Dirac series of $U(p,q)$ for general $p,q$

The Dirac series of a real reductive group is the set of irreducible unitary representations with nonzero Dirac cohomology. It has been determined completely for $U(n,2)$ by Wei–Du–Zhang–Zhao, building on the Wong–Zhang classification of the unitary dual of $U(n,2)$ and on Dong–Wong's computation of Dirac indices of weakly fair $A_{\mathfrak{q}}(\lambda)$-modules of $U(p,q)$. The open problem is to extend this classification to $U(p,q)$ for arbitrary $p$ and $q$: to list all fully supported scattered (FS-scattered) building-block representations, to show every remaining Dirac-series member is cohomologically induced in the weakly good range from lower-rank Dirac-series factors, and to decide whether Dirac cohomology is always detected by the Dirac index (no even/odd cancellation), as it is for $U(n,2)$.

Background

Let $G=U(p,q)$ be the unitary group of the Hermitian form of signature $(p,q)$ on $\mathbb{C}^{p+q}$, with maximal compact subgroup $K \cong U(p)\times U(q)$, and let $\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{p}_0$ be the Cartan decomposition of its Lie algebra. Representations of $G$ are studied through Harish-Chandra modules: a $(\mathfrak{g},K)$-module is a complex representation of the complexified Lie algebra $\mathfrak{g}$ that integrates to $K$ and has finite $K$-multiplicities; it is unitary if it carries a compatible positive-definite Hermitian form, and the set of irreducible unitary ones is the unitary dual $\widehat{G}$. Parthasarathy's Dirac operator $D$ on $G/K$, a square root of the Casimir, acts on the tensor product of any $(\mathfrak{g},K)$-module $\pi$ with the spin module $S$ (the irreducible Clifford-algebra module for $\mathfrak{p}$); Vogan's Dirac cohomology is $H_D(\pi)=\ker D/\operatorname{im} D$, a finite-dimensional $\widetilde{K}$-module ($\widetilde{K}$ is the spin double cover of $K$). The foundational theorem of Huang–Pandžić (proving a conjecture of Vogan) says that whenever $H_D(\pi)\neq 0$, the infinitesimal character of $\pi$ (the parameter classifying its annihilator on the center of the universal enveloping algebra) is conjugate to $\gamma+\rho_c$ for any highest weight $\gamma$ of a $\widetilde{K}$-type in $H_D(\pi)$, where $\rho_c$ (resp. $\rho$) is half the sum of the compact (resp. all) positive roots. The Dirac series $\widehat{G}^{\,d}$ — the term coined by Huang — is the set of irreducible unitary $(\mathfrak{g},K)$-modules with nonzero Dirac cohomology; it contains the discrete series and other fundamental unitary families, and its classification is a recognized open problem.

When $G$ and $K$ have equal rank, as for $U(p,q)$, the Dirac index $DI(\pi)=H_D(\pi)^{\text{even}}-H_D(\pi)^{\text{odd}}$ (the formal difference of the $\widetilde{K}$-types of even and odd Clifford degree) is well defined and behaves well under coherent continuation, but cancellations between the even and odd parts can make $DI$ vanish while $H_D\neq 0$: this happens for $E_{7(7)}$ (Ding–Dong–Wei), while Dong–Wong verified computationally that no such cancellation occurs on unitary groups $U(p,q)$ of small rank.

The classification of Dirac series is organized by cohomological induction: for a $\theta$-stable parabolic subalgebra $\mathfrak{q}=\mathfrak{l}\oplus\mathfrak{u}$ ($\theta$ the Cartan involution, $\mathfrak{l}$ the Levi), and a one-dimensional $\mathfrak{l}$-module of weight $\lambda$, the derived functor module $A_{\mathfrak{q}}(\lambda)$ is the basic building block of the theory; the induction is in the good (resp. weakly good, weakly fair) range when $\lambda$ is shifted far enough into dominance, these ranges being successively larger and the induced modules being better behaved the larger the range. Dong's finiteness theorem shows that for a real reductive group of finite center, all but finitely many Dirac-series members (the string representations) are cohomologically induced in the good range from Dirac-series members of proper $\theta$-stable Levi subgroups; the remaining exceptions are the scattered representations, and those that are not cohomologically induced from any proper $\theta$-stable parabolic in the weakly good range are fully supported, called FS-scattered. The FS-scattered representations are the building blocks of the whole Dirac series, so classifying them is the heart of the problem.

