T. Kobayashi and G. Mano,
The inversion formula and holomorphic extension of the minimal representation of the conformal group, Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: In Honour of Roger E. Howe (Jian-Shu Li, Eng-Chye Tan, Nolan Wallach, and Chen-Bo Zhu, eds.), Singapore University Press and World Scientific Publishing, 2007, pp. 159-223. ISBN 978-9812770783. DOI: 10.1142/9789812770790_0006. math.RT/0607007..
The minimal representation π of the indefinite orthogonal group O(m+1,2) is realized on the Hilbert space of square integrable functions on Rm with respect to the measure |x|-1 dx1... dxm. This article gives an explicit integral formula for the holomorphic extension of π to a holomorphic semigroup of O(m+3, C) by means of the Bessel function. Taking its 'boundary value', we also find the integral kernel of the 'inversion operator' corresponding to the inversion element on the Minkowski space Rm,1.

Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory

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