We construct unitary representations of simple Lie groups which, up to finite covering, occur as conformal groups of simple Jordan algebras. The annihilator ideals of these representations are completely prime, and the associated varieties are the closures of the minimal complex nilpotent coadjoint orbits with real points. In particular, for split Jordan algebras with conformal Lie algebra not of type An the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction provides a unified way to realize the minimal representations of the Lie groups in question as Schrödinger models in L2-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits such that the Lie algebra representations are given explicitly by differential operators of order at most two. The key new ingredient in this realization are the (generalized) Bessel operators naturally defined in terms of the Jordan structure. For Jordan algebras of split rank one our construction gives L2-models for all complementary series representations of SO(n,1)0.
© Toshiyuki Kobayashi