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Multiplicity formulas for representations of transformation groupoids

Giżycki A., Pysiak L.

Demonstratio Mathematica

2017

Vol. 50, nr 1

42--50

We study the representations of transitive transformation groupoids with the aim of generalizing the Mackey theory. Using the Mackey theory and a bijective correspondence between the imprimitivity systems and the representations of a transformation groupoid we derive the irreducibility theory. Then we derive the direct sum decomposition for representations of a groupoid together with the formula for the multiplicity of subrepresentations. We discuss a physical interpretation of this formula. Finally, we prove the claim analogous to the Peter–Weyl theorem for a noncompact transformation groupoid. We show that the representation theory of a transitive transformation groupoids is closely related to the representation theory of a compact groups.