The Salamanca-Riba–Vogan unitarily-small infinitesimal-character conjecture for real reductive groups beyond $U(p,q)$

A central open problem in the representation theory of real reductive Lie groups $G$ is the classification of the unitary dual $\widehat{G}$ , the set of irreducible unitarizable $(\mathfrak{g},K)$ -modules. In 1998 Salamanca-Riba and Vogan proposed reducing $\widehat{G}$ by partitioning irreducible modules according to their lowest $K$ -types; the linchpin of their program is Conjecture 5.7: an irreducible unitary module whose lowest $K$ -type $\delta$ is unitarily small must have real infinitesimal character $\Lambda$ in the convex region $\lambda_u(\delta)+\operatorname{conv}(W(\mathfrak{g},\mathfrak{h})\cdot\rho(\mathfrak{g}))$ . Wong (2024) proved this conjecture, together with Vogan's fundamental parallelepiped (FPP) conjecture, for $G=U(p,q)$ , by a case-by-case study of fundamental cases using deformation of Hermitian forms, Jantzen filtrations, and Dirac inequality techniques. The FPP sibling has since been proved for all real reductive groups by Davis and Mason-Brown (2024) with Hodge-theoretic methods, but their argument bounds the infinitesimal character alone and does not address the lowest- $K$ -type statement. The surviving open core is the following: prove, or refute by an explicit counterexample, Conjecture 5.7 for connected real reductive groups beyond the family $U(p,q)$ . A positive resolution would, via Theorem 5.8 of the 1998 paper, make the Salamanca-Riba–Vogan reduction of $\widehat{G}$ unitarity-preserving for arbitrary $G$ .

Background

For a connected real reductive Lie group $G$ with maximal compact subgroup $K$ , representations are studied through Harish-Chandra modules: a $(\mathfrak{g},K)$ -module is a simultaneous representation of the complexified Lie algebra $\mathfrak{g}$ and of $K$ , compatible with the two actions. A $(\mathfrak{g},K)$ -module is unitary if it carries a positive-definite $G$ -invariant Hermitian form, and the unitary dual $\widehat{G}$ is the set of irreducible unitarizable $(\mathfrak{g},K)$ -modules up to equivalence. The Langlands classification parametrizes all irreducible admissible modules by tempered data on Levi subgroups plus a positivity condition, but unitarity is not preserved by this parametrization, and determining which irreducible modules are unitarizable remains, in the words of the source paper, a main unsolved problem of the field. The atlas software of Adams, van Leeuwen, Trapa, and Vogan can decide unitarity of any specified module, so the obstacle is structural: a finite, group-independent organizing principle for $\widehat{G}$ .

Fix a Cartan involution $\theta$ and Cartan decomposition $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ . A $K$ -type is an irreducible representation $\delta$ of $K$ , with highest weight $\mu(\delta)$ ; a lowest $K$ -type of a $(\mathfrak{g},K)$ -module $X$ (a notion of Vogan) is a $K$ -type of $X$ minimizing $\|\mu(\delta)+2\rho(\mathfrak{k})\|$ , where $\rho(\mathfrak{k})$ is the half-sum of the positive roots of $\mathfrak{k}$ . For a $K$ -dominant weight $\mu$ , Salamanca-Riba and Vogan defined $\lambda_a(\mu):=P(\mu+2\rho(\mathfrak{k})-\rho(\mathfrak{g}))$ , where $P$ is the projection to the dominant Weyl chamber for a positive system $\Delta^+(\mathfrak{g},\mathfrak{t})$ and $\rho(\mathfrak{g})$ is the half-sum of its positive roots. The weight $\lambda_a$ determines a $\theta$ -stable parabolic subalgebra $\mathfrak{q}(\lambda_a)$ with Levi subgroup $G(\lambda_a)$ , which is always quasisplit, and their Annals paper establishes a bijection between irreducible modules having a lowest $K$ -type $\delta$ with $\lambda_a(\delta)=\lambda_a$ and certain modules for the smaller group $G(\lambda_a)$ , the bijection being implemented by cohomological induction. Cohomological induction (Knapp and Vogan) is the functor $\mathcal{L}_{\mathfrak{q},S}$ building $\mathfrak{g}$ -modules from $\mathfrak{l}$ -modules for a $\theta$ -stable parabolic $\mathfrak{q}=\mathfrak{l}+\mathfrak{u}$ ; in the good range of infinitesimal characters it preserves irreducibility and unitarity, so unitary duals of larger groups are built from those of their $\theta$ -stable Levis in this range. For $U(p,q)$ the relevant Levis are products of general linear groups and smaller indefinite unitary groups, and the base of the induction is supplied by Vogan's classification of the unitary dual of $GL(n)$ over an archimedean field.

