Conformal Geometry and Branching Laws for Unitary Representations Attached to Minimal Nilpotent Orbits

For the unitary representation of O(p; q) attached to the minimal nilpotent coadjoint orbit, we explicitly calculate its restriction to certain natural dual pairs in O(p; q). We furthermore show how the results are compatible with the orbit method, in particular when viewing the minimal nilpotent orbit as belonging to the limit set of semisimple orbits. R esum e Pour la repr esentation unitaire de O(p; q) associ ee a l'orbite coadjointe minimale nilpotente on donnera une description explicite de la restriction a certaines paires duales naturelles. Les resultats sont compatibles avecs la m ethode d'orbites, en particulier on peut regarder l'orbite minimale nilpotente comme appartenant a une limite des orbites semisimples.