TITLE: Representation of Groupoids
QUESTION [15 upvotes]: The title is vague, my actuall question is the following:
Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? (Either brand new things or pitfalls for those who too familiar with representations of groups). Does this point of view simplified any proof of theorems in representation of groups?
I've been only think of this for 15 mins. But I feel like it might be helpful to think of representation of groupoids for the following reasons:

When one talk about local systems, thinking of it as "representation of the fundamental groupoid" seems more natural than talking about "representation of the fundamental group".
When we talk about modules on stacks, if we choose a presentation of the stack (which is a groupoid), can we treat a given module as a representation of the groupoid? (Just as modules on BG gives representations of G.) Studying when two representations give the "same" module will be interesting.

REPLY [3 votes]: There is a theory of representation of groupoids, that runs parallel to the classical theory of locally compact groups. A locally compact Hausdorff group $G$ always has a left (as well as a right) invariant measure. Using this measure one can define the convolution *-algebra of the group. A classical result is, that there is a natural bijection between the irreducible unitary representations of $G$ and non-degenerate *-representations of the convolution algebra of $G$.
On the similar lines, J. Renault proves, that if $G$ is a locally compact, Hausdorff groupoid with a left invariant continuous family of measures, then one can form a convolution *-algebra of $G$. An irreducible unitary representation of $G$ are defined using a measurable bundle of Hilbert spaces over the space of units of $G$ and a quasi-invariant measure on the space of units of $G$. Then he proves that there is a natural bijection between the irreducible unitary representations of $G$ and the convolution *-algebra of $G$.
When $G$ is a group, this theory is same as the classical theory representation theory.
Indeed, proving this theory for groupoids is very hard and there are many issues to take care of. Groupoids behave differently than groups. For example, a locally compact Hausdorff groupoid need not carry an invariant continuous family of measures. One has to demand the family of measures, as data. As mentioned earlier, a good reference for this purpose is Renault's book,``A groupoid approach to $C^*$-algebras. This theory is very useful for studying dynamical systems and associated $C^*$-algebras.