The inversion formula and holomorphic extension of the minimal   representation of the conformal group
  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 $|x|^{-1} dx_1... dx_m$.   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 $\mathbb R^{m,1}$. 
Toshiyuki Kobayashi and Gen Mano