Atlas of Lie Groups and Representations

The unitary dual

The unitary dual

A fundamental tenet of physics is that the laws governing the universe are everywhere the same. Beginning in the nineteenth century, this tenet has been understood as a law of symmetry: that physics is unchanged by certain variations in point of view. For Newton, these variations included translation and rotation of space: what we would now call the (six-dimensional) group of rigid motions. Newton also allowed a point of view moving at a constant velocity, and required that physical laws change in a very simple way in that setting. Implicitly he assumed that physics did not change with time. We would say that the symmetry group of Newtonian physics is ten-dimensional: the six-dimensional Euclidean motion group, times the three-dimensional group of constant velocity motions times a one-dimensional group of translations in time.

For Einstein, the setting for physics was explicitly a four-dimensional space-time. Motion at a constant velocity was still allowed, but for Einstein it changed physical laws in a more subtle way: the symmetries were now the “Lorentz transformations.” The symmetry group was again ten-dimensional, but more complicated: the semidirect product of the six-dimensional group SO(3,1) with the four-dimensional space-time. The nature of this Lorentz group constrained Einstein’s description of physics.

By the 1930s, physicists were building quantum physics in (usually infinite-dimensional) Hilbert spaces. Eugene Wigner’s 1931 paper “Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren” argued that symmetry in quantum physics meant an action of the symmetry group had to act on these Hilbert spaces by unitary representations. In these early applications, the symmetry groups (like the group SO(3) of rotations of an atom around its nucleus) were usually compact. The mathematical theory of unitary representations of compact groups was developed in the paper “Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe” by Fritz Peter and Hermann Weyl.

A unitary representation of a group G is an action of G on a Hilbert space H by unitary operators (invertible linear maps preserving the inner product). Equivalently, it is a group homomorphism π: G → U(H), with U(H) the group of unitary linear transformations of H. (This is exactly like the statement that an action of G on a set X is the same thing as a group homomorphism from G to the group Perm(X) of permutations of X. The difference is that only very special permutations of X are considered: those by linear maps preserving length.) A unitary representation is called irreducible if H cannot be written as an orthogonal direct sum of two nonzero subspaces each preserved by G.

The mathematical theory of unitary representations of compact groups was developed in the paper “Die Vollständigkeit der primitiven Darstellungen einer geschlossenen kontinuierlichen Gruppe” by Fritz Peter and Hermann Weyl. The main theorems are simple and beautiful: any irreducible unitary representation of a compact G must be on a finite-dimensional Hilbert space; and any unitary representation has a unique decomposition as an orthogonal direct sum of pieces, each of which is isomorphic to a multiple of an irreducible representation. This theorem shows the importance of the set Ĝ of (equivalence classes of) irreducible unitary representations of G, the unitary dual.

Because of the Peter–Weyl theorem, it was important to understand in detail the unitary duals of compact groups. Deep information about this problem comes from Élie Cartan’s 1913 paper “Les groupes projectifs qui ne laissent invariante aucune multiplicité plane.” These results were translated to the setting of compact Lie groups in Hermann Weyl’s 1926–27 papers “Theorie der Darstellungen kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen I–III.” With these roots, the Cartan–Weyl highest weight theory has played an important part in non-relativistic quantum theory ever since; the “eightfold way” in the standard model refers to the eight-dimensional adjoint representation of SU(3).

Wigner’s 1939 paper “On Unitary Representations of the Inhomogeneous Lorentz Group” studied relativistic quantum particles using unitary representations of the Lorentz group. The difference from earlier work is that the Lorentz group is noncompact. The difference that makes in unitary representation theory is like the difference between the theory of Fourier series and the Fourier transform: formally the analogies are deep and meaningful, but every detail is more complicated and subtle.

To be non-chronological for a moment: group theorists since the 19th century have studied a group G by looking for a normal subgroup N, then trying to “piece together” the two smaller groups N and G/N. The Lorentz group has space-time translations N = ℝ3,1 as an abelian normal subgroup N; the quotient is the six-dimensional simple group G/N = SO(3,1).

Wigner initiated the idea of studying the unitary dual Ĝ in terms of the unitary duals N̂ and (G/N)∧ of the smaller groups. A bit more precisely, in this setting the group G/N acts on N̂. A unitary irreducible representation π of G is constructed from a unitary irreducible πN of N, and… well, G/N acts on N̂, so there is a subgroup L of G/N stabilizing the representation πN. The group L is the Mackey–Wigner “little group.” The additional information needed to construct π from πN is an irreducible unitary representation of the little group L. (This sounds simple, and it has a lot of the flavor of the truth, but it is not the truth.)

