Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}

Details Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1} Analysis on the minimal representation of O(p,q) -- III. ultrahyperbolic equations on R^{p-1,q-1}

2001-11-07

arxiv.org/abs/math/0111086v1

Adv. Math., 180 (2003), pp.551-595

Mathematics

Representation Theory

Scientific paper

10.1016/S0001-8708(03)00014-8

For the group O(p,q) we give a new construction of its minimal unitary representation via Euclidean Fourier analysis. This is an extension of the q = 2 case, where the representation is the mass zero, spin zero representation realized in a Hilbert space of solutions to the wave equation. The group O(p,q) acts as the Moebius group of conformal transformations on R^{p-1, q-1}, and preserves a space of solutions of the ultrahyperbolic Laplace equation on R^{p-1, q-1}. We construct in an intrinsic and natural way a Hilbert space of ultrahyperbolic solutions so that O(p,q) becomes a continuous irreducible unitary representation in this Hilbert space. We also prove that this representation is unitarily equivalent to the representation on 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).

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