Unitary Representations of Lie Groups

Updated 14 November 2025

Unitary representations of Lie groups are strongly continuous homomorphisms mapping groups to unitary operators on Hilbert spaces, connecting geometry with analysis.
They employ methodologies like holomorphic induction, harmonic analysis, and dynamical systems to construct realizations with applications in quantum field theory and operator algebras.
Classification frameworks such as the Langlands classification and Dirac cohomology provide structured insights into both finite and infinite-dimensional cases.

A unitary representation of a Lie group is a strongly continuous homomorphism from the group into the unitary operators on a Hilbert space. Unitary representation theory of Lie groups is a foundational subject spanning harmonic analysis, complex geometry, operator algebras, and quantum field theory. It encompasses the construction, classification, realization, and analysis of representations for finite- and infinite-dimensional, as well as real, complex, and p-adic Lie groups. The topic connects geometric constructions (e.g., coadjoint orbits, Bergman spaces), functional analytic methods (e.g., Hilbert and Banach spaces of sections, semigroup theory), and deep algebraic structures (e.g., highest-weight modules, Dirac and Lie algebra cohomology).

1. Foundational Definitions and Classes of Unitary Representations

A unitary representation of a Lie group $G$ on a Hilbert space $\mathcal{H}$ is a homomorphism $\pi: G \to U(\mathcal{H})$ that is strongly continuous: for each $v\in \mathcal{H}$ , $g \mapsto \pi(g)v$ is continuous. The most prominent classes of Lie groups in this context include:

Nilpotent and solvable groups: All irreducible unitary representations of finite-dimensional nilpotent groups can be realized geometrically via the orbit method and ergodic dynamical systems (Beltita et al., 2015).
Semisimple and reductive groups: The Langlands classification parametrizes irreducible admissible $(\mathfrak{g},K)$ -modules, allowing a systematic enumeration of unitary duals (Adams et al., 2012).
Infinite-dimensional and Banach-Lie groups: Unitary representations are constructed via holomorphic/analytic induction, semigroup methods, and spectral analysis (Neeb, 2010, Neeb, 2010, Merigon et al., 2011).

The Hilbert module carries a natural action of the Lie algebra via the derived representation $d\pi(X) := \frac{d}{dt}|_{t=0}\pi(\exp tX)$ , generally realized by unbounded skew-adjoint operators with common invariant analytic vectors.

2. Geometric and Analytic Realizations

2.1 Induced Representations and Homogeneous Spaces

Holomorphic induction: Given a Banach–Lie group $G$ , split subroup $H$ , and a $G$ -invariant complex structure on $M=G/H$ , holomorphic bundle structures on $G\times_H V$ are classified by extensions of the representation of $H$ to a subalgebra of $\mathfrak{g}_\mathbb{C}$ . The associated holomorphic induction yields reproducing kernel Hilbert spaces of global sections, with $G$ acting by translation (Neeb, 2010).
Bergman spaces: For a real analytic unimodular $G$ embedded in its complexification $G_\mathbb{C}$ , the Bergman space $A^2(M_\varepsilon)$ of holomorphic $L^2$ -functions on a strongly pseudoconvex, $G$ -invariant domain $M_\varepsilon \subset G_\mathbb{C}$ provides a natural infinite-dimensional Hilbert space for a nontrivial unitary representation of $G$ . Unitarity follows from the $G$ -invariance of the volume form, and the representation is faithful if $G$ has no nontrivial compact subgroups (Sala et al., 2010).

2.2 Harmonic and Cohomological Constructions

Harmonic realizations: For classical low-rank groups—e.g., $\mathrm{SU}(2)$ , $\mathrm{SU}(1,1)$ , $\mathrm{SL}(2,\mathbb{C})$ , $E_2$ —all unitary irreducible representations can be realized as harmonic eigenfunctions on parameterizing manifolds equipped with invariant measures and Laplace–Beltrami operators proportional to the quadratic Casimir (Campoamor-Stursberg et al., 2014).
Positive definite analytic extensions: Given a local positive definite analytic function on a simply-connected Banach–Lie group, there exists a unique global analytic extension corresponding to a cyclic analytic vector in a unitary representation, constructed using GNS and analytic functional calculus (Neeb, 2010).