The same Annals paper proposed enlarging the Levi by projecting more of $\mu$ to zero: define $\lambda_u(\mu):=P(\mu+2\rho(\mathfrak{k})-2\rho(\mathfrak{g}))$ and call a $K$ -type $\delta$ unitarily small if $G(\lambda_u(\delta))=G$ . Their suggested reduction of $\widehat{G}$ is to the modules whose lowest $K$ -types are unitarily small, together with good-range cohomological induction. For this reduction to preserve unitarity they conjectured (Conjecture 5.7) that unitarity of an irreducible Hermitian module $X$ with unitarily small lowest $K$ -type $\delta$ and real infinitesimal character $\Lambda=(\lambda_a(\delta),\nu)$ forces $\Lambda$ into the convex region $\lambda_u(\delta)+(\text{convex hull of }W(\mathfrak{g},\mathfrak{h})\cdot\rho(\mathfrak{g}))$ ; when $\Lambda$ leaves this region, the invariant Hermitian form of $X$ should take opposite signatures on two unitarily small $K$ -types of $X$ , so $X$ cannot be unitary. Their Theorem 5.8 shows that Conjecture 5.7 implies the unitarity-preservation of the $\lambda_u$ -bijection, which is the reduction step. Between 1998 and Wong's 2022–2024 work there was, by Wong's account, nearly no progress on this conjecture; Wong proved it for $G=U(p,q)$ (Advances in Mathematics, 2024) through a detailed study of what he calls fundamental cases, the modules not reachable from smaller data, handled by deforming the Hermitian form along a parameter and detecting sign changes via Jantzen filtrations (a canonical filtration of a standard module whose layer-wise signatures expose the signature character of the invariant Hermitian form) and Parthasarathy's Dirac inequality (a lower bound, coming from the Dirac operator of representation theory, that the infinitesimal character of any unitary module must satisfy).

A parallel conjecture of Vogan (2023; formulated with Adams and Miller, refining polyhedral bounds of Barbasch and Ciubotaru for spherical principal series) asserts a different necessary condition, the fundamental parallelepiped (FPP) conjecture: an irreducible unitary module that is not cohomologically induced in the good range from a proper $\theta$ -stable parabolic must have infinitesimal character in the fundamental parallelepiped, that is $\langle\Lambda,\alpha^{\vee}\rangle\leq 1$ for every simple root $\alpha$ . Wong proved the FPP conjecture for $U(p,q)$ in the same paper. Since then the FPP conjecture has been fully resolved: Dong and Wong proved it for complex simple Lie groups (Transactions of the AMS, 2025), and Davis and Mason-Brown proved it for all real reductive groups, case-free, by an argument mixing Beilinson–Bernstein localization of the module on the flag variety with the Hodge filtration of mixed Hodge modules; a $p$ -adic analogue has since been formulated and proved under local Langlands assumptions. The Davis–Mason-Brown theorem bounds the infinitesimal character in absolute terms and does not control how $\Lambda$ sits relative to the lowest $K$ -type, so the Salamanca-Riba–Vogan Conjecture 5.7 — the lowest- $K$ -type-sensitive half of the pair of necessary conditions — remains open for every family of real reductive groups other than $U(p,q)$ .

Problem Statement

Prove or refute, for connected real reductive Lie groups beyond the family $U(p,q)$ , the following conjecture of Salamanca-Riba and Vogan (Conjecture 5.7 of their 1998 Annals paper). Let $G$ be a connected real reductive Lie group with maximal compact subgroup $K$ , let $\mathfrak{h}=\mathfrak{t}+\mathfrak{a}$ be a fundamental Cartan subalgebra, and let $X$ be an irreducible Hermitian $(\mathfrak{g},K)$ -module having a unitarily small lowest $K$ -type $\delta$ , that is, a lowest $K$ -type with $G(\lambda_u(\delta))=G$ , where $\lambda_u(\delta)=P(\mu(\delta)+2\rho(\mathfrak{k})-2\rho(\mathfrak{g}))$ with $P$ the projection to the dominant chamber. Suppose $X$ has real infinitesimal character $\Lambda=(\lambda_a(\delta),\nu)\in\mathfrak{h}^{*}$ , where $\lambda_a(\delta)=P(\mu(\delta)+2\rho(\mathfrak{k})-\rho(\mathfrak{g}))$ and $\nu\in\mathfrak{a}^{*}$ is the continuous Langlands parameter in the decomposition $\mathfrak{h}^{*}=\mathfrak{t}^{*}+\mathfrak{a}^{*}$ . Then: if $X$ is unitary, $\Lambda$ must lie in the convex region

