The Research and Postgraduate Centre congratulates Assoc. Prof. Dr. Ma Jiajun from the Department of Mathematics at Xiamen University Malaysiafor publishing the research "Special unipotent representations of real classical groups: Counting and reduction" in the Journal of the European Mathematical Society.

This work was conducted in collaboration with Prof. Dan Barbasch (Cornell University), Prof. Sun Binyong  (Zhejiang University), and Prof. Zhu Chengbo  (National University of Singapore), 

Special unipotent representations are a distinct class of representations of real reductive groups. In the 1980s, Arthur-Barbasch-Vogan conjectured that these representations are unitary and form the building blocks for other unitary representations. This research utilizes Kazhdan-Lusztig-Vogan theory to provide combinatorial counting formulas for special unipotent representations of classical groups. The authors also reduce the problem of constructing special unipotent representations to the case of "good parity orbits." This work is the first in a series of two papers on the classification of special unipotent representations of classical groups. Based on the counting formula established here, the second paper (preprint: https://arxiv.org/abs/1712.05552) shows that iterated theta lifting constructs all unipotent representations for good parity orbits and all unipotent representations of classical groups are unitary, i.e., settling Arthur-Barbasch-Vogan's conjecture for classical groups. The rest cases of Arthur-Barbasch-Vogan's conjecture are for the exceptional groups, which can be solved by the atlas of Lie group software system (http://liegroups.org/) in principle.

This article can be accesssed at https://doi.org/10.4171/jems/1609