Annihilators and Associated Varieties of A$ q $(λ) Modules for U(p, q)

Abstract

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.