Square Integrable Representation of Groupoids

A notion of an irreducible representation, as well as of a square integrable representationon an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur’s lemmaon a locally compact groupoid is given. This is used in order to extend some well-known results fromlocally compact groups to the case of locally compact groupoids. Indeed, we have proved that if Lis a continuous irreducible representation of a compact groupoid G defined by a continuous Hilbertbundle ? = {H _u }u∈¸G ⁰, then each H _u is finite dimensional. It is also shown that if L is an irreduciblerepresentation of a principal locally compact groupoid defined by a Hilbert bundle (G ⁰, {H _u }, μ), thendimH _u = 1 (u ∈¸ G ⁰). Furthermore it is proved that every square integrable representation of alocally compact groupoid is unitary equivalent to a subrepresentation of the left regular representation.Furthermore, for r-discrete groupoids, it is shown that every irreducible subrepresentation of the leftregular representation is square integrable.