Are all Dirac series members of $U(p,q)$ weakly fair $A_{\mathfrak{q}}(\lambda)$-modules? (Conjecture 1.3 of Dong–Wong)

For the indefinite unitary group $G=U(p,q)$, the Dirac series is the set of irreducible unitary $(\mathfrak{g},K)$-modules with nonzero Dirac cohomology. Dong and Wong (Math. Z. 2021) computed the Dirac index of all weakly fair $A_{\mathfrak{q}}(\lambda)$-modules of $U(p,q)$ and verified by computer up to $U(5,5)$ that every Dirac series member arises this way, leading to their Conjecture 1.3: every irreducible unitary representation of $U(p,q)$ with nonzero Dirac cohomology must be a weakly fair $A_{\mathfrak{q}}(\lambda)$-module satisfying the combinatorial restrictions (28) on its infinitesimal character. The conjecture remains open for general ranks; subsequent work confirmed it for $U(n,2)$, refuted a companion vanishing conjecture (Remark 5.8), and proved adjacent unitary-dual conjectures, but no proof or counterexample is known for general $U(p,q)$.

Background

Let $G=U(p,q)$ be the group of complex linear transformations of $\mathbb{C}^{p+q}$ preserving a Hermitian form of signature $(p,q)$ — a real reductive Lie group that is indefinite (non-compact) for $p,q\geq 1$. Its unitary representation theory is encoded in the unitary dual, the set of irreducible unitary $(\mathfrak{g},K)$-modules (Harish-Chandra modules: representations of the complexified Lie algebra $\mathfrak{g}$ compatible with the maximal compact subgroup $K\cong U(p)\times U(q)$), and classifying it is a major open problem. Vogan introduced in 1997 the Dirac cohomology $H_D(\pi)$ of such a module $\pi$: one forms the cubic Dirac operator $D$, a square root of the Casimir acting on $\pi\otimes S$, where $S$ is the spin module for the complexified $-1$-eigenspace of the Cartan involution, and $H_D(\pi)=\ker D/\ker D^2$ graded by the action of $K$. Huang and Pandžić proved Vogan's conjecture that whenever $H_D(\pi)\neq 0$, it reveals the infinitesimal character of $\pi$ (the $W(\mathfrak{g},\mathfrak{h})$-orbit of a weight $\Lambda$ that determines much of the representation). This made the Dirac series — the irreducible unitary modules with $H_D(\pi)\neq 0$ — a structured, tractable subproblem of the unitary dual, pursued group by group. A rich source of unitary modules is cohomological induction: for a $\theta$-stable parabolic subalgebra $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ ($\theta$ the Cartan involution) and a one-dimensional unitary $(\mathfrak{l},L\cap K)$-module $\lambda$, the derived functor module $A_{\mathfrak{q}}(\lambda)$ is irreducible or zero when $\lambda$ lies in the weakly fair range (a range of parameters, between the good range and the fair range, where unitarity is preserved but irreducibility is delicate). For $U(p,q)$, Vogan conjectured that weakly fair $A_{\mathfrak{q}}(\lambda)$-modules exhaust the unitary modules whose infinitesimal character is a weight translate of $\rho$ (half the sum of positive roots); Dong and Wong found counterexamples to this on $U(p,p)$, but all counterexamples have zero Dirac cohomology. Their main theorem (Theorem 5.11) computes the Dirac index $DI$ (the formal difference of the even and odd $K$-type graded parts of Dirac cohomology) of every weakly fair $A_{\mathfrak{q}}(\lambda)$: it never vanishes when the infinitesimal character satisfies condition (28) — in the standard coordinates $\nu=\lambda+\rho$ attached to the $A_{\mathfrak{q}}$-chain of $\mathfrak{q}$, no entry of $\nu$ appears more than twice and at most $\min\{p,q\}$ distinct entries appear twice — and a certain linear system (30) in nonnegative integers is solvable. Since Lemma 5.2 shows (28) is necessary for nonzero Dirac cohomology, proving that every Dirac series member is a weakly fair $A_{\mathfrak{q}}(\lambda)$-module satisfying (28) would complete the classification of the Dirac series of $U(p,q)$ via their theorem. Computer verification with the atlas software found no gap up to $U(5,5)$.

Problem Statement

Prove or disprove Conjecture 1.3 of Dong–Wong ('On the Dirac series of $U(p,q)$', Math. Z. 298 (2021), 839–859): for the group $G=U(p,q)$ with arbitrary $p,q\geq 1$, every irreducible unitary $(\mathfrak{g},K)$-module $\pi$ with nonzero Dirac cohomology $H_D(\pi)\neq 0$ (i.e., every member of the Dirac series of $U(p,q)$) is a weakly fair $A_{\mathfrak{q}}(\lambda)$-module whose infinitesimal character satisfies condition (28) of Lemma 5.2 of that paper: writing $\nu=\lambda+\rho$ in the coordinates attached to the $A_{\mathfrak{q}}$-chain of the $\theta$-stable parabolic $\mathfrak{q}$, no entry of $\nu$ appears more than twice, and at most $\min\{p,q\}$ distinct entries appear twice. An affirmative answer, combined with Theorem 5.11 of the same paper, yields the complete classification of the Dirac series of $U(p,q)$ in all ranks; an explicit irreducible unitary $U(p,q)$-module with nonzero Dirac cohomology that is not a weakly fair $A_{\mathfrak{q}}(\lambda)$-module satisfying (28) would refute it.

References

1. Chao-Ping Dong, Kayue Daniel Wong, 'On the Dirac series of U(p,q)', Mathematische Zeitschrift 298 (2021), 839–859. DOI: 10.1007/s00209-020-02641-2; arXiv:2003.07165. https://arxiv.org/abs/2003.07165
2. Lin Wei, Chengyu Du, Hongfeng Zhang, Xuanchen Zhao, 'Dirac series of U(n,2)', Journal of Algebra 689 (2026), 551–577. DOI: 10.1016/j.jalgebra.2025.09.043; arXiv:2408.10817. https://arxiv.org/abs/2408.10817
3. Chengyu Du, 'On the non-vanishing condition for $A_{\mathfrak{q}}(\lambda)$ of $U(p,q)$ in the mediocre range', International Journal of Mathematics 36, no. 13 (2025). DOI: 10.1142/S0129167X2550051X; arXiv:2405.03216. https://arxiv.org/abs/2405.03216
4. Kayue Daniel Wong, 'On some conjectures of the unitary dual of U(p,q)', Advances in Mathematics 442 (2024), 109584. DOI: 10.1016/j.aim.2024.109584; arXiv:2210.08684. https://arxiv.org/abs/2210.08684
5. Kayue Daniel Wong, Hongfeng Zhang, 'The Unitary Dual of U(n,2)', International Mathematics Research Notices 2024, no. 14, 10678–10707. DOI: 10.1093/imrn/rnae097; arXiv:2401.15660. https://arxiv.org/abs/2401.15660
6. Jing-Song Huang, Pavle Pandžić, 'Dirac cohomology, unitary representations and a proof of a conjecture of Vogan', Journal of the American Mathematical Society 15 (2002), 185–202. DOI: 10.1090/S0894-0347-01-00383-6. https://doi.org/10.1090/S0894-0347-01-00383-6