I am studying Geometric invariant theory and wonder how I should understand linearly reductive algebraic group. We say that an affine algebraic group $G$ is linearly reductive if all finite dimensional $G$-modules are semi-simple.
I am not sure if linearly reductive groups are the same as reductive groups, which are defined as algebraic groups $G$ over algebraically closed field such that the unipotent radical of $G$ is trivial. But this definition is still beyond my intuition.
Are there any good way to understand (linearly) reductive groups? It would especially be nice if reductive (Lie) groups can be characterized in geometry.