In this paper, we complete the ADE-like classification of simple transitive 2-representations of Soergel bimodules in finite dihedral type, under the assumption of gradeability. In particular, we use bipartite graphs and zigzag algebras of ADE type to give an explicit construction of a graded (non-strict) version of all these 2-representations.

Moreover, we give simple combinatorial criteria for when two such 2-representations are equivalent and for when their Grothendieck groups give rise to isomorphic representations.

Finally, our construction also gives a large class of simple transitive 2-representations in infinite dihedral type for general bipartite graphs.

We establish a character multiplicity duality for a certain natural class of nonlinear (nonalgebraic) groups arising as two-fold covers of simply laced real reductive algebraic groups. This allows us to extend part of the formalism of the local Langlands conjecture to such groups.

Vogan has conjectured that the cohomologically induced modules A$_ q$(λ) in the weakly fair range exhaust all unitary representations of U(p, q) with certain kinds of real integral infinitesimal character. To prove a statement like this, it is essential to identify these modules among the set of all irreducible Harish-Chandra modules. Barbasch and Vogan have parametrized this latter set in terms of their annihilators and asymptotic supports (or, equivalently, associated varieties). In this paper, we identify the weakly fair A$_ q$(λ) in this parametrization by combining known results about their asymptotic supports together with an explicit computation of their annihilators. In particular, this determines all vanishing and coincidences among the A$_ q$(λ) in the weakly fair range, and gives the Langlands parameters of these modules.