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, to appear. 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 R^m with respect to the measure |x|^-1 dx₁... dx_m. 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 R^m,1.

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