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Analysis on the minimal representation of O(\(p,\) \(q\)). III: Ultrahyperbolic equations on \(\mathbb{R}^{p-1,q-1}\). (English) Zbl 1039.22005
A new construction of minimal unitary representation [B. Kostant, Prog. Math. 92, 85–124 (1990; Zbl 0739.22012), B. Binegar and R. Zierau, Commun. Math. Phys. 138, 245–258 (1991; Zbl 0748.22009)] of the noncompact orthogonal group \(O(p, q)\) via Euclidean Fourier analysis is given, which is an extension of the \(q=2\) case, where the representation is the mass zero, spin zero representation realized in the Hilbert space of solutions to the corresponding wave equation (1970, Todorov). It is taken into account that the group \(O(p,q)\) acts as the Möbius group of conformal transformations on the real finite-dimensional vector space \(R^{p-1,q-1}\), and preserves a space of solutions of the ultrahyperbolic Laplace equation on \(R^{p-1,q-1}\). In the paper under review a Hilbert space of solutions is constructed in an intrinsical way, so that \(O(p,q)\) becomes a continuous irreducible unitary representation in this Hilbert space. It is also proved that this representation is unitarily equivalent to the representation of \(L^2(C)\), where \(C\) is the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra of \(O(p,q)\). The paper is organized as follows:
1. Introduction. 2. Ultrahyperbolic equation on \(R^{p-1,q-1}\) and conformal group. 3. Square integrable functions on the cone. 4. Green function and inner product. 5. Bessel function and an integral formula of spherical functions. 6. Explicit inner product on solutions \(\square_{R^{p-1,q-1}} f=0\).

MSC:
22E45 Representations of Lie and linear algebraic groups over real fields: analytic methods
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