Dirac series of real reductive Lie groups: classifying unitary modules with nonzero Dirac cohomology beyond the tempered case

Dong and Huang (Amer. J. Math. 2015) proved that Dirac cohomology commutes with cohomological induction in the weakly good range and, as an application, classified the tempered irreducible unitary representations with nonzero Dirac cohomology. The general problem they left open is to determine, for an arbitrary real reductive Lie group, all irreducible unitary $(\mathfrak g,K)$-modules with nonzero Dirac cohomology — the set now called the Dirac series of the group — together with the companion question of how far the weakly good range hypothesis can be weakened in their induction identity. A decade of follow-up work has produced finiteness theorems (every Dirac series member is good-range cohomologically induced from a Levi factor with nonzero Dirac cohomology, up to finitely many scattered exceptions) and complete group-by-group classifications for many classical and exceptional groups, but no uniform classification; even the $U(p,q)$ case is only conjectural, and the induction formula beyond the weakly good range remains unsolved. The problem is open.

Background

Let $G$ be a real reductive Lie group in the Harish-Chandra class (a broad class that includes the groups of real points of connected complex reductive algebraic groups and allows $G$ to be disconnected). Fix a Cartan involution $\theta$ with fixed-point subgroup $K$, a maximal compact subgroup, and write the Cartan decomposition of the Lie algebra as $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$. The algebraic avatar of a representation of $G$ is a $(\mathfrak g,K)$-module: a complex $\mathfrak g$-module carrying a compatible, locally finite action of $K$. Parthasarathy (1972) introduced the algebraic Dirac operator $D=\sum_i Z_i\otimes Z_i\in U(\mathfrak g)\otimes C(\mathfrak p)$, where $U(\mathfrak g)$ is the universal enveloping algebra, $C(\mathfrak p)$ is the Clifford algebra of $\mathfrak p$, and the $Z_i$ run over an orthonormal basis of $\mathfrak p_0$. If $S$ is a spin module for $C(\mathfrak p)$, then $D$ acts on $X\otimes S$ for any $(\mathfrak g,K)$-module $X$, and $S$ is naturally a module for the pin double cover $\widetilde K$ of $K$. The Dirac cohomology of $X$, introduced by Vogan in the late 1990s, is the $\widetilde K$-module

$H_D(X)=\operatorname{Ker} D/(\operatorname{Im} D\cap\operatorname{Ker} D).$

For unitary $X$ the operator $D$ is self-adjoint, so $H_D(X)=\operatorname{Ker} D$. Recall also that the infinitesimal character of $X$ is the character by which the center $Z(\mathfrak g)$ of the enveloping algebra acts; by Harish-Chandra it is parametrized, up to Weyl group orbit, by an element of a Cartan subalgebra dual.

Two facts make Dirac cohomology a powerful invariant of unitary representations. First, Vogan's conjecture — proved by Huang and Pandžić (2002) for connected groups and extended to possibly disconnected groups by Dong and Huang (2015) — states that if $H_D(X)\neq 0$ and $X$ has infinitesimal character $\lambda$, then some irreducible $\widetilde K$-module $E_\mu$ occurring in $H_D(X)$ recovers $\lambda$ as the Weyl orbit of $\mu+2\rho_c$, where $\rho_c$ denotes half the sum of the positive compact roots: nonzero Dirac cohomology pins down the infinitesimal character. Second, Parthasarathy's Dirac inequality gives a lower bound on the eigenvalues of $D^2$ depending only on the infinitesimal character, with equality on some $\widetilde K$-type exactly when $H_D(X)\neq 0$; consequently the unitary modules with nonzero Dirac cohomology are precisely the extreme points of the unitary dual $\widehat G$ (the set of all irreducible unitary representations, itself one of the great open classification problems of the field) relative to this inequality. The subset $\widehat G^{\mathrm d}$ of irreducible unitary $(\mathfrak g,K)$-modules with nonzero Dirac cohomology is called the Dirac series of $G$.

