Dirac series of connected complex simple Lie groups: Dong's Conjecture B on the classification of unitary representations with non-zero Dirac cohomology

For a connected complex simple Lie group $G$ viewed as a real reductive group, the Dirac series $\widehat{G}^{\mathrm{d}}$ is the set of irreducible unitary representations with non-zero Dirac cohomology. Dong's Conjecture B (2019) predicts that $\widehat{G}^{\mathrm{d}}$ consists exactly of the unitary representations $J(\lambda,-s\lambda)$, where $s$ is an involution in the Weyl group and $\lambda$ is a weight with $2\lambda$ dominant integral and regular, $\lambda+s\lambda$ integral, and $\lambda-s\lambda$ a non-negative integer combination of simple roots. At the source time the conjecture was verified for low-rank classical types, $G_2$, $F_4$ and $E_6$, leaving the remaining exceptional types and high ranks open. Later literature settled the classical groups in all ranks by a conceptual proof (2022) and complex $E_7$ (2022), while the complex $E_8$ classification (preprint, 2023/2026) is conditional on the unproven unitarity of two spherical unipotent representations. The surviving problem is to close Conjecture B unconditionally, or to find a counterexample.

Background

A connected complex simple Lie group $G$ can be regarded as a real reductive group: its Lie algebra $\mathfrak{g}_0$ has a Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0\oplus\mathfrak{p}_0$, where $K=G^{\theta}$ is a maximal compact subgroup for the Cartan involution $\theta$, and representations of $G$ are studied through Harish-Chandra modules — actions of the complexified Lie algebra $\mathfrak{g}$ compatible with the action of $K$. By Zhelobenko's classification of the irreducible Harish-Chandra modules of a complex group, every such module is a subquotient of a principal series and is written $J(\lambda_L,\lambda_R)$, parametrized by a pair of weights.

Parthasarathy's Dirac operator $D\in U(\mathfrak{g})\otimes C(\mathfrak{p})$ (with $C(\mathfrak{p})$ the Clifford algebra of $\mathfrak{p}_0$) acts on $\pi\otimes S$, where $S$ is a spin module for $C(\mathfrak{p})$. Vogan introduced the Dirac cohomology of an irreducible $(\mathfrak{g},K)$-module $\pi$ as $H_D(\pi)=\ker D/(\ker D\cap\operatorname{im} D)$, a module for $\widetilde K$, the spin double cover of $K$. Vogan conjectured, and Huang and Pandžić proved (2002), that whenever $H_D(\pi)\neq 0$ the infinitesimal character of $\pi$ (the character through which the center of the universal enveloping algebra $U(\mathfrak{g})$ acts) is obtained from the highest weight of any $\widetilde K$-type occurring in $H_D(\pi)$ by adding half the sum of the compact positive roots. The set $\widehat{G}^{\mathrm{d}}$ of irreducible unitary $(\mathfrak{g},K)$-modules with non-zero Dirac cohomology — called the Dirac series — contains every unitary module with non-zero $(\mathfrak{g},K)$-cohomology (for instance the modules $A_{\mathfrak{q}}(\lambda)$, cohomologically induced from a $\theta$-stable parabolic subalgebra $\mathfrak{q}$, which are unitary whenever the inducing parameter lies in the good range — the range in which cohomological induction preserves unitarity) and every unitary module with strongly regular infinitesimal character, so classifying it is a recognized step toward the unitary dual $\widehat{G}$, which is unknown for the exceptional complex groups.

For complex $G$, Barbasch and Pandžić (2011) deduced from the Huang–Pandžić theorem that every member of $\widehat{G}^{\mathrm{d}}$ must be of the form $J(\lambda,-s\lambda)$, where $s$ is an involution (an element of order two) in the Weyl group $W$ of $G$, and $\lambda$ satisfies: (i) $2\lambda$ is a dominant integral and regular weight (regular meaning it lies on no root hyperplane); (ii) $\lambda+s\lambda$ is integral; (iii) $\lambda-s\lambda$ is a non-negative integer combination of simple roots. These necessary conditions delimit a finite-by-families candidate set. Barbasch and Pandžić also conjectured (their Conjecture 1.1) that every member of $\widehat{G}^{\mathrm{d}}$ with regular half-integral infinitesimal character is cohomologically induced from an essentially unipotent representation with non-zero Dirac cohomology, where an essentially unipotent representation means a unipotent representation tensored with a unitary character; unipotent representations are the conjectural building blocks of the unitary dual attached by Arthur and Barbasch–Vogan to nilpotent orbits.

