On reflexive groups and function spaces with a Mackey group topology

We prove that every reflexive abelian group $G$ such that its dual group $G^{\wedge }$ has the $qc$-Glicksberg property is a Mackey group. We show that a reflexive abelian group of finite exponent is a Mackey group. We prove that, for a realcompact space $X$, the space $C_k(X)$ is barrelled if and only if it is a Mackey group.