Minimal representations via Bessel operators

Abstract

We construct an $L^2$-model of “very small” irreducible unitary representations of simple Lie groups $G$ which, up to finite covering, occur as conformal groups Co$(V)$ of simple Jordan algebras $V$. If $V$ is split and $G$ is not of type $A_n$, then the representations are minimal in the sense that the annihilators are the Joseph ideals. Our construction allows the case where $G$ does not admit minimal representations. In particular, applying to Jordan algebras of split rank one we obtain the entire complementary series representations of $SO(n,1)_0$. A distinguished feature of these representations in all cases is that they attain the minimum of the Gelfand-Kirillov dimensions among irreducible unitary representations. Our construction provides a unified way to realize the irreducible unitary representations of the Lie groups in question as Schrödinger models in $L^2$-spaces on Lagrangian submanifolds of the minimal real nilpotent coadjoint orbits. In this realization the Lie algebra representations are given explicitly by differential operators of order at most two, and the key new ingredient is a systematic use of specific second-order differential operators (Bessel operators) which are naturally defined in terms of the Jordan structure.