On the non-vanishing condition for A𝔮(λ) of U(p,q) in the mediocre range

The modules [Formula: see text] of [Formula: see text] can be parameterized by their annihilators and asymptotic supports, both of which can be identified using Young tableaux. Trapa developed an algorithm for determining the tableaux of the modules [Formula: see text] in the mediocre range, along with an equivalent condition to determine non-vanishing. The condition involves a combinatorial concept called the overlap, which is not straightforward to compute. In this paper, we establish a formula for the overlap and simplify the condition for ease of use. We then apply it to [Formula: see text]-types and the Dirac index of [Formula: see text].