The main machine for producing unitary representations is cohomological induction (Knapp–Vogan). Given a $\theta$-stable parabolic subalgebra $\mathfrak q=\mathfrak l\oplus\mathfrak u$ with Levi factor $\mathfrak l$ and nilradical $\mathfrak u$, and $L=N_G(\mathfrak q)$, the cohomological induction functors $\mathcal L_j$ and $\mathcal R^j$ lift $(\mathfrak l,L\cap K)$-modules to $(\mathfrak g,K)$-modules; put $d=\dim(\mathfrak u\cap\mathfrak k)$. If $Z$ is an $(\mathfrak l,L\cap K)$-module with infinitesimal character $\lambda$, then $Z$ is called weakly good (relative to $\mathfrak q$) when $\operatorname{Re}\langle\lambda+\rho(\mathfrak u),\alpha\rangle\ge 0$ for every root $\alpha\in\Delta(\mathfrak u,\mathfrak h)$, where $\rho(\mathfrak u)$ is half the sum of the roots of $\mathfrak u$; the good range (the corresponding strict inequalities) guarantees that $\mathcal L_d(Z)$ is irreducible and unitary whenever $Z$ is, with infinitesimal character $\lambda+\rho(\mathfrak u)$. When $Z$ is one-dimensional, $\mathcal L_d(Z)$ is the classical $A_{\mathfrak q}(\lambda)$ module. Huang, Kang and Pandžić (2009) computed Dirac cohomology for good-range $A_{\mathfrak q}(\lambda)$ modules. Dong and Huang (2015) then proved (their Theorem B) that Dirac cohomology commutes with cohomological induction in the weakly good range: for an irreducible unitary $(\mathfrak l,L\cap K)$-module $Z$ with weakly good infinitesimal character,

$H_D(\mathcal L_d(Z))\cong \mathcal L_d^{\widetilde K}\bigl(H_D(Z)\otimes\mathbb C_{-\rho(\mathfrak p\cap\mathfrak u)}\bigr),$

where the right-hand side is cohomological induction performed at the level of $\widetilde K$-modules, and $\mathbb C_{-\rho(\mathfrak p\cap\mathfrak u)}$ is the corresponding one-dimensional twist. As an immediate application they classified the tempered irreducible unitary modules (tempered means the matrix coefficients lie in $L^{2+\varepsilon}$ for every $\varepsilon>0$; tempered representations are the building blocks from which general unitary representations are induced) having nonzero Dirac cohomology. What happens beyond the tempered case, and outside the weakly good range, is the open problem recorded here.

The subsequent literature added several structural notions relevant to this problem. Finiteness theorems of Ding–Dong (complex groups) and Dong (real groups) show that $\widehat G^{\mathrm d}$ consists of strings of representations obtained by good-range cohomological induction from Levi factors — one-parameter families arising when a common Levi datum is tensored with a varying one-dimensional character — plus finitely many exceptional scattered representations, meaning members not obtained by such induction. The Dirac index is the Euler characteristic $H_D^+-H_D^-$ of the $\mathbb Z/2$-graded Dirac cohomology; it is computable but strictly weaker, since the even and odd parts can cancel (first observed for $F_{4(4)}$ in 2018). A $K$-type is called unitarily small (u-small, in the sense of Salamanca-Riba and Vogan) if its highest weight satisfies a certain convexity condition relative to the spin norm $\lVert\mu+\rho_c\rVert$ along Vogan pencils (the lines of $K$-type highest weights generated by Weyl-group moves); the Salamanca-Riba–Vogan program conjectures that the whole unitary dual is controlled by modules containing u-small $K$-types.

Problem Statement

Let $G$ be a real reductive Lie group in the Harish-Chandra class (possibly disconnected), and let $\widehat G^{\mathrm d}$ be the set of equivalence classes of irreducible unitary $(\mathfrak g,K)$-modules with nonzero Dirac cohomology (the Dirac series of $G$). The tempered members of $\widehat G^{\mathrm d}$ are classified by Dong–Huang, as an immediate application of their weakly good induction identity; determine the rest. The problem has two linked components, both required, sharing one objective — extending the computation of Dirac cohomology from the tempered, weakly good range fragment of the unitary dual to all of it.

(1) Classification. Classify $\widehat G^{\mathrm d}$: determine which irreducible unitary $(\mathfrak g,K)$-modules have nonzero Dirac cohomology, covering non-tempered representations. By the known finiteness theorems, every member is either obtained by good-range cohomological induction from a Levi factor with nonzero Dirac cohomology or belongs to the finite scattered part, so the classification reduces, for each group, to a finite calculation pinning down the scattered part. The reduction is a known consequence of the finiteness theorems and does not by itself resolve the component; the classification sought is of $\widehat G^{\mathrm d}$ for an arbitrary real reductive $G$. Dong's Conjecture B is one instance of that finite calculation and settles only a part of component (1): any $K$-type of a scattered member that contributes to its Dirac cohomology is unitarily small (in the sense of Salamanca-Riba–Vogan).

