Representation theory of infinite groups?

Question

I am familiar with the representation theory of finite groups (at least of the symmetric groups over the field of complexes)

And I know that the group algebra of an infinite group is not semisimpe (lets say again over the complex field).

I would like to know if any work has been done in this area, I mean regarding the classification of simple modules of an infinite dimensional associative group algebra ? (which I guess it has been done but I am not aware of it).

Answer 1

The simplest case is when the group is equipped with a topology with respect to which it is compact (Hausdorff), so the proof of Maschke's theorem still works and the Peter-Weyl theorem is available. In particular, the representation theory of compact Lie groups is very well understood. The representation theory of noncompact Lie groups is still a major area of modern study, with ties to algebraic geometry, harmonic analysis and number theory. The representation theory of profinite groups such as Galois groups is also a major area.

The representation theory of infinite discrete groups is, as far as I know, extremely hard in general. Some work has been done on representations into $\text{SL}_2$ and related groups; see character variety and the discussion and references here, for example.

For some indication of how hard these questions are, the representation theory of $F_2$ is essentially a mystery: that is, no one knows a reasonable way to classify pairs of matrices up to simultaneous conjugation.

Answer 2

There are few interesting directions in which unitary (including infinite-dimensional) representation theory of infinite discrete groups is developed:

1. Property T: Isolation phenomenon of the trivial representation among all irreducible unitary representations. For instance, every discrete group with this property has to be finitely-generated; the group $SL(n,Z)$ has this property if and only if $n\ge 3$; there are many other classes of groups satisfying Property T. On the other hand, for instance, infinite free groups, fundamental groups of compact surfaces of positive genus and discrete infinite solvable groups never have this property. Taken together with the extreme flexibility of free groups (mentioned by Qiaochu Yuan), this suggests that having a "meaningful" unitary representation theory is most realistic in the setting of groups satisfying Property T.

2. There is one very nontrivial rigidity result about infinite-dimensional unitary representation theory of discrete subgroups of $SL(2, R)$, namely a theorem of Bishop and Steger which allows one to tell apart such subgroups up to conjugation by looking at certain classes of their infinite-dimensional unitary representation. Thus, maybe there is some hope for such groups as well.