## 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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### References:

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