T. Kobayashi and G. Mano,
Integral formula of the unitary inversion operator for the minimal representation of O(p,q), Proc. Japan Acad. Ser. A 83 (2007), no. 3, 27-31.

The indefinite orthogonal group G = O(p,q) has a distinguished infinite dimensional unitary representation π, called the minimal representation for p+q even and greater than 6. The Schrödinger model realizes π on a very simple Hilbert space, namely, L2(C) consisting of square integrable functions on a Lagrangean submanifold C of the minimal nilpotent coadjoint orbit, whereas the G-action on L2(C) has not been well-understood. This paper gives an explicit formula of the unitary operator π(w0) on L2(C) for the 'conformal inversion' w0 as an integro-differential operator, whose kernel function is given by a Bessel distribution. Our main theorem generalizes the classic Schrödinger model on L2(Rn) of the Weil representation, and leads us to an explicit formula of the action of the whole group O(p,q) on L2(C). As its corollaries, we also find a representation theoretic proof of the inversion formula and the Plancherel formula for Meijer's G-transforms.

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