Irreducible Characters and Semisimple Coadjoint Orbits

When G(R) in is a real, linear algebraic group, the orbit method predicts that nearly all of the unitary dual of G(R) in consists of representations naturally associated to orbital parameters (O, Gamma). If G(R) in is a real, reductive group and O is a semisimple coadjoint orbit, the corresponding unitary representation pi(O, Gamma) may be constructed utilizing Vogan and Zuckerman's cohomological induction together with Mackey's real parabolic induction. In this article, we give a geometric character formula for such representations pi(O, Gamma). Special cases of this formula were previously obtained by Harish-Chandra and Kirillov when G(R) in is compact and by Rossmann and Duflo when pi(O, Gamma) is tempered.