Structural theory of the FS-scattered part of the Dirac series for real exceptional Lie groups: u-small K-types, unipotent representations, and cancellation in the Dirac index

The 2020 classification of the Dirac series (irreducible unitary representations with nonzero Dirac cohomology) of the real exceptional groups $E_{6(6)}$ , $E_{6(-26)}$ , $F_{4(4)}$ , $F_{4(-20)}$ and $G_{2(2)}$ left a precise structural agenda: the computations suggest an unexplained connection between the fully supported scattered (FS-scattered) members of the Dirac series and the unitarily small (u-small) K-types of the group; the identification of those scattered members that are genuinely unipotent representations is incomplete; and the discovery of a representation of $F_{4(4)}$ whose Dirac index vanishes while its Dirac cohomology is nonzero refuted the expected no-cancellation principle without supplying a replacement criterion. Later work classified the Dirac series of every real exceptional form except the split $E_{8(8)}$ (the intermediate real $E_6$ form $E_{6(2)}={\rm EII}$ being among those classified, via the non-vanishing criterion of ref. 9), embedded the u-small connection into the Salamanca-Riba-Vogan unified conjecture and Vogan's fundamental parallelepiped (FPP) conjecture (proved so far only for complex simple groups and $U(p,q)$ ), confirmed the Barbasch-Pandzic unipotent conjecture for complex classical groups, and gave a parity criterion for Dirac-index cancellation conditional on the spin-lowest K-types. The structural core remains open: a theorem-level account, for real exceptional groups, of the FS-scattered part - which u-small K-types realize it, which of its members are unipotent, and when Dirac cohomology cancels in the Dirac index.

Background

Let $G$ be a connected simple real Lie group in the Harish-Chandra class, $K$ a maximal compact subgroup, and $\mathfrak g_0=\mathfrak k_0\oplus\mathfrak p_0$ the Cartan decomposition. The unitary dual $\widehat G$ - the set of equivalence classes of irreducible unitary representations - is the central unresolved object of the representation theory of real reductive groups; it is known in full only for groups of small rank and a few families. Dirac cohomology, introduced by Vogan in 1997, is a tool for studying it: for a $(\mathfrak g,K)$ -module $V$ one forms the cubic Dirac operator $D$ acting on $V\otimes S$ , where $S$ is the spin module for $\mathfrak p$ , and defines $H_D(V)=\ker D/(\operatorname{im}D\cap\ker D)$ , a module for the spin double cover $\widetilde K$ of $K$ . Huang and Pandzic proved Vogan's conjecture that if $H_D(V)\neq0$ contains a $\widetilde K$ -type of highest weight $\gamma$ , then the infinitesimal character of $V$ is conjugate to $\gamma+\rho_c$ , where $\rho_c$ is the half-sum of the positive compact roots. The representations with $H_D\neq0$ - the Dirac series $\widehat G^{d}$ - are exactly those on which Parthasarathy's Dirac operator inequality becomes equality on some K-type, and they contain all unitary representations with nonzero $(\mathfrak g,K)$ -cohomology, so they carry the representation-theoretic information used in the study of automorphic forms.

Dong proved a finiteness theorem: $\widehat G^{d}$ splits into a finite scattered part - members not cohomologically induced in the good range from any proper $\theta$ -stable parabolic subgroup - and finitely many strings obtained by cohomological induction (the $A_{\mathfrak q}(\lambda)$ construction, with good, fair, or weakly fair range conditions controlling unitarity) from scattered members of Levi subgroups. A computationally convenient subcollection is the FS-scattered part: members whose KGB element (the parametrization of Cartan subgroups used by the atlas software of the Atlas of Lie Groups project) has full support. Salamanca-Riba and Vogan introduced unitarily small (u-small) K-types: K-types whose highest weight lies in the convex hull $R(\Delta(\mathfrak p,\mathfrak t_f))$ of the noncompact roots with coefficients in $[0,1]$ ; there are finitely many, and their 1998 unified conjecture (including Conjecture 5.7') proposes that the unitary dual is generated from representations built on u-small K-types. Unipotent representations - the most singular unitary representations, attached by Barbasch and Vogan to nilpotent orbits and believed to be building blocks of $\widehat G$ - are expected, by Conjecture 1.1 of Barbasch and Pandzic (2011), to be precisely the indecomposable pieces from which the Dirac series of a complex group is cohomologically induced; this was confirmed for complex classical groups by Barbasch, Dong, and Wong in 2022. For equal-rank $G$ ( $\operatorname{rank}K=\operatorname{rank}G$ ), the Dirac index $H_D^{+}(\pi)-H_D^{-}(\pi)$ , the difference of the even and odd parts of Dirac cohomology under the grading of the spin module, is a virtual $\widetilde K$ -module refining the character-theoretic index used in the theory of elliptic representations and endoscopy.

