Special unipotent representations and the coadjoint orbit method

Abstract

For any linear real reductive group or metapletic group, this article gives a geometric classification of special unipotent representations in the weak Arthur/Adams-Barbasch-Vogan (ABV) packets attached to quasi-distinguished nilpotent orbits in the Langlands or metaplectic dual Lie algebra in terms of their associated cycles. We provide a uniform construction of the Harish-Chandra modules of these representations via deformation quantization of admissible vector bundles over certain Lagrangian subvarieties of the affinizations of the universal covers of special nilpotent orbits in question, which aligns with the coadjoint orbit method philosophy of Kirillov, Kostant, and Vogan. As consequences, all such Harish-Chandra modules have irreducible associated cycles, and are unitarizable by results from the theory of mixed Hodge modules. Conjecturally, the unitarity of all ABV packets can be reduced to the case when the dual orbits are distinguished. Our approach highlights the application of Lusztig's conjecture on the geometry of special pieces, which has been proven for classical Lie algebras by Kraft and Procesi, and for all cases by Juteau, Levy, Sommers and the author.