Special unipotent representations of real classical groups: Counting and reduction

Let G be a real reductive group in Harish-Chandra’s class. We derive some consequences of the theory of coherent continuation representations to the counting of irreducible representations of G with a given infinitesimal character and a given bound of the complex associated variety. When G is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of G attached to \widecheck{\mathcal{O}} , in the sense of Arthur and Barbasch–Vogan. Here \widecheck{\mathcal{O}} is a nilpotent adjoint orbit in the Langlands dual of G (or the metaplectic dual of G when G is a real metaplectic group). We give a precise count for the number of special unipotent representations of G attached to \widecheck{\mathcal{O}} . We also reduce the problem of constructing special unipotent representations attached to \widecheck{\mathcal{O}} to the case when \widecheck{\mathcal{O}} is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups.