A reductive group over an algebraically closed field is a smooth connected affine algebraic
group whose unipotent radical is trivial. The unipotent radical is its largest
connected normal unipotent subgroup. Over a general field,
reductivity is tested after extension to an algebraic
closure. For example, 
is reductive, and its center consists of scalar matrices.
A connected semisimple algebraic group is reductive
with finite center.
These groups occur on both sides of the geometric Langlands correspondence. In positive characteristic, that correspondence can require additional Lie-theoretic hypotheses beyond reductivity.
