The inversion formula and holomorphic extension of the minimal representation of the conformal group

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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.