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Square integrable representation of groupoids. (English) Zbl 1114.22003

Summary: A notion of an irreducible representation, as well as of a square integrable representation on an arbitrary locally compact groupoid, is introduced. A generalization of a version of Schur’s lemma on a locally compact groupoid is given. This is used in order to extend some well-known results from locally compact groups to the case of locally compact groupoids. Indeed, we have proved that if \(L\) is a continuous irreducible representation of a compact groupoid \(G\) defined by a continuous Hilbert bundle \(\mathbb H = \{H _{ u }\}_{u\in G^{0}}\), then each \(H_{ u }\) is finite dimensional. It is also shown that if \(L\) is an irreducible representation of a principal locally compact groupoid defined by a Hilbert bundle \((G ^{0}, \{H _{ u }\}, \mu )\), then \(\dim H_{ u } = 1\) \((u \in G^{0})\). Furthermore it is proved that every square integrable representation of a locally 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 left regular representation is square integrable.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
22A25 Representations of general topological groups and semigroups
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