Yes if $k=1$. No if $k>1$; the adjoint representation is a counterexample.
For the edited question: isn't the vector representation of SL(n) (and its dual) a counterexample for $k=2$?
Yes if $k=1$. No if $k>1$; the adjoint representation is a counterexample.
For the edited question: isn't the vector representation of SL(n) (and its dual) a counterexample for $k=2$?