T. Kobayashi and B. Ørsted,

*Analysis on the minimal representations of O*(*p*,*q*),
*III. — Ultra-hyperbolic equations on* **R**^{p-1,q-1},

Adv. Math.**180** (2003), 551-595,
math.RT/0111086..

Adv. Math.

For the groupO(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of theq= 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The groupO(p,q) acts as the Moebius group of conformal transformations onR^{p-1, q-1}, and preserves a space of solutions of the ultrahyperbolic Laplace equation onR^{p-1, q-1}. We construct in an intrinsic and natural way a Hilbert space of ultrahyperbolic solutions so thatO(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation onL^{2}(C), whereCis the conical subvariety of the nilradical of a maximal parabolic subalgebra obtained by intersecting with the minimal nilpotent orbit in the Lie algebra ofO(p,q).

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© Toshiyuki Kobayashi