For $G=U(p,q)$ — the archetypal group of Hermitian symmetric type, for which Dirac cohomology of unitary modules can be identified with $\mathfrak{p}^+$-cohomology up to a twist — the state of the art is: (i) Dong–Wong computed the Dirac index of all weakly fair $A_{\mathfrak{q}}(\lambda)$-modules of $U(p,q)$ and showed it is nonzero exactly when a combinatorial H.P.-condition on the infinitesimal character holds; they found counterexamples $\Sigma'_p$ (Siegel-parabolic inductions on $U(p,p)$) to Vogan's conjecture that weakly fair $A_{\mathfrak{q}}(\lambda)$ exhaust the unitary modules with infinitesimal character a weight-translate of $\rho$, but these have zero Dirac cohomology, and they posed Conjecture 1.3: every Dirac-series member of $U(p,q)$ is a weakly fair $A_{\mathfrak{q}}(\lambda)$-module satisfying the H.P.-condition, verified up to $U(5,5)$ by computation with the atlas software (a software system for exact computations on real reductive groups and their representations); (ii) Wong proved the Salamanca-Riba–Vogan and fundamental-parallelepiped conjectures for $U(p,q)$, giving necessary conditions on infinitesimal characters for unitarity, and with Zhang classified the full unitary dual of $U(n,2)$; (iii) Wei–Du–Zhang–Zhao then determined the complete Dirac series of $U(n,2)$: Theorem 4.11 lists all FS-scattered Dirac-series representations, Theorem 4.12 realizes every string representation as cohomological induction in the weakly good range from FS-scattered factors of smaller $U(p,2)$'s and finite-dimensional modules, the spin lowest $K$-types (those capable of contributing to Dirac cohomology) are multiplicity free, and Dirac cohomology is consistent with the Dirac index, i.e. no even/odd cancellation occurs. Their proof partitions a $\theta$-stable combinatorial datum (a block-and-arrow diagram encoding the lowest $K$-type and infinitesimal character) into fundamental pieces, and they note explicitly that this partition trick breaks down for general $U(p,q)$, where the fundamental data are more complicated. The full unitary dual of general $U(p,q)$, unlike $U(n,2)$, is not known, which blocks the route of filtering a known unitary dual by Dirac cohomology.

Problem Statement

Determine the Dirac series $\widehat{U(p,q)}^{\,d}$ of $U(p,q)$ for arbitrary $p$ and $q$: give an explicit classification of all irreducible unitary $(\mathfrak{g},K)$-modules of $U(p,q)$ with nonzero Dirac cohomology, in the same structural form established for $U(n,2)$, namely: (i) a complete, explicitly parametrized list of the FS-scattered (fully supported scattered) Dirac-series representations, in terms of their $\theta$-stable combinatorial data or $A_{\mathfrak{q}}(\lambda)$-chains; (ii) a theorem realizing every remaining (string) Dirac-series member as cohomological induction in the weakly good range from Dirac-series factors of proper $\theta$-stable Levi subgroups, identifying which lower-rank groups and which factors occur; and (iii) for each member, the computation of its Dirac cohomology, including whether Dirac cohomology is detected by the Dirac index — that is, whether any even/odd cancellation can occur, as it does for $E_{7(7)}$ but provably does not for $U(n,2)$. The answer sought is the classification (i)–(iii) above.

References

1. Lin Wei, Chengyu Du, Hongfeng Zhang, Xuanchen Zhao, "Dirac series of U(n,2)", Journal of Algebra (2025), DOI 10.1016/j.jalgebra.2025.09.043; arXiv:2408.10817, https://arxiv.org/abs/2408.10817
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8. Yi-Hao Ding, Chao-Ping Dong, Lin Wei, "Dirac series of $E_{7(7)}$", Israel Journal of Mathematics 265 (2024), 347–378, DOI 10.1007/s11856-024-2658-1, https://doi.org/10.1007/s11856-024-2658-1; arXiv:2210.15833, https://arxiv.org/abs/2210.15833