The fundamental parallelepiped (FPP) conjecture of Adams, van Leeuwen, Miller, and Vogan — proved in general by Davis and Mason-Brown (2024) — yields a structural description of the unitary dual Πu(G(R))\Pi_u(G(\mathbb{R})) of any real reductive group, including non-linear finite coverings, as a finite union of cohomologically induced representations built from the sets Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) of irreducible unitary representations of Levi subgroups whose infinitesimal characters lie in the fundamental parallelepiped. The explicitly noted remaining task is to make this description effective: develop an efficient uniform algorithm that computes Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) for every Levi subgroup and carries out the required cohomological inductions, thereby determining the complete unitary dual of arbitrary real reductive groups; the motivating obstruction is the exceptional real forms, such as the split real form of E8E_8, whose unitary duals have remained out of computational reach. As of August 2026 this algorithmic completion is still open: the Adams–van Leeuwen–Miller–Vogan atlas algorithm is under development (their exceptional-groups monograph is listed as in preparation in June 2026), and the newest related work still treats E8(8)E_{8(8)} principal series case by case.

Background

A central problem in the representation theory of real reductive groups, open since Harish-Chandra's work in the 1950s–60s, is the determination of the unitary dual Πu(G(R))\Pi_u(G(\mathbb{R})): the set of equivalence classes of irreducible unitary representations. By Harish-Chandra's admissibility theorem this equals the set of irreducible unitary (g,K)(\mathfrak{g},K)-modules — algebraic objects on which the complexified Lie algebra g\mathfrak{g} and a maximal compact subgroup KK act compatibly, carrying a positive-definite invariant Hermitian form. The Langlands classification parametrizes all irreducible admissible representations by triples consisting of a parabolic subgroup, a discrete-series representation of its Levi factor (an irreducible unitary representation that is square-integrable modulo the center), and a real continuous parameter ν\nu, so the unitary dual is the subset of a well-parametrized admissible dual on which the invariant Hermitian form is positive definite; unitarity can change only across the reducibility hyperplanes of the Langlands parameters, which carve the parameter space into polyhedral pieces.

The fundamental parallelepiped (FPP) is the region in hR∗\mathfrak{h}^{*}_{\mathbb{R}} (the real form of the dual of a Cartan subalgebra) defined by 0≤⟨λ,αˇ⟩≤10\le\langle\lambda,\check{\alpha}\rangle\le 1 for every simple coroot αˇ\check{\alpha}; the coroot associated to a root α\alpha is αˇ=2α/⟨α,α⟩\check{\alpha}=2\alpha/\langle\alpha,\alpha\rangle, and ⟨λ,αˇ⟩\langle\lambda,\check{\alpha}\rangle is the natural duality pairing between hR∗\mathfrak{h}^{*}_{\mathbb{R}} and the real Cartan. Via the Harish-Chandra isomorphism, an infinitesimal character — the character by which the center of the universal enveloping algebra acts — is represented by such a λ\lambda modulo the Weyl group. Cohomological induction is the Zuckerman construction that builds (g,K)(\mathfrak{g},K)-modules from modules of a θ\theta-stable parabolic subalgebra q=l+u\mathfrak{q}=\mathfrak{l}+\mathfrak{u} (a parabolic subalgebra stable under the Cartan involution — the involution of the group defining its symmetric-space decomposition — with Levi factor l\mathfrak{l}); when the infinitesimal character shift lies in the weakly good range, cohomological induction is nonzero in a single degree, is irreducible or zero, and — by theorems of Vogan cited in the Davis–Mason-Brown paper — preserves unitarity exactly.

Building on polyhedral unitary bounds of Barbasch and Ciubotaru for spherical representations (representations containing a nonzero vector fixed by the maximal compact subgroup KK), Adams, van Leeuwen, Miller, and Vogan formulated the FPP conjecture (presented in Vogan's 2023 lecture “How to compute the unitary dual”): every irreducible unitary representation either has infinitesimal character in the FPP, or is cohomologically induced in the weakly good range from a unitary representation of the Levi subgroup of a proper θ\theta-stable parabolic. Davis and Mason-Brown proved this in full in 2024, case-free and for non-linear finite coverings of real reductive groups, using mixed Hodge module and D\mathcal{D}-module methods (sheaf-theoretic machinery from Hodge theory and from the theory of algebraic differential operators) on the flag variety (the projective variety parametrizing Borel subgroups). The theorem implies the structural formula Πu(G(R))=⋃Q=LUCohIndqg(Πu,FPP(L(R)))\Pi_u(G(\mathbb{R}))=\bigcup_{Q=LU}\mathrm{CohInd}^{\mathfrak{g}}_{\mathfrak{q}}(\Pi_{u,\mathrm{FPP}}(L(\mathbb{R}))), the union over the finitely many KK-conjugacy classes of θ\theta-stable parabolics, where Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) is the set of irreducible unitary L(R)L(\mathbb{R})-representations satisfying the FPP condition — for a Levi L(R)L(\mathbb{R}), this is the FPP condition of the theorem applied to LL itself: the infinitesimal character is real, represented by λ∈hR∗\lambda\in\mathfrak{h}^{*}_{\mathbb{R}} dominant for LL, with 0≤⟨λ,αˇ⟩≤10\le\langle\lambda,\check{\alpha}\rangle\le 1 for every simple coroot αˇ\check{\alpha} of LL; integrality of λ\lambda is not required. Since the unitary set attached to a fixed discrete Langlands parameter is a compact rational polyhedron — a finite union of facets of the affine coroot hyperplane arrangement (Adams–van Leeuwen–Trapa–Vogan) — the sets Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) are polyhedral families of representations rather than finite lists; the FPP confines the candidate real infinitesimal characters to a compact region meeting only finitely many facets, so unitarity inside the FPP is decided by a finite facet-by-facet computation. This turns the classification of the unitary dual into a finite recursion over Levi subgroups, finite at the level of facets.