In the Lorentz group, N is just ℝ3,1, so its unitary dual is the set of unitary characters of ℝ3,1: another four-dimensional vector space carrying an indefinite form of signature (3,1). So πN is just a four-vector. The corresponding little group is isomorphic to

For Wigner the interesting cases were massive and massless. In those cases he understood the irreducible unitary representations of the little group, and so could describe the corresponding irreducible unitary representations of G.

Mathematicians had no interest in stopping with what Wigner said was of first physical importance. They wished to understand all irreducible unitary representations of the Lorentz group G. This required understanding the irreducible unitary representations of the two remaining little groups: SO(2,1) and SO(3,1). These groups are locally isomorphic to SL(2,ℝ) and SL(2,ℂ) respectively: the smallest of the simple Lie groups (to which the Mackey–Wigner little group method does not apply). Their unitary duals were published in 1947 by Valentine Bargmann (“Irreducible unitary representations of the Lorentz group”) and Israel Gelfand and Mark Naimark (“Unitary representations of the Lorentz group”).

Ever since that time, the determination of the unitary duals of other Lie groups G has been vigorously pursued. In the 1950s, George Mackey extended Wigner’s “little group” method enormously. This led to a great body of work (not discussed here) on the unitary representations of solvable Lie groups, and more generally on groups with interesting normal subgroups. As in the work of Bargmann and Gelfand–Naimark, this focused attention on the case of simple Lie groups. Even in the absence of normal subgroups, the theory makes constant use of other subgroups, like SO(p) × SO(q) inside SO(p+q); so one needs to work in a class of Lie groups which are close to simple, but allow lots of such subgroups. A typical setting is real reductive groups. This notion is so convenient that there are many inequivalent definitions in use; but for the atlas software, a real reductive group is the group of real points of a connected reductive algebraic group. Examples include GL(n,ℝ) (the n2-dimensional Lie group of invertible n × n matrices); SO(p,q) (orientation-preserving isometries of a quadratic form on ℝp+q of signature (p,q), a Lie group of dimension (p+q)(p+q−1)/2); and E8(ℝ), the 248-dimensional exceptional Lie group. The simplest reductive groups are the abelian ones, called tori: products of circles, ℝ×, and ℂ×. A Cartan subgroup of a real reductive group G is a maximal torus in G. Inside any real reductive G, there are finitely many conjugacy classes of Cartan subgroups, and understanding them is the key to understanding G.

A central goal of the atlas project is to compute the unitary dual of any real reductive group G. Here is a brief outline of work on that problem since Bargmann and Gelfand–Naimark.

In the 1950s, Harish-Chandra began his monumental work on the unitary representations of a real reductive group G, which culminated in his Plancherel formula in “Harmonic analysis on real reductive groups I–III,” published in 1975–1976. The main theorem states that the unitary representations appearing in the decomposition of L2(G) are indexed by the (one-dimensional) unitary characters of Cartan subgroups of G. This is precisely the shape of the Peter–Weyl theorem, together with the Cartan–Weyl classification of irreducible representations, in case G is compact.

So why was Harish-Chandra’s theorem not the end of the subject?

The Peter–Weyl theorem says that all irreducible unitary representations of a compact group K appear in L2(K), so in that case the Plancherel theorem describes the unitary dual K̂ completely. Harmonic analysis on other homogeneous spaces K/S can be described in terms of the restriction of irreducible K-representations to S.

For noncompact G, none of that is true. There are many irreducible unitary representations not appearing in L2(G) (like the trivial representation), which nevertheless have important roles to play. Here is how they appear in Harish-Chandra’s work.

Each Cartan subgroup H of G is a direct product (as an algebraic group) of a copies of ℝ×, b copies of ℂ×, and c circles. As a topological group, it is therefore a product of the real vector space ℝa+b, a compact torus Tb+c, and (ℤ/2ℤ)a. Its group of unitary characters is therefore ℝa+b × ℤb+c × (ℤ/2ℤ)a: analytically, a countable disjoint union of real vector spaces. Harish-Chandra’s unitary representations are therefore a family πH(ν,λ,ε), with ν ∈ ℝa+b, λ ∈ ℤb+c, and ε ∈ (ℤ/2ℤ)a. If the discrete parameters λ0 and ε0 are fixed, then all the representations πH(ν,λ0,ε0) can be realized on a common Hilbert space Vλ0,ε0. In this picture, each collection of unitary operators πH(ν,λ0,ε0)(g) (for g in G) depends real analytically on ν. Even more, the operators can be analytically continued to all νℂ in the complex vector space ℂa+b.

To repeat: Harish-Chandra’s family of unitary representations πH(ν,λ,ε), for ν in the real vector space ℝa+b, is part of a complex analytic family of (no longer unitary) representations πH(νℂ,λ,ε).