Ding, Dong, and Yang classified $\widehat G^{d}$ for ${\rm EI}=E_{6(6)}$ , ${\rm EIV}=E_{6(-26)}$ , ${\rm FI}=F_{4(4)}$ , ${\rm FII}=F_{4(-20)}$ , with $G_{2(2)}$ as an illustrative toy case: the FS-scattered members number 13, 2, 22, 2 respectively (3 for $G_{2(2)}$ ), all of their spin-lowest K-types (the K-types contributing to $H_D$ ) are u-small, and the counts $N_{\rm us}$ of u-small K-types are 484, 37, 544, 27 (16 for $G_{2(2)}$ ). Along the way they disproved a 2015 conjecture of Huang asserting that $H_D^{+}$ and $H_D^{-}$ never share a $\widetilde K$ -type for irreducible modules of equal-rank groups, by exhibiting an FS-scattered representation of $F_{4(4)}$ with $H_D\neq0$ but vanishing Dirac index.

Problem Statement

Provide a structural, theorem-level account of the fully supported scattered (FS-scattered) part of the Dirac series $\widehat G^{d}$ of real exceptional Lie groups, replacing the current case-by-case atlas computations, by resolving the following linked questions. Items (1)-(3) are the questions raised by the 2020 classification of Ding, Dong, and Yang for $E_{6(6)}$ , $E_{6(-26)}$ , $F_{4(4)}$ , $F_{4(-20)}$ and $G_{2(2)}$ ; item (4) is the continuation of the same program recognized by the subsequent classification papers, which record that the split $E_8$ case 'is still very challenging' (Introduction of the $E_{8(-24)}$ paper, arXiv:2402.00286) and that some FS-scattered members are not $A_{\mathfrak q}(\lambda)$ -modules, leaving their uniform, non-computational construction open (Section 6 of the $E_{7(-25)}$ paper, arXiv:2204.07902).

(1) u-small connection. Prove or refute, for real exceptional $G$ , a precise structural relation between the FS-scattered members of $\widehat G^{d}$ and the u-small K-types of $G$ , i.e., which u-small K-types occur as spin-lowest K-types of FS-scattered members and with which multiplicities.

(2) Unipotent identification. Determine which FS-scattered members of $\widehat G^{d}$ are genuinely unipotent representations - that is, special unipotent in the sense of Barbasch and Vogan - and prove or refute that the scattered part of the Dirac series exhausts the unipotent spectrum (the set of all special unipotent representations) of real exceptional groups - the real-group counterpart of Conjecture 1.1 of Barbasch and Pandzic, established so far only for complex groups.

(3) Cancellation criterion. Give a criterion, expressible in terms of the Langlands (atlas/KGB) parameter of an irreducible unitary representation $\pi$ of an equal-rank real group $G$ , that decides whether $H_D^{+}(\pi)$ and $H_D^{-}(\pi)$ share $\widetilde K$ -types - so that the Dirac index vanishes although $H_D(\pi)\neq0$ - without first computing the full Dirac cohomology.

(4) Completing the landscape. Classify the Dirac series of the remaining real exceptional form, the split $E_{8(8)}$ , and give a uniform cohomological-induction mechanism explaining which FS-scattered members arise as limits of strings.

The common scientific objective is a theory of the scattered part as the conjectured unipotent core of the unitary dual; the components above jointly serve it, the classifications in (4) being the computational substrate that the theorems in (1)-(3) must explain. Each of (1)-(4) is an independently posed sub-question that can be proved or refuted on its own.

References

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