Unitary representations with nonzero Dirac cohomology of complex Lie groups: the Barbasch–Pandžić conjecture and its remaining complex E8 case

For a connected complex simple Lie group $G$ viewed as a real reductive group, Barbasch and Pandžić (2011) conjectured that an irreducible unitary representation $\pi$ with $2\chi_\pi$ regular integral has nonzero Dirac cohomology if and only if $\pi$ is induced from an essentially unipotent representation — a unipotent representation tensored with a unitary character — that itself has nonzero Dirac cohomology; moreover the Dirac cohomology, when nonzero, should consist of a single $K$ -type occurring with explicit multiplicity $[\mathrm{Spin}:E(\rho)]$ (Conjectures 1.1, 3.4 and 4.1 of that paper). Subsequent work proved the conjecture for all complex classical Lie groups and their Spin covers, and for the exceptional types $G_2$ , $F_4$ , $E_6$ and $E_7$ , completing the classification of the Dirac series — the unitary representations with nonzero Dirac cohomology — for every complex simple group except one. The single unresolved case is complex $E_8$ : Barbasch and Wong (preprint, 2023, revised 2026) classify its Dirac series assuming the unitarity of three spherical unipotent modules, of which two are nontrivial — the model representation (orbit $4A_1$ ) and the module attached to the orbit $3A_1$ — a unitarity that is widely believed but unproven, since the unitary dual of complex $E_8$ is unknown. The open core is to establish the unitarity of these two modules unconditionally (or disprove it and update the classification accordingly), thereby completing the Barbasch–Pandžić program for all complex simple Lie groups.

Background

Let $G$ be a connected complex simple Lie group viewed as a real reductive group (for example $GL(n,\mathbb{C})$ considered as a real Lie group). Its maximal compact subgroup is $K=G^\theta$ for a Cartan involution $\theta$ (an involutive automorphism of the real Lie algebra whose fixed-point set is the Lie algebra of $K$ ), and complex representations of $G$ are studied through their $(\mathfrak{g},K)$ -modules — vector spaces carrying both an action of the complexified Lie algebra $\mathfrak{g}$ and a compatible locally finite action of $K$ . The Dirac operator $D=\sum_i b_i\otimes d_i$ , built from a basis of the $(-1)$ -eigenspace of $\theta$ and its dual, acts on $\pi\otimes S$ , where $S$ is a spin module for the Clifford algebra of that eigenspace; Dirac cohomology is the $K$ -module $H_D(\pi)=\ker D/(\ker D\cap\operatorname{im}D)$ , introduced by Vogan to sharpen Parthasarathy's Dirac inequality, which bounds the infinitesimal character of a unitary representation (the parameter by which the center of the universal enveloping algebra acts) in terms of the $K$ -types (irreducible representations of $K$ ) occurring in $\pi$ . Vogan conjectured, and Huang and Pandžić proved in 2002, that whenever $H_D(\pi)\neq 0$ the infinitesimal character of $\pi$ is determined by a $K$ -type in $H_D(\pi)$ , making Dirac cohomology an effective invariant for studying the unitary dual $\widehat{G}$ — the classification of irreducible unitary representations, which remains open in general.

A necessary condition for $H_D(\pi)\neq 0$ is that twice the infinitesimal character, $2\chi_\pi$ , be regular and integral (fixed by no simple reflection of the Weyl group and lying in the weight lattice). When $\chi_\pi$ itself is regular integral, the representation has nonzero $(\mathfrak{g},K)$ -cohomology and is cohomologically induced from a unitary character (an $A_{\mathfrak{q}}(\lambda)$ -module), so the genuinely new regime is the regular half-integral one. Unipotent representations, introduced by Barbasch and Vogan (1985) in connection with Arthur's packets of representations attached to nilpotent orbits (orbits of nilpotent elements of the Lie algebra under the adjoint action), form a class of conjectural building blocks of the unitary dual; an essentially unipotent representation means a unipotent representation tensored with a unitary character. Barbasch and Pandžić (2011) proved that unitary induction preserves nonzero Dirac cohomology whenever $2\chi$ is regular integral, computed $H_D$ explicitly for representations unitarily induced from unitary characters at infinitesimal character $\rho/2$ ( $\rho$ = half-sum of positive roots), verified the resulting conjecture in type A and computationally in several other classical and exceptional types, and computed the Dirac cohomology of the unipotent representations of types A–E. They left open their Conjecture 1.1 — the structural classification of unitary representations with nonzero Dirac cohomology — together with the precise multiplicity predictions of Conjecture 3.4 (at infinitesimal character $\rho/2$ the Dirac cohomology is the trivial $K$ -type with multiplicity $[\mathrm{Spin}:E(\rho)]$ , the multiplicity of the representation of highest weight $\rho$ in the spin module, or else zero) and Conjecture 4.1 (for $GL(n,\mathbb{C})$ , $H_D(\pi)=E(2\lambda-\rho)$ with that same multiplicity). Notably, they could not establish the unitarity of one $E_7$ unipotent representation and of the two $E_8$ unipotent representations attached to the nilpotent orbits $4A_1$ (the model representation, the unique one containing almost all $K$ -types) and $3A_1$ .

Subsequent work transformed the problem. Dong (2013) reduced the classification for a complex group to spherical representations on Levi components. Ding and Dong (2020) proved a finiteness theorem: the set $\widehat{G}^{\mathrm{d}}$ of unitary representations with nonzero Dirac cohomology (the Dirac series) consists of finitely many scattered representations (members not obtained by such induction) together with strings (families of members parametrized by unitary characters) cohomologically induced from lower-rank groups in the good range (the parameter range in which cohomological induction is irreducible and preserves unitarity); they determined $\widehat{F_4}^{\mathrm{d}}$ (ten scattered representations plus thirty strings) and $\widehat{G_2}^{\mathrm{d}}$ (the model representation alone). Dong (2019) determined $\widehat{E_6}^{\mathrm{d}}$ , and Dong and Wong (2022) determined $\widehat{E_7}^{\mathrm{d}}$ , deducing that Conjecture 1.1 holds for complex $E_7$ . Finally Barbasch, Dong and Wong (2022) proved Conjecture 1.1, together with the multiplicity-one theorem refining Conjecture 4.1 and the explicit formula $H_D(\pi)=2^{\lfloor l/2\rfloor}V_k(2\lambda-\rho)$ , for all complex connected classical Lie groups and their Spin covers; the same paper notes that the corresponding statements for $G_2$ , $F_4$ , $E_6$ and $E_7$ follow from the earlier papers. These results settle Conjectures 1.1, 3.4 and 4.1 in peer-reviewed work for every complex simple group except $E_8$ , whose treatment so far exists only in a preprint whose classification is conditional on an unproven unitarity assumption.

Problem Statement

The source problem is the Barbasch–Pandžić classification conjecture for Dirac cohomology of complex groups. Conjecture 1.1 (Barbasch–Pandžić 2011): let $G$ be a complex Lie group viewed as a real group and let $\pi$ be an irreducible unitary representation such that $2\chi_\pi$ is regular and integral; then $\pi$ has nonzero Dirac cohomology if and only if $\pi$ is cohomologically induced from an essentially unipotent representation (a unipotent representation tensored with a unitary character) with nonzero Dirac cohomology. The companion predictions sharpen it: at infinitesimal character $\rho/2$ the Dirac cohomology is the trivial $K$ -type with multiplicity $[\mathrm{Spin}:E(\rho)]$ or zero (Conjecture 3.4), and for $GL(n,\mathbb{C})$ one has $H_D(\pi)=E(2\lambda-\rho)^{\oplus[\mathrm{Spin}:E(\rho)]}$ whenever $2\lambda$ is regular integral (Conjecture 4.1).

By published results this is now proved for the complex classical groups, their Spin covers, and the exceptional types $G_2$ , $F_4$ , $E_6$ , $E_7$ . The surviving task, and the content of this problem, is to complete the classification unconditionally for the one remaining type, complex $E_8$ :

Establish, without assumption, the unitarity — or the non-unitarity — of the two nontrivial spherical unipotent $E_8$ -modules with Zhelobenko parameters — the coordinates $2\nu$ labeling the Hermitian spherical modules $J(\nu,\nu)$ in Zhelobenko's classification of the spherical principal series of the complex group — given by $2\nu=(1,1,1,1,1,1,1,1)$ (the model representation, attached to the nilpotent orbit $4A_1$ ) and $2\nu=(1,1,1,1,1,1,2,2)$ (attached to the nilpotent orbit $3A_1$ ); the third candidate, $2\nu=(2,2,2,2,2,2,2,2)$ , is the trivial representation.
With that step discharged in either direction, finish the unconditional classification of the Dirac series of complex $E_8$ : determine all irreducible unitary representations with nonzero Dirac cohomology; confirm that each is the lowest $K$ -type subquotient — the subquotient containing the lowest $K$ -type, i.e. a $K$ -type of minimal norm in Vogan's sense — of a module parabolically induced from a unitary character tensored with a unipotent representation with nonzero Dirac cohomology (the Conjecture 1.1 structure, with the classification updated if either module turns out non-unitary); and verify that each such representation contributes Dirac cohomology consisting of a single spin-lowest $K$ -type (a $K$ -type $\mu$ of $\pi$ minimizing the spin norm $\|\{\mu-\rho\}+\rho\|$ ; when $H_D(\pi)\neq 0$ exactly the spin-lowest $K$ -types contribute to it) with multiplicity $2^{\lfloor\operatorname{rank}(\mathfrak{g})/2\rfloor}=16$ .

References

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