What is known about the representation theory of free groups? In particular, I am interested in character theory and faithful representations. Many results in this theory seem to depend on the group being finite, though it seems many generalizations exist (compact, locally compact, etc).
More concretely, I am interested in whether the following fact translates to free groups. It is known that in free groups, the characters of irreducible representations form a (orthonormal) basis of the vector space of all class functions.
I am looking for any and all references in this area. Thanks!