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Integral formula of the unitary inversion operator for the minimal representation of $$\text{O}(p,q)$$. (English) Zbl 1230.22007
Summary: The indefinite orthogonal group $$G = \text{O}(p,q)$$ has a distinguished infinite dimensional unitary representation $$\pi$$, called the minimal representation for $$p+q$$ even and greater than 6. The Schrödinger model realizes $$\pi$$ on a very simple Hilbert space, namely, $$L^2(C)$$ consisting of square integrable functions on a Lagrangian 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 $$\pi(w_0)$$ 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 $$L^2(\mathbb R^n)$$ of the Weil representation, and leads us to an explicit formula of the action of the whole group $$\text{O}(p,q)$$ 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.

##### MSC:
 22E30 Analysis on real and complex Lie groups 22E46 Semisimple Lie groups and their representations 43A80 Analysis on other specific Lie groups
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