Special unipotent representations of real classical groups: Counting and reduction

Abstract

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 $\check{\mathcal{O}}$, in the sense of Arthur and Barbasch–Vogan. Here $\check{\mathcal{O}}$ 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 $\check{\mathcal{O}}$. We also reduce the problem of constructing special unipotent representations attached to $\check{\mathcal{O}}$ to the case when $\check{\mathcal{O}}$ 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.