Removing the integrality hypothesis in the improved Helgason–Johnson bound for the real exceptional group $E_{7(-5)}$

For the connected linear real exceptional group $E_{7(-5)}$ , Ding, Dong and Li (2022) proved an improved Helgason–Johnson bound: any irreducible unitary $(\mathfrak g,K)$ -module whose infinitesimal character $\Lambda$ is a non-negative integral combination of the fundamental weights, and whose continuous part $\nu$ satisfies $\|\nu\|\ge\sqrt{165/2}$ , must be either the trivial representation or the unique minimal representation. In Remark 7.2 the authors state that they believe the integrality assumption on $\Lambda$ is removable, but that the proof would be much harder; their argument rests on a finite atlas search over 1113 integral infinitesimal characters, which cannot touch the non-integral locus. The problem is to prove, or refute by an explicit module, the integrality-free statement. Such a result would upgrade the main computational sieve of the Dirac-series program for the unitary duals of real exceptional groups, where every subsequent sharpened bound (for $E_{7(-25)}$ , $E_{7(7)}$ , $E_{8(-24)}$ ) still carries an integrality-type hypothesis.

Background

Let $G$ be a connected semisimple real Lie group with maximal compact subgroup $K$ . Irreducible unitary representations of $G$ are studied through their Harish-Chandra modules: irreducible $(\mathfrak g,K)$ -modules, where $\mathfrak g$ is the complexified Lie algebra of $G$ . Fix a maximally split $\theta$ -stable Cartan subgroup $H=TA$ ; each irreducible $(\mathfrak g,K)$ -module has an infinitesimal character $\Lambda\in\mathfrak h^*$ (Harish-Chandra's analogue of a highest weight). Following Vogan, one writes $\Lambda=(\lambda_a(\mu),\nu)$ relative to the decomposition $\mathfrak h=\mathfrak t\oplus\mathfrak a$ , where $\mu$ runs over the lowest $K$ -types of the module (the $K$ -types minimal for Vogan's lambda norm) and the component $\nu\in\mathfrak a^*$ is the continuous part of $\Lambda$ ; in the setting below, $\|\nu\|$ coincides with the norm of the $\nu$ -part of the module's atlas final parameter. A classical theorem of Helgason and Johnson (1969) bounds the continuous part of any irreducible unitary representation: $\|\nu\|\le\|\rho(G)\|$ , where $\rho(G)$ is the half-sum of the positive restricted roots. For $G=E_{7(-5)}$ — the connected linear real form EVI of the complex exceptional group $E_7$ , with $\dim\mathfrak k-\dim\mathfrak p=69-64=5$ in magnitude and center $\mathbb Z/2\mathbb Z$ , denoted E7_q by the atlas software — this gives $\|\nu\|\le\sqrt{399/2}$ . The modern refinement comes from the theory of Dirac cohomology: the Dirac operator $D$ acts on $\pi\otimes S_G$ ( $S_G$ a spin module for the complexified Clifford algebra of $\mathfrak p_0$ ), and Vogan's Dirac cohomology $H^D(\pi)=\ker D/(\ker D\cap\operatorname{im}D)$ was shown by Huang and Pandžić to determine the infinitesimal character whenever it is nonzero. Parthasarathy's Dirac operator inequality $\|\gamma+\rho_c\|\ge\|\Lambda\|$ (for $\rho_c$ the half-sum of positive compact roots) underlies both the classification of the Dirac series — the irreducible unitary modules with nonzero Dirac cohomology, computed with the atlas software of Adams, van Leeuwen, Trapa and Vogan, whose unitarity test is a proven algorithm — and Dong's sharpened Helgason–Johnson bound $\|\nu\|\le\sqrt{227/2}$ for all infinite-dimensional irreducible unitary $(\mathfrak g,K)$ -modules of $E_{7(-5)}$ , which is attained by the minimal representation (the nonzero irreducible representation of smallest Gelfand–Kirillov dimension, where Gelfand–Kirillov dimension measures the polynomial growth rate of a $(\mathfrak g,K)$ -module; it equals 17 for $E_{7(-5)}$ ). Ding–Dong–Li pushed the constant further down under an integrality hypothesis: choosing the fundamental weights $\zeta_1,\dots,\zeta_7$ for a positive root system of the equal-rank group $E_{7(-5)}$ , their Proposition 7.1 assumes that $\Lambda$ is a non-negative integer combination of $\zeta_1,\dots,\zeta_7$ and concludes that $\|\nu\|\ge\sqrt{165/2}$ forces $\pi$ to be the trivial representation or the minimal representation. The proof enumerates the 1113 integral infinitesimal characters in the relevant window and runs exhaustive atlas unitarity searches; since the non-integral infinitesimal characters form a continuum, this finite-search method has no direct extension. In Remark 7.2 the authors record: "We believe that the condition $\Lambda$ being integral is removable. But the proof should be much harder."

