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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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Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$
In view of Laurent's answer, one may ask more generally for a criterion on a connected reductive $K$-group $G$ to ensure that for all compact open subgroups $U$ of $G(\mathbf{A}_K^S)$ (for a finite se …