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

Abstract

The minimal representation $\pi$ of the indefinite orthogonal group $O(m+1,2)$ is realized on the Hilbert space of square integrable functions on ${\mathbb{R}}^m$ with respect to the measure $|x{|}^{-1}d{x}_1...d{x}_m$. This article gives an explicit integral formula for the holomorphic extension of $\pi$ to a holomorphic semigroup of $O(m+3,\mathbb{C})$ 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 ${\mathbb{R}}^{m,1}$.