## 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,
*L*^{2}(*C*) consisting of square integrable
functions on a Lagrangean submanifold *C* of the minimal nilpotent coadjoint orbit,
whereas the *G*-action on *L*^{2}(*C*) has not been well-understood.
This paper gives an explicit formula of
the unitary operator π(*w*_{0}) on *L*^{2}(*C*) for the 'conformal inversion'
*w*_{0} as an integro-differential operator, whose kernel function
is given by a Bessel distribution.
Our main theorem generalizes the classic Schrödinger model
on *L*^{2}(*R*^{n}) of the Weil representation,
and leads us to an explicit formula
of the action of the whole group *O*(*p*,*q*) on
*L*^{2}(*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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The original publication is available at projecteuclid.org.

© Toshiyuki Kobayashi