Abstract
In this chapter we introduce the basic concepts of representation theory of locally compact groups. Classically, a representation of a group G is an injective group homomorphism from G to some \({\rm GL}_n({\mathbb C})\), the idea being that the “abstract” group G is “represented” as a matrix group.
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- 1.
There is a set-theoretic problem here, since it is not clear why the equivalence classes should form a set. It is, however, not difficult to show that there exists a cardinality \({\alpha}\), depending on G, such that every irreducible unitary representation \((\pi,V_\pi)\) of G satisfies \({\rm dim} V_\pi\le{\alpha}\). This means that one can fix a Hilbert space H of dimension \({\alpha}\) and each irreducible unitary representation π can be realized on a subspace of H. Setting the representation equal to 1 on the orthogonal complement one gets a representation on H, i.e., a group homomorphism \(G\to{\rm GL}(H)\). Indeed, since every irreducible representation has a cyclic vector by Schur’s lemma, one can choose α as the cardinality of G. Therefore, each equivalence class has a representative in the set of all maps from G to \({\rm GL}(H)\) and so \(\hat G\) forms a set.
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© 2014 Springer International Publishing Switzerland
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Deitmar, A., Echterhoff, S. (2014). Representations. In: Principles of Harmonic Analysis. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-05792-7_6
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DOI: https://doi.org/10.1007/978-3-319-05792-7_6
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