Analysis on the minimal representation of \(O(p,q)\). I: Realization via conformal geometry.
For a semisimple Lie group \(G\) an interesting unitary irreducible representation (called the minimal representation) is the one corresponding via geometric quantization to the minimal nilpotent coadjoint orbit. This representation coincides with the minimal representation studied by Kostant [\textit{B. Kostant}, The vanishing scalar curvature and the minimal representation of \(SO(4,4)\). Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 85--124 (1990; Zbl 0739.22012)] and Binegar-Zierau [\textit{B. Binegar} and \textit{R. Zierau}, Commun. Math. Phys. 138, No. 2, 245--258 (1991; Zbl 0748.22009)]. The present paper deals with the study of the minimal unitary representation of the semisimple Lie group \(G = O (p,q)\) and plays a basic role for subsequent papers by the authors on this topic. The authors work on a single unitary representation and analyze it by examining its restrictions to natural subgroups of the semisimple Lie group. They study the symmetries of the representation space by breaking the large symmetry originally present in the Lie group \(O (p, q)\) by passing to a subgroup. In this way they consider the restriction from the conformal group \(O (p, q)\) to the subgroup of isometries, where different geometries (which are all locally conformal equivalent) are associated to different choices of the subgroup. Changing the subgroup, one obtains different models of the representation and one can compute the spectrum of the subgroup. Applying methods from conformal geometry of pseudo-Riemannian manifolds (like the functorial properties of the Yamabe operator) they obtain different models of the unitary representation and new proofs of unitarity and irreducibility.
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