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

TL;DR

This paper constructs a new minimal unitary representation of the group O(p,q) using Euclidean Fourier analysis, extending previous work on the q=2 case and connecting it to ultrahyperbolic equations and conformal transformations.

Contribution

It introduces an intrinsic construction of the minimal representation of O(p,q) on solutions to ultrahyperbolic equations, generalizing known cases and establishing unitary equivalence with an L^2 space on a conical variety.

Findings

Constructed a Hilbert space of ultrahyperbolic solutions for O(p,q)

Proved the representation is irreducible and unitary

Established equivalence with L^2 space on a conical subvariety

Abstract

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).