A finiteness theorem of Ding and Dong structures $\widehat{G}^{\mathrm{d}}$ into a finite scattered part (representations not induced from proper Levi subgroups, detected by $s$ involving every simple reflection in any reduced expression) and finitely many strings — families cohomologically induced, in the good range, from scattered representations of Levi subgroups tensored with unitary characters; classification therefore reduces to the scattered representations of $G$ and its Levis. A $K$-type is called unitarily small (u-small) if its highest weight lies in a certain convex hull introduced by Salamanca-Riba and Vogan; empirically the spin-lowest $K$-types of scattered Dirac series are always u-small.

The conjecture has since been settled for the complex classical groups in all ranks by a conceptual proof (Barbasch–Dong–Wong, 2022) and for complex $E_7$ (Dong–Wong, 2022); the complex $E_8$ classification of Barbasch–Wong (preprint arXiv:2305.03254) covers $E_8$ but is conditional on the unproven unitarity of the spherical unipotent representations attached to the nilpotent orbits $4A_1$ and $3A_1$ (the spherical modules $J_{E_8}(\nu_8,\nu_8)$ with $2\nu_8=[1,1,1,1,1,1,1,1]$ and $[1,1,1,1,1,1,2,2]$, the brackets listing Dynkin coefficients on the $E_8$ weight lattice). As of 2026 this is the only unsettled case-by-case gap.

In the source paper, Dong classified $\widehat{E_6}^{\mathrm{d}}$ completely (33 scattered representations, all with u-small spin-lowest $K$-types, plus 213 strings) and, observing that every computation pointed the same way, posed:

Problem Statement

Conjecture B (Dong, 2019). Let $G$ be a connected complex simple Lie group. The Dirac series $\widehat{G}^{\mathrm{d}}$ consists exactly of the unitary representations $J(\lambda,-s\lambda)$, where $s$ is an involution in the Weyl group of $G$ and $\lambda$ is a weight such that:

• $2\lambda$ is dominant integral and regular;
• $\lambda+s\lambda$ is an integral weight;
• $\lambda-s\lambda$ is a non-negative integer combination of simple roots.

Prove or disprove this conjecture for every connected complex simple Lie group.

References

1. Chao-Ping Dong, Unitary representations with non-zero Dirac cohomology for complex E6, Forum Mathematicum 31 (2019), no. 1, 69–82. DOI: 10.1515/forum-2018-0132, https://doi.org/10.1515/forum-2018-0132. arXiv:1707.01380, https://arxiv.org/abs/1707.01380 (source paper)
2. Dan Barbasch, Pavle Pandžić, Dirac cohomology and unipotent representations of complex groups, Contemporary Mathematics 546, American Mathematical Society (2011), 1–22. DOI: 10.1090/conm/546/10782, https://doi.org/10.1090/conm/546/10782. arXiv:1007.1289, https://arxiv.org/abs/1007.1289
3. 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), no. 1, 185–202. DOI: 10.1090/S0894-0347-01-00383-6, https://doi.org/10.1090/S0894-0347-01-00383-6
4. Jian Ding, Chao-Ping Dong, Unitary representations with Dirac cohomology: a finiteness result for complex Lie groups, Forum Mathematicum 32 (2020), no. 4, 941–964. DOI: 10.1515/forum-2019-0295, https://doi.org/10.1515/forum-2019-0295. arXiv:1702.01876, https://arxiv.org/abs/1702.01876
5. Chao-Ping Dong, Kayue Daniel Wong, Scattered representations of SL(n, C), Pacific Journal of Mathematics 309 (2020), no. 2, 289–312. DOI: 10.2140/PJM.2020.309.289, https://doi.org/10.2140/PJM.2020.309.289. arXiv:1910.02737, https://arxiv.org/abs/1910.02737
6. Chao-Ping Dong, Kayue Daniel Wong, Scattered Representations of Complex Classical Lie Groups, International Mathematics Research Notices 2022 (2022), no. 14, 10431–10457 (first published online 2021). DOI: 10.1093/imrn/rnaa388, https://doi.org/10.1093/imrn/rnaa388. arXiv:2006.07806, https://arxiv.org/abs/2006.07806
7. Dan Barbasch, Chao-Ping Dong, Kayue Daniel Wong, Dirac series for complex classical Lie groups: A multiplicity-one theorem, Advances in Mathematics 403 (2022), 108370. DOI: 10.1016/j.aim.2022.108370, https://doi.org/10.1016/j.aim.2022.108370. arXiv:2010.01584, https://arxiv.org/abs/2010.01584
8. Chao-Ping Dong, Kayue Daniel Wong, Dirac series for complex E7, Forum Mathematicum 34 (2022), 1033–1049. DOI: 10.1515/forum-2021-0274, https://doi.org/10.1515/forum-2021-0274. arXiv:2110.00694, https://arxiv.org/abs/2110.00694
9. Dan Barbasch, Kayue Daniel Wong, Dirac series for complex E8, preprint (2023, revised April 2026). arXiv:2305.03254, https://arxiv.org/abs/2305.03254