On the nonvanishing condition for $A_{\mathfrak{q}}(\lambda )$ of $U(p,q)$ in the mediocre range

Abstract

The modules $A_{\mathfrak{q}}(\lambda )$ of $U(p,q)$ 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 $A_{\mathfrak{q}}(\lambda )$ 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 $K$-types and the Dirac index of $A_{\mathfrak{q}}(\lambda )$.