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Let $G$ be an algebraic group (or a group scheme) over a field $\Bbbk$, and let $V$ be an algebraic $G-$representation (I mean, corresponding to a homomorphism of $\Bbbk-$group schemes $G\rightarrow \mathrm{GL}(n,\Bbbk)$). Then $V$ can be seen as a scheme (isomorphic to $\mathbb{A}^n_{\Bbbk}$) with an action of $G$ by algebraic automorphisms. Question:

Is there an example of non-equivalent $G-$representations $V,V'$ that are $G-$equivariantly isomorphic as schemes over $\Bbbk$?

I would be most curious of an exmple (if it exists) with $\Bbbk=\mathbb{C}$ and $G$ reductive.

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    $\begingroup$ No, such example does not exist. Namely both schemes $V$ and $V'$ have $G-$invariant points and the tangent spaces at any of these points are $G-$isomorphic to $V$ and $V'$ respectively. The $G-$equivariant isomorphism of schemes will give us isomorphism of tangent spaces and we are done. $\endgroup$
    – Victor Ostrik
    Commented Aug 24, 2014 at 3:47

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