Problem Statement

For the connected linear real exceptional group $G=E_{7(-5)}$ (real form EVI of complex $E_7$ , the group E7_q in atlas), prove or disprove the following integrality-free strengthening of Proposition 7.1 of arXiv:2205.11999: every irreducible unitary $(\mathfrak g,K)$ -module $\pi$ of $G$ whose continuous part $\nu$ — the second component of the infinitesimal character $\Lambda=(\lambda_a(\mu),\nu)$ in the coordinates of Eq. (18) of that paper, whose norm equals the norm of the $\nu$ -part of the atlas final parameter of $\pi$ — satisfies $\|\nu\|\ge\sqrt{165/2}$ , must be either the trivial representation or the unique minimal representation (of Gelfand–Kirillov dimension 17). No assumption may be made that $\Lambda$ is an integral (non-negative integer) combination of the fundamental weights $\zeta_1,\dots,\zeta_7$ . A disproof consists of an explicit irreducible unitary $(\mathfrak g,K)$ -module with $\|\nu\|\ge\sqrt{165/2}$ that is neither trivial nor minimal.

References

Yi-Hao Ding, Chao-Ping Dong, Ping-Yuan Li, "Dirac series of $E_{7(-5)}$ ", Indagationes Mathematicae 34 (2023) 54–77; DOI 10.1016/j.indag.2022.09.002; arXiv:2205.11999, https://arxiv.org/abs/2205.11999
Chao-Ping Dong, "On the Helgason-Johnson bound", Israel Journal of Mathematics 254 (2022) 373–397; DOI 10.1007/s11856-022-2403-6; arXiv:2012.13474, https://arxiv.org/abs/2012.13474
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; arXiv:2204.07902, https://arxiv.org/abs/2204.07902
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; arXiv:2210.15833, https://arxiv.org/abs/2210.15833
Yi-Hao Ding, Chao-Ping Dong, Chengyu Du, Yongzhi Luan, Liang Yang, "Dirac Series of $E_{8(-24)}$ ", International Mathematics Research Notices (2025), article rnaf058; DOI 10.1093/imrn/rnaf058; arXiv:2402.00286, https://arxiv.org/abs/2402.00286
Chao-Ping Dong, "A non-vanishing criterion for Dirac cohomology", Transformation Groups 29 (2024) 935–958; DOI 10.1007/s00031-022-09758-0; arXiv:2107.09220, https://arxiv.org/abs/2107.09220
Lin Wei, Chengyu Du, Hongfeng Zhang, Xuanchen Zhao, "Dirac series of $U(n,2)$ " (2024); arXiv:2408.10817, https://arxiv.org/abs/2408.10817
Sigurdur Helgason, Kenneth Johnson, "The bounded spherical functions on symmetric spaces", Advances in Mathematics 3 (1969) 586–593; DOI 10.1016/0001-8708(69)90010-3, https://doi.org/10.1016/0001-8708(69)90010-3
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) 185–202; DOI 10.1090/S0894-0347-01-00383-6, https://doi.org/10.1090/S0894-0347-01-00383-6