(2) Dirac cohomology under cohomological induction beyond the weakly good range. Determine to what extent the weakly good hypothesis can be weakened in the Dong–Huang identity

$H_D(\mathcal L_d(Z))\cong \mathcal L_d^{\widetilde K}\bigl(H_D(Z)\otimes\mathbb C_{-\rho(\mathfrak p\cap\mathfrak u)}\bigr):$

for which inducing modules $Z$ outside the weakly good range (for example, in the weakly fair range, or with nonunitary $Z$) does Dirac cohomology still commute with cohomological induction, and what correction terms or case distinctions are required where it does not?

Component (2) is a question of independent interest that bears on component (1), since the classification is propagated through cohomological induction by such an identity, which is currently available only in the weakly good range.

References

1. Chao-Ping Dong, Jing-Song Huang, Dirac cohomology of cohomologically induced modules for reductive Lie groups, American Journal of Mathematics 137 (2015), no. 1, 37–60. DOI: 10.1353/ajm.2015.0007. https://doi.org/10.1353/ajm.2015.0007
2. 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
3. R. Parthasarathy, Dirac Operator and the Discrete Series, Annals of Mathematics (2) 96 (1972), no. 1, 1–17. DOI: 10.2307/1970892. https://doi.org/10.2307/1970892
4. Anthony W. Knapp, David A. Vogan, Cohomological Induction and Unitary Representations (PMS-45), Princeton University Press, 1995. DOI: 10.1515/9781400883936. https://doi.org/10.1515/9781400883936
5. Susana A. Salamanca-Riba, David A. Vogan, On the classification of unitary representations of reductive Lie groups, Annals of Mathematics (2) 148 (1998), no. 3, 1067–1133. DOI: 10.2307/121036. https://doi.org/10.2307/121036
6. Jing-Song Huang, Yi-Fang Kang, Pavle Pandžić, Dirac cohomology of some Harish-Chandra modules, Transformation Groups 14 (2009), no. 1, 163–173. DOI: 10.1007/s00031-008-9036-7. https://doi.org/10.1007/s00031-008-9036-7
7. Dan Barbasch, Pavle Pandžić, Dirac cohomology and unipotent representations of complex groups, preprint (2010). arXiv:1007.1289. https://arxiv.org/abs/1007.1289
8. Chao-Ping Dong, On the Dirac cohomology of complex Lie group representations, Transformation Groups 18 (2013), no. 1, 61–79. DOI: 10.1007/s00031-013-9206-0. https://doi.org/10.1007/s00031-013-9206-0
9. Jing-Song Huang, Pavle Pandžić, David Vogan, On classifying unitary modules by their Dirac cohomology, Science China Mathematics 60 (2017), no. 9, 1937–1962. DOI: 10.1007/s11425-017-9097-8. https://doi.org/10.1007/s11425-017-9097-8
10. 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 (preprint arXiv:1702.01876)
11. Chao-Ping Dong, Unitary Representations with Dirac Cohomology: Finiteness in the Real Case, International Mathematics Research Notices 2020 (2020), no. 24, 10277–10316. DOI: 10.1093/imrn/rny293. https://doi.org/10.1093/imrn/rny293 (preprint arXiv:1708.00383)
12. Chao-Ping Dong, Unitary representations with non-zero Dirac cohomology for complex $E_6$, Forum Mathematicum 31 (2019), no. 1, 69–82. DOI: 10.1515/forum-2018-0132. https://doi.org/10.1515/forum-2018-0132 (preprint arXiv:1707.01380)
13. Dan Barbasch, Chao-Ping Dong, Kayue Daniel Wong, Dirac series for complex classical Lie groups: a multiplicity-one theorem, Advances in Mathematics 403 (2022), Article 108370. DOI: 10.1016/j.aim.2022.108370. https://doi.org/10.1016/j.aim.2022.108370 (preprint arXiv:2010.01584)
