Work

Special unipotent representations of real classical groups: Counting and reduction

Year: 2025

Type: article

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 \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.

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1609

Authors: Dan Barbasch, Jia-Jun Ma, Binyong Sun, Chen-Bo Zhu

Institutions: Cornell University, Xiamen University, Xiamen University Malaysia, Zhejiang University, National University of Singapore

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond