Minimal representations via Bessel operators

TL;DR

This paper constructs a unified L^2-model for minimal and complementary series representations of certain simple Lie groups using Bessel operators, revealing explicit differential operator realizations.

Contribution

It introduces a novel construction of irreducible unitary representations via Bessel operators on Jordan algebra structures, covering cases without minimal representations.

Findings

Realizes representations as Schroedinger models on Lagrangian submanifolds.

Provides explicit differential operator formulas of order at most two.

Achieves minimal Gelfand--Kirillov dimensions among irreducible unitary representations.

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 Schroedinger 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.