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.


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


[1] DOI: 10.1063/1.1664833 · Zbl 0184.54601 · doi:10.1063/1.1664833
[2] DOI: 10.1137/0515056 · Zbl 0578.42007 · doi:10.1137/0515056
[3] DOI: 10.1063/1.526072 · doi:10.1063/1.526072
[4] DOI: 10.1007/BF01646483 · Zbl 0214.38203 · doi:10.1007/BF01646483
[5] DOI: 10.1007/BF01645091 · Zbl 0243.22016 · doi:10.1007/BF01645091
[6] DOI: 10.1016/0022-1236(76)90079-3 · Zbl 0317.43013 · doi:10.1016/0022-1236(76)90079-3
[7] DOI: 10.1017/S0004972700036728 · Zbl 0327.22008 · doi:10.1017/S0004972700036728
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