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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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How to translate the representation theory of semisimple to reductive groups?

The derived group $G'$ of $G$ always works for your second question (i.e., $G'(F)Z(F)$ is closed and cocompact in $G(F)$). Indeed, by using local class field theory and Kneser-Bruhat-Tits we know that …
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