By a construction of Tits, Kantor, and Koecher, every simple real Jordan algebra may be identified with the nilradical of a maximal parabolic subgroup \(P = LN\) of a semisimple Lie group \(G\). Moreover, such a parabolic is characterized by the two conditions that (i) it has an abelian nilradical, and (ii) it is conjugate to its opposite \(\overline {P}\).
In this paper we study the representations of \(G\) on sections of line bundles on \(G/ \overline {P}\), and determine their composition series and unitarizable constituents. Of particular interest are the small unitarizable constituents which are “unipotent”. In a sequel we will show that these have natural realizations on appropriate \(L^ 2\)-spaces of \(L\)-orbits in \(N\).