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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.

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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