\lambda_u(\delta)+\bigl(\text{convex hull of }W(\mathfrak{g},\mathfrak{h})\cdot\rho(\mathfrak{g})\bigr);

equivalently, when $\Lambda$ lies outside this region, the invariant Hermitian form of $X$ must take opposite signatures on two unitarily small $K$ -types of $X$ , so $X$ is not unitary. The conjecture is a theorem for $G=U(p,q)$ (Wong, 2024) and is open for every other family of real reductive groups — split and quasisplit forms, complex groups viewed as real groups, other classical families such as $Sp(2n,\mathbb{R})$ and $SO(p,q)$ , exceptional groups, and non-linear coverings. The intended target generality is all connected real reductive groups. A positive answer at full generality would, via Theorem 5.8 of the source paper, also settle that the $\lambda_u$ -bijection of Salamanca-Riba and Vogan preserves unitarity, completing their reduction of the unitary dual to unitarily small lowest $K$ -types plus good-range cohomological induction.

References

S. Salamanca-Riba and D. A. Vogan Jr., "On the Classification of Unitary Representations of Reductive Lie Groups", Annals of Mathematics. Second Series 148 (1998), no. 3, 1067–1133. DOI: 10.2307/121036, https://doi.org/10.2307/121036
Kayue Daniel Wong, "On some conjectures of the unitary dual of U(p,q)", Advances in Mathematics 442 (2024), article 109584 (arXiv:2210.08684; v1 circulated as "On the unitary dual of U(p,q): proof of a conjecture of Salamanca-Riba and Vogan"). DOI: 10.1016/j.aim.2024.109584, https://arxiv.org/abs/2210.08684
D. A. Vogan Jr., "How to compute the unitary dual", talk given at Zhejiang University, 2023, https://math.mit.edu/~dav/zhejiang23HO.pdf
Dougal Davis and Lucas Mason-Brown, "The FPP Conjecture for Real Reductive Groups", arXiv:2411.01372 (2024), https://arxiv.org/abs/2411.01372
Chao-Ping Dong and Kayue Daniel Wong, "Vogan's FPP conjecture for complex Lie groups", Transactions of the American Mathematical Society 379 (2025), 867–884. DOI: 10.1090/tran/9626, https://arxiv.org/abs/2407.16512
Kayue Daniel Wong and Hongfeng Zhang, "The unitary dual of U(n,2)", arXiv:2401.15660 (2024), https://arxiv.org/abs/2401.15660
Chao-Ping Dong, Chengyu Du, and Haojun Xu, "Non-decreasable K-types are unitarily small", arXiv:2505.06969 (2025), https://arxiv.org/abs/2505.06969
Dihua Jiang, Baiying Liu, Chi-Heng Lo, and Lucas Mason-Brown, "The FPP Conjecture for p-adic Groups", arXiv:2509.18320 (2025), https://arxiv.org/abs/2509.18320
Stephen D. Miller, "Non-unitarity outside the fundamental parallelepiped", arXiv:2608.11355 (2026), https://arxiv.org/abs/2608.11355
J. Adams, M. van Leeuwen, P. Trapa, and D. A. Vogan Jr., "Unitary representations of real reductive groups", Astérisque 417 (2020). DOI: 10.24033/ast.1119, https://doi.org/10.24033/ast.1119
A. W. Knapp and B. Speh, "The role of basic cases in classification: Theorems about unitary representations applicable to SU(N,2)", in Non-Commutative Harmonic Analysis and Lie Groups (Marseille, 1982), Lecture Notes in Mathematics 1020, Springer, Berlin, 1983, pp. 119–160. DOI: 10.1007/bfb0071500, https://doi.org/10.1007/bfb0071500
A. W. Knapp and D. A. Vogan Jr., "Cohomological Induction and Unitary Representations (PMS-45)", Princeton Mathematical Series 45, Princeton University Press, Princeton, NJ, 1995. DOI: 10.1515/9781400883936, https://doi.org/10.1515/9781400883936
D. Barbasch, "A reduction theorem for the unitary dual of U(p,q)", in Noncommutative Harmonic Analysis, Progress in Mathematics 220, Birkhäuser Boston, Boston, MA, 2004, pp. 21–60. DOI: 10.1007/978-0-8176-8204-0_2, https://doi.org/10.1007/978-0-8176-8204-0_2
D. A. Vogan Jr., "The unitary dual of GL(n) over an archimedean field", Inventiones Mathematicae 83 (1986), no. 3, 449–505. DOI: 10.1007/BF01394418, https://doi.org/10.1007/BF01394418
Chao-Ping Dong and Zhan Ying, "Distribution of spin norm along pencils: the $Sp(p,q)$ case", arXiv:2605.06015 (2026), https://arxiv.org/abs/2605.06015