THE INVERSION FORMULA AND HOLOMORPHIC EXTENSION OF THE MINIMAL REPRESENTATION OF THE CONFORMAL GROUP

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|⁻¹dx₁ ⋯ 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}.