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.


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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