when i'm finished! Thank you for doing this. $\endgroup$
    – user062295
    Sep 22, 2016 at 16:51

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I am not closely familiar with algebraic groups, but the corresponding question for Lie groups has a simple postive answer: Take a point $x_0\in X$, and denote by $G_{x_0}$ its stabilizer in $G$. Then $g\mapsto g\cdot x_0$ induces a diffeomorphism $G/G_{x_0}\to X$. Since $f$ is equivariant and bijective, you see that for $y_0=f(x_0)$, you get $G_{y_0}=G_{x_0}$ and $g\mapsto g\cdot y_0$ induces a diffeomorphism $G/G_{y_0}\to Y$, so $X$ and $Y$ are $G$-equivariantly diffeomorphic.

On the other hand, for your motivating example, the two constructions for $G/B$ you describe are essentially the same. The natural choice for the representation $W$ in Chevalley's construction is that the highest weight of $W$ is the sum of all fundamental weights. The natural construction of $W$ is a the $G$-invariant subspace in the tensor product of all fundamental representations. These fundamental representations are just $\mathbb C^n$, $\Lambda^2\mathbb C^n$, ..., $\Lambda^{n-1}\mathbb C^n$. So $P(W)$ is naturally a subspace of $P(\mathbb C^n\otimes\dots\otimes \Lambda^{n-1}\mathbb C^n)$. By definition, the embedding of $G/B$ into $P(W)$ is induced by $g\mapsto g\cdot w_0$, where $w_0$ is a highest weight vector in $W$. The natural choice for $w_0$ is the tensor product of the highest weight vectors in the fundamental representations. But you can also obtain the Plücker embedding of the Grassmannian $Gr(m,\mathbb C^n)$ similarly to Chevalley's construction, but starting from the highest weight vector in $\Lambda^m\mathbb C^m$. So you readily see that the embedding you construct via PlÜcker actually has values in $P(W)$ and coincides with the Chevalley embedding.

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