A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for nonabelian groups?

I can guess what this means for a compact (Hausdorff) group $G$: the category of unitary representations of $G$ should be discrete in the sense that every one-parameter family of unitary representations consists of isomorphic representations, or something like that. Is this true? Is the converse true?

I am less sure what this means for a discrete group $G$. What does it mean for the category of unitary representations to be compact?

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for nonabelian groups?

I can guess what this means for a compact (Hausdorff) group $G$: the category of unitary representations of $G$ should be discrete in the sense that every one-parameter family of unitary representations consists of isomorphic representations, or something like that. Is this true? Is the converse true?

I am less sure what this means for a discrete group $G$. What does it mean for the category of unitary representations to be compact? I suppose that $\text{Hom}(G, \text{U}(n))$ is a closed subspace of $\text{U}(n)^{G}$, hence compact, hence so is the appropriate quotient space of it...

A standard property of Pontryjagin duality is that a group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for nonabelian groups?

I can guess what this means for a compact (Hausdorff) group $G$: the category of unitary representations of $G$ should be discrete in the sense that every one-parameter family of unitary representations consists of isomorphic representations, or something like that. Is this true? Is the converse true?

I am less sure what this means for a discrete group $G$. What does it mean for the category of unitary representations to be compact?

A standard property of Pontrjagin duality is that a locally compact Hausdorff abelian group is discrete iff its dual is compact (and vice versa). In what senses, if any, is this still true for nonabelian groups?

I can guess what this means for a compact (Hausdorff) group $G$: the category of unitary representations of $G$ should be discrete in the sense that every one-parameter family of unitary representations consists of isomorphic representations, or something like that. Is this true? Is the converse true?

I am less sure what this means for a discrete group $G$. What does it mean for the category of unitary representations to be compact?