TITLE: Discrete-compact duality for nonabelian groups
QUESTION [17 upvotes]: 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...

REPLY [3 votes]: Just an example, from which I learned a lot. Consider the free group $F$ on two generators. It makes a lot of sense to think of $n \mapsto hom(F,U(n))$ as some sort of dual. It comes with a natural conjugation action of $U(n)$ and natural operations of $\oplus$ and $\otimes$.
However, if one considers the bi-dual, which in this context would be the set of natural transformation $F^{xx}$ from the functor $n \mapsto hom(F,U(n))$ to $n \mapsto U(n)$, compatible with conjugation, $\oplus$ and $\otimes$, then this turns out to be too big and not discrete. First of all, $F^{xx}$ is a polish group and there is a natural homomorphism $F \to F^{xx}$. The whole construction goes under the name Chu duality and works just as in the case of Pontrjagin duality or Tannaka-Krein duality. But $F \to F^{xx}$ is not a homeomorphism. 
This follows from the fact that for fixed $n \in \mathbb N$ and a fixed neighborhood $V$ of $1_n \in U(n)$, there exists $w \in F \setminus \lbrace e\rbrace$, such that $\phi(w) \in V$, for all homomorphisms $\phi \colon F \to U(n)$. The same problem appears for every finitely generated group, which is not virtually abelian.
The problem can be cured completely if one takes the infinite-dimensional representations into account. Then, the appropriate bi-dual is equal to $F$ (and the same holds for any locally compact group). An important step in the proof of this assertion is the Gelfand-Raikov theorem.