The unitary dual
The unitary dual
For a real reductive group G(ℝ), such as SL(n,ℝ), Sp(2n,ℝ), SO(p,q) or the exceptional groups up to E8(ℝ), the unitary dual is the set of its irreducible unitary representations. Describing it has been a central open problem in representation theory for decades. We now have an algorithm that computes it, and we have carried out the computation for all groups up to and including E7.
The problem
Lie groups are continuous groups of symmetries, such as the group SO(3) of rotations of the sphere. A common way to study a group acting on a geometric space X is to linearize: let it act on the space L2(X) of functions on X. This action preserves length, so it is a unitary representation, and the tools of linear algebra become available. Unitary representations on Hilbert spaces also arise in quantum mechanics and in number theory, for example on spaces such as L2(G(ℝ)/Γ).
The basic building blocks are the irreducible unitary representations, and the problem is to classify them for every reductive group. For compact groups this goes back to Weyl in the 1920s, and for SL(2,ℝ) to Bargmann in 1947. For general reductive groups the problem is well known to be hard, and the answer is complicated. In his 1987 Bourbaki lecture on recent progress toward the classification, Laurent Clozel ended with the hope that the classification of the unitary dual of an arbitrary reductive group would soon be complete.
From analysis to algebra
Harish-Chandra turned the analytic problem into an algebraic one: a representation of G(ℝ) on a Hilbert space gives a representation of the Lie algebra on a complex vector space, a (g,K)-module. Through the work of Harish-Chandra, Langlands, and Knapp, Zuckerman and Vogan, all the irreducible admissible representations are classified, and so are the Hermitian ones: those carrying an invariant Hermitian form, which need not be positive definite. The unitary representations sit in a chain
Ĝdisc ⊂ Ĝtemp ⊂ Ĝu ⊂ Ĝherm ⊂ Ĝadm,
and the problem becomes: given a Hermitian representation, decide whether its invariant form is positive definite, and describe the set of all representations for which it is.
The Atlas project
The Atlas of Lie Groups and Representations began in 2002, at a meeting held during a conference on computers and mathematics in Montreal organized by Bill Casselman. Jeffrey Adams, Fokko du Cloux, John Stembridge, Peter Trapa and David Vogan discussed using computers to study the unitary dual. Over the next few years we turned the theory into algorithms: root data, real forms, structure theory, the Langlands classification of representations, and the Kazhdan–Lusztig–Vogan polynomials that relate irreducible and standard representations. Fokko du Cloux joined the team in 2004, after finishing his Coxeter software for Kazhdan–Lusztig polynomials, and in 2007 we computed the Kazhdan–Lusztig–Vogan polynomials for E8.
Deciding whether one representation is unitary
Vogan’s approach, from the 1980s, is to deform. A representation is described by a parameter (x,λ,ν), where ν lies in a finite-dimensional rational vector space. Move ν to 0: the Hermitian form changes sign only at finitely many points, and the changes can be computed from Kazhdan–Lusztig–Vogan polynomials, until one reaches ν = 0, where the representation is tempered and hence unitary. Two problems stood in the way: a standard module might have no invariant Hermitian form at all, and when it has one, the form is not canonical.
The solution was to change the form. Replacing the real form σ by a compact real form σc gives the c-invariant Hermitian form. In Unitary representations of real reductive groups (Adams, Trapa, van Leeuwen and Vogan; Astérisque 2020, arXiv 2012) we proved that every irreducible representation has a canonical c-invariant form, positive on its lowest K-types, and that the ordinary Hermitian form can be computed from it. The result is an explicit algorithm to decide whether a given representation is unitary, and it is implemented in atlas.
Hodge theory
In the 1980s Schmid and Vilonen observed that every representation carries a canonical Hodge filtration, coming from Saito’s theory of mixed Hodge modules, related in an intriguing but complicated way to the invariant Hermitian form. In 2011 they made a precise conjecture relating the Hodge filtration to the c-invariant form. In 2020, based on some conjectures about the Hodge filtration, we formulated an algorithm to compute it and implemented it in atlas. The statement we needed, that the Hodge grading reduced mod 2 is the grading given by the c-form, was proved by Dougal Davis and Kari Vilonen (arXiv:2309.13215).
The whole unitary dual: the FPP
Knowing how to test one representation is not the same as describing all of them. Fix the infinitesimal character γ. The fundamental parallelepiped (FPP) is the set of γ with 0 ≤ ⟨γ,α∨⟩ ≤ 1 for every simple coroot α∨. It is a union of finitely many facets, and unitarity is constant on each facet. The FPP conjecture says that a unitary representation whose infinitesimal character lies outside the FPP is cohomologically induced from a Levi subgroup, and is unitary exactly when the representation it is induced from is.
Dougal Davis and Lucas Mason-Brown proved the FPP conjecture (The FPP Conjecture for Real Reductive Groups), in a beautiful application of Hodge theory. As a result
Ĝ(ℝ)u = ∪Q CohIndQG( L̂(ℝ)FPP ),
the union over the finitely many θ-stable parabolic subgroups Q = LU up to conjugacy. Computing the unitary dual is therefore a finite calculation: for each Levi subgroup L, test one representation on each facet of its FPP.
The computations
We call this the FPP algorithm. Groups of rank up to 6 run quickly on a laptop. The split real form of E7 has about two million parameters, of which 237,641 are unitary in the FPP; the computation takes about 16 hours on a laptop, or an hour on a parallel machine. For split E8, Steve Miller computed the FPP unitary set over about 18 months and found 3,075,281 unitary representations. We have checked independently that these are unitary; confirming that none are missing is much harder, since there are billions of non-unitary facets, and is still in progress.
The answer can also be explored. For example, the spherical unitary dual of E8 has 9,282 unitary facets, and their closure relations describe its topology. See the FPP Facet Explorer and the Spherical Unitary Explorer.
Arthur’s conjectures
An algorithm is not the same as understanding. A parallel line of work gives a conceptual description of a large part of the unitary dual. At a conference at the University of Maryland in 1983, Arthur conjectured that the representations in his packets are unitary. For real groups these packets are defined by the theory of Adams, Barbasch and Vogan. Arthur representations account for most, but not all, of the unitary dual. Many people have contributed to proving the conjecture, among them Arthur, Mœglin and Renard, Barbasch, Ma, Sun and Zhu, and Adams, Miller, van Leeuwen and Vogan for the exceptional groups using atlas. Davis and Mason-Brown gave a uniform proof for complex groups using Hodge theory, and in The Unitarity of Arthur Packets for Real Reductive Groups Adams, Ionov, Mason-Brown and Vogan prove that all unipotent Arthur packets are unitary.
Looking back
Solving the problem brought together representation theory, number theory, Hodge theory and computer science. The FPP theorem is a mathematical theorem, proved without computers, and could in principle have been found without the Atlas project. In practice it probably would not have been: the c-invariant Hermitian form, which made everything else possible, came from thinking algorithmically.
For more, see the talks, especially The Unitary Dual (Maryland, September 2025), and the recent papers.