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A reductive group is an algebraic group $G$ over an algebraically closed field such that the unipotent radical of $G$ is trivial

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Centralizer of a reductive subgroup

$\DeclareMathOperator\GL{GL}$No. Take $H=\GL_2$ embedded diagonally into $G=\GL_2\times \GL_2$ and take $\rho$ equal to $\mathbb C^2 \otimes (\mathbb C^2)^*$ with the natural action of $\GL_2$ on $\ma …
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