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

TL;DR

This paper simplifies the non-vanishing condition for modules of $U(p,q)$ in the mediocre range by providing a formula for the overlap, facilitating easier analysis of their properties and applications.

Contribution

We derive a formula for the overlap in the non-vanishing condition, simplifying the combinatorial criteria for modules $A_rak{q}(\lambda)$ of $U(p,q)$ in the mediocre range.

Findings

Established a formula for the overlap in the non-vanishing condition.

Simplified the combinatorial criterion for module analysis.

Applied the results to $K$-types and Dirac index computations.

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 $A_\mathfrak{q}(\lambda)$.