The computational backdrop is the atlas software of the Atlas of Lie Groups and Representations project. Adams, van Leeuwen, Trapa, and Vogan (Astérisque 417, 2020) gave a finite algorithm, implemented in atlas, that decides unitarity by tracking the signature of the invariant Hermitian form across the facets (alcoves and their faces) cut out by the affine coroot hyperplanes; it was validated on thousands of representations of groups of rank up to about eight. But the number of facets grows exponentially with rank, and the authors state the software is “not fast enough” for arbitrary representations of the split real form of E8E_8 (the maximally noncompact real form, denoted E8(8)E_{8(8)}, of the complex exceptional Lie algebra E8E_8); Vogan's 2023 lecture calls the facet-by-facet approach “inaccessible to existing computers for the largest exceptional groups.” The FPP theorem is what collapses the search region to a bounded, recursively structured one — which is why Davis and Mason-Brown note that Adams, van Leeuwen, Miller, and Vogan are developing an atlas algorithm to compute Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) for any real reductive group, and that the FPP conjecture “may be the key missing ingredient in the computation of the unitary dual.”

Problem Statement

Make the FPP structural description of the unitary dual of real reductive groups algorithmically effective. Concretely: develop an efficient uniform algorithm that, for an arbitrary real reductive group G(R)G(\mathbb{R}) — including non-linear finite coverings —

  1. computes, for every Levi subgroup L(R)L(\mathbb{R}) of a θ\theta-stable parabolic subgroup of GG, the set Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})) of irreducible unitary L(R)L(\mathbb{R})-representations whose Harish-Chandra parameters lie in the fundamental parallelepiped of LL (the region defined by 0≤⟨λ,αˇ⟩≤10\le\langle\lambda,\check{\alpha}\rangle\le 1 for every simple coroot αˇ\check{\alpha} of LL, with λ\lambda real and dominant for LL; integrality of λ\lambda is not required) — equivalently, since this region is compact and meets only finitely many facets of the affine coroot hyperplane arrangement, decides unitarity on those finitely many facets and thereby produces a finite facet-level (polyhedral) description of Πu,FPP(L(R))\Pi_{u,\mathrm{FPP}}(L(\mathbb{R})); and

  2. carries out the cohomological inductions CohIndqg\mathrm{CohInd}^{\mathfrak{g}}_{\mathfrak{q}} in the weakly good range from these sets, so that the formula

Πu(G(R))=⋃Q=LUCohIndqg(Πu,FPP(L(R)))\Pi_{u}(G(\mathbb{R}))=\bigcup_{Q=LU}\mathrm{CohInd}^{\mathfrak{g}}_{\mathfrak{q}}\big(\Pi_{u,\mathrm{FPP}}(L(\mathbb{R}))\big)

(the union over KK-conjugacy classes of θ\theta-stable parabolics Q=LUQ=LU) determines the complete unitary dual of G(R)G(\mathbb{R}) as a finite union of facet-level families (parameters without real infinitesimal character being recovered by unitary parabolic induction, as in the Knapp–Zuckerman reduction underlying the atlas algorithm).

The algorithm must run uniformly over all real reductive groups and their finite coverings. The exceptional real forms — prominently the split real form of E8E_8 — lie beyond the reach of the current facet-by-facet atlas computation and are the groups motivating the problem.

References

  1. Dougal Davis and Lucas Mason-Brown, 'The FPP Conjecture for Real Reductive Groups', arXiv:2411.01372 (2024), DOI 10.48550/arXiv.2411.01372, https://arxiv.org/abs/2411.01372
  2. Jeffrey Adams, Marc van Leeuwen, Peter Trapa, and David Vogan, 'Unitary representations of real reductive groups', Astérisque 417 (2020), DOI 10.24033/ast.1119, arXiv:1212.2192, https://arxiv.org/abs/1212.2192
  3. David Vogan, 'How to compute the unitary dual', representation theory seminar lecture slides, February 2023, https://math.mit.edu/~dav/zhejiang23HO.pdf
  4. Dan Barbasch and Dan Ciubotaru, 'Spherical Unitary Principal Series', Pure and Applied Mathematics Quarterly 1(4) (2005), DOI 10.4310/pamq.2005.v1.n4.a3, https://doi.org/10.4310/pamq.2005.v1.n4.a3
  5. Dan Barbasch, Dan Ciubotaru, and Alessandra Pantano, 'Unitarizable minimal principal series of reductive groups', Contemporary Mathematics 472 (2008), DOI 10.1090/conm/472/09237, https://doi.org/10.1090/conm/472/09237
  6. Chao-Ping Dong and Kayue Daniel Wong, 'Vogan's FPP conjecture for complex Lie groups', arXiv:2407.16512 (2024), https://arxiv.org/abs/2407.16512
  7. Kayue Daniel Wong, 'On some conjectures of the unitary dual of U(p,q)', Advances in Mathematics 442 (2024), article 109584, DOI 10.1016/j.aim.2024.109584, arXiv:2210.08684, https://arxiv.org/abs/2210.08684
  8. Stephen D. Miller, 'Non-unitarity outside the fundamental parallelepiped', arXiv:2608.11355 (2026), https://arxiv.org/abs/2608.11355
  9. 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
  10. Jeffrey Adams, Andrei Ionov, Lucas Mason-Brown, and David Vogan, 'The Unitarity of Arthur Packets for Real Reductive Groups', arXiv:2606.01609 (2026), https://arxiv.org/abs/2606.01609