TITLE: Representation theory of infinite groups?
QUESTION [5 upvotes]: 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).

REPLY [2 votes]: There are few interesting directions in which unitary (including infinite-dimensional) representation theory of infinite discrete groups is developed:

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. 
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.