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Integral formulas for the minimal representation of \(O(p,2\)). (English) Zbl 1100.22007
Summary: The minimal representation \(\pi\) of \(O(p,q)\) (\(p+q\) even) is realized on the Hilbert space of square integrable functions on the conical subvariety of \(\mathbb R^{p+q-2}\). This model presents a close resemblance of the Schrödinger model of the Segal–Shale–Weil representation of the metaplectic group. We shall give explicit integral formulas for the “inversion” together with the analytic continuation to a certain semigroup of \(O(p+2,\mathbb C\)) of the minimal representation of \(O(p,2)\) by using Bessel functions.

MSC:
22E30 Analysis on real and complex Lie groups
22E46 Semisimple Lie groups and their representations
20M20 Semigroups of transformations, relations, partitions, etc.
43A80 Analysis on other specific Lie groups
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