Special unipotent representations of real classical groups: construction and unitarity
Dan Barbasch, Jia-Jun Ma, Binyong Sun, Chen-Bo Zhu
TL;DR
This work classifies and constructs all special unipotent representations of real classical groups $G$ attached to a nilpotent orbit in the Langlands dual, establishing their unitarity in line with the Arthur–Barbasch–Vogan conjecture. The authors develop a complete combinatorial parametrization via painted bipartitions, pair these data with a descent along Howe duals, and realize the representations by theta lifting, complemented by geometric theta lifts to control associated cycles. A central achievement is proving that the associated cycles of the constructed representations match the combinatorial data, enabling an exhaustion and bijection between extended painted bipartitions and the unipotent representations; this is underpinned by a careful analysis of unitarity through matrix-coefficient integrals and the oscillator representation. The results, together with prior work, verify unitarity in the classical real setting and provide a detailed description of associated/wave-front cycles, underpinning the ABV framework for real groups of classical type and informing endoscopic classifications and dual-pair correspondences.
Abstract
Let $G$ be a real classical group (including the real metaplectic group). We consider a nilpotent adjoint orbit $\check{\mathcal O}$ of $\check G$, the Langlands dual of $G$ (or the metaplectic dual of $G$ when $G$ is a real metaplectic group). We classify all special unipotent representations of $G$ attached to $\check{\mathcal O}$, in the sense of Arthur and Barbasch-Vogan. When $\check{\mathcal O}$ has good parity in the sense of Moeglin, we construct all such representations of $G$ via the method of theta lifting. As a consequence of the construction and the classification, we conclude that all special unipotent representations of $G$ are unitarizable, as predicted by the Arthur-Barbasch-Vogan conjecture. We also determine precise structure of the associated cycles of special unipotent representations of $G$. The paper is the second in a series of two papers on the classification of special unipotent representations of real classical groups.