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

Chengyu Du

TL;DR

The paper investigates when cohomologically induced modules $A_\mathfrak{q}(\lambda)$ for $U(p,q)$ fail to vanish in the mediocre range, building on Trapa's tableaux framework. It derives an explicit formula for the overlap between adjacent skew columns, $\mathrm{overlap}(S_i,S_{i+1})=\min\{p_i,q_{i+1}\}+\min\{p_{i+1},q_i\}$, and uses this to simplify the nonvanishing criterion in the nice range to $\lambda_{i+1}-\lambda_i \le \min\{p_i,q_{i+1}\}+\min\{p_{i+1},q_i\}$. The authors extend the analysis to Dirac cohomology by introducing a strengthened HP-condition, establishing a practical nonvanishing criterion for $\mathrm{DI}(A_\mathfrak{q}(\lambda))$ in the nice range, and showing that the Dirac index remains nonzero in many cases, with concrete implications for $K$-types. Overall, the work provides actionable, computation-friendly criteria for nonvanishing and Dirac-index questions in the mediocre-to-nice range for $U(p,q)$. These results facilitate applications to representation-theoretic invariants and may aid future progress on Vogan–Trapa-type conjectures in broader settings.

Abstract

The modules $A_\mathfrak{q}(λ)$ 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 $A_\mathfrak{q}(λ)$ 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 $A_\mathfrak{q}(λ)$.

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

TL;DR

The paper investigates when cohomologically induced modules for fail to vanish in the mediocre range, building on Trapa's tableaux framework. It derives an explicit formula for the overlap between adjacent skew columns, , and uses this to simplify the nonvanishing criterion in the nice range to . The authors extend the analysis to Dirac cohomology by introducing a strengthened HP-condition, establishing a practical nonvanishing criterion for in the nice range, and showing that the Dirac index remains nonzero in many cases, with concrete implications for -types. Overall, the work provides actionable, computation-friendly criteria for nonvanishing and Dirac-index questions in the mediocre-to-nice range for . These results facilitate applications to representation-theoretic invariants and may aid future progress on Vogan–Trapa-type conjectures in broader settings.

Abstract

The modules of 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 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 -types and the Dirac index of .
Paper Structure (13 sections, 19 theorems, 80 equations)

This paper contains 13 sections, 19 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $K$-conjugacy classes of $\theta$-stable parabolic subalgebras of $\mathfrak{g}$
  4. The module $A_\mathfrak{q}(\lambda)$ and ranges of positivity
  5. Young Tableaux and $A_q(\lambda)$ modules of $U(p,q)$
  6. Dirac cohomology and Dirac index
  7. A formula for the overlap
  8. New notations for tableaux
  9. Proof of the overlap formula
  10. Non-vanishing criterion for $A_\mathfrak{q}(\lambda)$ in the nice range
  11. Application to Dirac index
  12. The strengthened H.P.-condition
  13. Non-vanishing criterion for $\mathop{\mathrm{DI}}\nolimits(A_\mathfrak{q}(\lambda))$ for $\lambda$ in the nice range

Key Result

Lemma 2.2

Trapa2001 Suppose $G=U(p,q)$. Let $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ be attached to $\{(p_1,q_1),\cdots,(p_r,q_r)\}$, and let $\mathbb{C}_\lambda$ be a 1-dimensional $(\mathfrak{l},L\cap K)$-module. Write where $n_i=p_i+q_i$. Then the definition of (weakly) good, (weakly) fair, and mediocre can be equivalently rephrased in the following way.

Theorems & Definitions (44)

  • Conjecture 1: Vogan
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Lemma 2.8
  • Example 2.9
  • ...and 34 more