14. Chao-Ping Dong, Kayue Daniel Wong, Dirac series of $GL(n,\mathbb R)$, International Mathematics Research Notices 2023 (2023), no. 12, 10702–10735. DOI: 10.1093/imrn/rnac150. https://doi.org/10.1093/imrn/rnac150 (preprint arXiv:2007.00913)
15. Jian Ding, Chao-Ping Dong, Liang Yang, Dirac series for some real exceptional Lie groups, Journal of Algebra 559 (2020), 379–407. DOI: 10.1016/j.jalgebra.2020.05.001. https://doi.org/10.1016/j.jalgebra.2020.05.001 (preprint arXiv:1809.06034)
16. Lin-Gen Ding, Chao-Ping Dong, Haian He, Dirac series for $E_{6(-14)}$, Journal of Algebra 590 (2022), 168–201. DOI: 10.1016/j.jalgebra.2021.10.002. https://doi.org/10.1016/j.jalgebra.2021.10.002 (preprint arXiv:1903.06861)
17. Chao-Ping Dong, Kayue Daniel Wong, On the Dirac series of $U(p,q)$, Mathematische Zeitschrift 298 (2021), no. 1–2, 839–859. DOI: 10.1007/s00209-020-02641-2. https://doi.org/10.1007/s00209-020-02641-2 (preprint arXiv:2003.07165)
18. Chao-Ping Dong, A Non-vanishing Criterion for Dirac Cohomology, Transformation Groups 29 (2024), no. 3, 935–958. DOI: 10.1007/s00031-022-09758-0. https://doi.org/10.1007/s00031-022-09758-0 (preprint arXiv:2107.09220)
19. Yi-Hao Ding, Chao-Ping Dong, Dirac series of $E_{7(-25)}$, Journal of Algebra 614 (2023), 670–694. DOI: 10.1016/j.jalgebra.2022.10.009. https://doi.org/10.1016/j.jalgebra.2022.10.009 (preprint arXiv:2204.07902)
20. Chao-Ping Dong, Kayue Daniel Wong, Dirac series for complex $E_7$, Forum Mathematicum 34 (2022), no. 4, 1033–1049. DOI: 10.1515/forum-2021-0274. https://doi.org/10.1515/forum-2021-0274 (preprint arXiv:2110.00694)
21. Yi-Hao Ding, Chao-Ping Dong, Ping-Yuan Li, Dirac series of $E_{7(-5)}$, Indagationes Mathematicae. New Series 34 (2023), no. 1, 54–77. DOI: 10.1016/j.indag.2022.09.002. https://doi.org/10.1016/j.indag.2022.09.002 (preprint arXiv:2205.11999)
22. Yi-Hao Ding, Chao-Ping Dong, Lin Wei, Dirac series of $E_{7(7)}$, Israel Journal of Mathematics 265 (2025), no. 1, 347–378. DOI: 10.1007/s11856-024-2658-1. https://doi.org/10.1007/s11856-024-2658-1 (preprint arXiv:2210.15833)
23. Yi-Hao Ding, Chao-Ping Dong, Chengyu Du, Yong-Zhi Luan, Liang Yang, Dirac series of $E_{8(-24)}$, International Mathematics Research Notices 2025 (2025), no. 6, Article rnaf058. DOI: 10.1093/imrn/rnaf058. https://doi.org/10.1093/imrn/rnaf058 (preprint arXiv:2402.00286)
24. Yihao Ding, Hongfeng Zhang, Dirac series of $\mathrm{GL}(n)$ over an Archimedean field, Mathematische Zeitschrift 311 (2025), no. 1. DOI: 10.1007/s00209-025-03813-8. https://doi.org/10.1007/s00209-025-03813-8 (preprint arXiv:2208.09903)
25. 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. https://doi.org/10.1016/j.jalgebra.2025.09.043 (preprint arXiv:2408.10817)
26. Chao-Ping Dong, Kayue Daniel Wong, Vogan's FPP conjecture for complex Lie groups, Transactions of the American Mathematical Society 379 (2026), no. 2, 867–884. DOI: 10.1090/tran/9626. https://doi.org/10.1090/tran/9626 (preprint arXiv:2407.16512)
27. Chao-Ping Dong, Zhan Ying, Distribution of spin norm along pencils: the $Sp(p,q)$ case, preprint (2026). arXiv:2605.06015. https://arxiv.org/abs/2605.06015
28. Dan Barbasch, Kayue Daniel Wong, Dirac series for complex $E_8$, preprint (2026). arXiv:2305.03254. https://arxiv.org/abs/2305.03254