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Transforms associated to square integrable group representations. I: General results. (English) Zbl 0571.22021

Let G be a locally compact group, which need not be unimodular. Let \(x\to U(x)\quad (x\in G)\) be an irreducible unitary representation of G in a Hilbert space \({\mathcal H}(U)\). Assume that U is square integrable, i.e., that there exists in \({\mathcal H}(U)\) at least one nonzero vector g such that \(\int | (U(x)g,g)|^ 2 dx<\infty\). We give here a reasonably self-contained analysis of the correspondence associating to every vector \(f\in {\mathcal H}(U)\) the function (U(x)g,f) on G, discussing its isometry, characterization of the range, inversion, and simplest interpolation properties. This correspondence underlies many properties of generalized coherent states.

MSC:

22E70 Applications of Lie groups to the sciences; explicit representations
22D10 Unitary representations of locally compact groups
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