To say the same thing in a coordinate free way: the real manifold Ĥ (of unitary characters of the Cartan subgroup H) is a group of real points a complex manifold Ĥℂ (of possibly non-unitary characters of H). The complex conjugation defining Ĥ sends a character ξ to (ξ̄)−1; that is, each value reiθ of ξ is replaced by r−1eiθ.

Harish-Chandra made extensive use of these analytic families of nonunitary representations, and of complex analysis on the spaces Ĥℂ; they are at the heart of his proof of the Plancherel formula for G. He found a nice category of nonunitary representations in which they lived, the quasisimple representations. (An irreducible representation is called quasisimple if the center of the universal enveloping algebra acts by scalars.) Following a suggestion of Chevalley, he proved that every such representation π has a very nice algebraic model X(π), the Harish-Chandra module or (𝔤,K)-module of π. He proved that π is unitarizable—that is, equivalent to a unitary representation—if and only if the Harish-Chandra module is unitarizable—that is, that X(π) admits a (𝔤,K)-invariant positive definite Hermitian form.

But it was Langlands who (about 1973) realized that Harish-Chandra’s unitary representations πH(ξ) in the Plancherel formula, when extended to the complex analytic family πH(ξℂ), gave all irreducible quasisimple representations of G. This statement is the “Langlands classification,” and it gives a very clean and simple list of all irreducible quasisimple π. This statement is entirely analogous to the statement that irreducibles in category 𝒪 are indexed exactly by weights in 𝔥*.

Where does that leave the study of all irreducible unitary representations of G? Anthony Knapp in the 1970s understood that the existence of an invariant Hermitian form on the irreducible corresponding to ξℂ is a simple algebraic condition on ξℂ: for fixed (λ,ε), Knapp’s condition is satisfied exactly on a finite set of real subspaces of Ĥℂ. He further understood that these Hermitian forms varied in a real-analytic way with ξℂ.

The idea of the algorithms in atlas is that Knapp’s forms (inside one of his subspaces Sξ(m)) must be definite at 0, and can change signature only on the hyperplanes in Ĥℂ where π(ξℂ) is reducible. Those hyperplanes were located by Birgit Speh and David Vogan in the 1980 paper “Reducibility of generalized principal series.” The Jeffrey Adams/Marc van Leeuwen/Peter Trapa/David Vogan 2020 paper “Unitary representations of real reductive groups” computes the signature change in the form at each reducibility wall, using deep results of Beilinson–Bernstein and Lusztig. Therefore it provides an algorithm to determine exactly for which ξℂ the invariant Hermitian form is positive definite.

To pass from this one-representation-at-a-time information to a complete description of the unitary dual is still a difficult task, but it has been completed and coded in the atlas software.

More about computing Ĝ

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.

The approach to computing the form on a single irreducible, as sketched in “The unitary dual,” is to deform. A representation is described in atlas by a parameter (x,λ,ν), where ν lies in a finite-dimensional real vector space; the parameters (x and λ) record what is called (H, λ, and ε) in “The unitary dual.” Deform ν in a straight line to 0: the Hermitian form changes sign only at finitely many reducibility points, and the changes can be computed from Kazhdan–Lusztig–Vogan polynomials. Finally one reaches ν = 0, where the representation is tempered and hence unitary.

Two problems impaired this idea: a standard module might have no invariant Hermitian form at all, and when it has one, the form is only canonical up to a real multiple.

The solution, carried out in the 2020 Adams/van Leeuwen/Trapa/Vogan paper, was to change the form. We replace the (antiholomorphic) involution σ of 𝔤 defining the real form, by a compact real form σc; and prove that any irreducible has a preferred σc-invariant form (called the c-form), characterized by being positive definite on the lowest K-types. Furthermore we show how to compute the signature of the σ-invariant form from the signature of the σc-invariant form. This is the algorithm that 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 the Hodge filtration (precisely, the K-multiplicities in each level) 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 in 2023 (arXiv:2309.13215).

The whole unitary dual: the FPP

Knowing how to test one representation is not the same as describing all of them. Each representation π has an infinitesimal character γ(π) (defined up to the action of the Weyl group W) in 𝔥*: the weight lattice of the algebraic group G, tensored with the complex numbers. Inside 𝔥* is the “canonical real form” 𝔥*ℝ: the weight lattice of G tensored with the real numbers. Knapp’s description of the invariant Hermitian forms showed that it is enough to parametrize irreducible unitary representations with infinitesimal character in 𝔥*ℝ: all others are obtained by unitary induction from such representations on Levi factors of real parabolics.

𝔥*ℝ has a locally finite hyperplane arrangement: the hyperplanes determined by affine coroots (α∨,n) with α∨ any coroot and n an integer. The hyperplane is {γ | ⟨γ,α∨⟩ = n}. 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.

[Still to come: the role of parabolic and cohomological induction in the unitary dual.]

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.