3. Classification and Structural Results

3.1 Langlands Classification for Real Reductive Groups

The Langlands classification provides a parametrization of all (admissible) irreducible $(\mathfrak{g},K)$ -modules by triples $(H,\gamma,R^+_{i\mathbb{R}})$ , where $H$ is a Cartan subgroup, $\gamma$ a character, and $R^+_{i\mathbb{R}}$ a choice of positive imaginary roots (Adams et al., 2012). The theory associates standard modules $I(\Gamma)$ with unique irreducible quotients $J(\Gamma)$ . The determination of unitarity proceeds via signature-tracking and Jantzen filtrations, using Kazhdan–Lusztig polynomials and signature-multiplicity algorithms.
For groups such as $\mathrm{SO}_0(2,m)$ , every irreducible unitary representation with non-zero $(\mathfrak{g},K)$ -cohomology must have trivial infinitesimal character (Wigner's theorem), leading to finiteness results for such representations and explicit formulas for their multiplicities and cohomological invariants (Pal et al., 2023).

3.2 Dirac Cohomology and Finiteness

For complex Lie groups, the set of irreducible unitary representations with nonzero Dirac cohomology decomposes into finitely many "scattered" representations and finitely many infinite "strings," the latter arising from (good range) cohomological induction from Levi subgroups (Ding et al., 2017).
Each such representation is detected via the Dirac operator $D$ and its cohomology, and the property of non-vanishing Dirac cohomology singles out a tractable, well-structured subset of the unitary dual.

3.3 Infinite-dimensional Extensions

For Banach–Lie or Fréchet–Lie groups, regularity and the existence of analytic or holomorphic vectors are critical for the integrability of representation-theoretic data from the Lie algebra to the group (Neeb, 2010, Merigon et al., 2011, Neeb, 2010).
For the Lie algebras of smooth or compactly supported sections with values in compact algebras, all bounded irreducible representations reduce to finite (or, in the noncompact case, possibly infinite) tensor products of "evaluation representations" at points, reducing the problem to the theory of UHF $C^*$ -algebras (Janssens et al., 2013).

4. Unitary Representations via Dynamical Systems and Cocycles

For nilpotent Lie groups and generalizations, unitary representations may be constructed via cocycles associated to ergodic, quasi-invariant actions on measure spaces. This yields a correspondence between ergodicity and irreducibility and recovers the orbit method and Stone–von Neumann theorem as special cases (Beltita et al., 2015).
In the infinite-dimensional Heisenberg group or abelian case, Gaussian quasi-invariant measures and Cameron–Martin theory facilitate explicit irreducible representations on $L^2$ -spaces with modulated translations and pointwise multiplicative cocycles.

5. Analytic and Holomorphic Techniques: Induction, Semigroups, and Cohomology

The machinery of analytic continuation from contraction semigroups (Olshanski semigroups) allows the construction of unitary representations of dual groups $G_c$ associated to symmetric Banach–Lie pairs $(G,\theta)$ , unifying approaches ranging from Weyl's unitary trick to the highest-weight module theory (Merigon et al., 2011). The central result is that a nondegenerate strongly continuous *-representation of the semigroup with dense smooth vectors yields, via analytic continuation, a unique unitary representation of $G_c$ that is precisely characterized as $(-iW)$ –semibounded.
Positive-energy and semibounded representations of Banach–Lie groups are always holomorphically induced from a representation of a subgroup fixing the generator of energy or its minimal spectral subspace, with the holomorphic induction functor providing an explicit realization in spaces of holomorphic sections (Neeb, 2010).

6. Special Constructions and Phenomena

6.1 Tensor Products and Intertwining

For rank-one groups (e.g., universal covering of $\mathrm{SU}(1,1)$ ), the tensor product of irreducible unitary representations decomposes into a direct integral over principal series plus a discrete sum of highest/lowest weight modules. The supports and intersection properties of smooth and analytic vectors exhibit new phenomena, such as the trivial intersection of the smooth vector subspaces in summands (Tomasini et al., 2011).

6.2 Projective and Central Extensions

Every smooth projective unitary representation of a 1-connected regular infinite-dimensional Lie group lifts uniquely to a linear representation of a central Lie group extension, which may be constructed explicitly using local cocycles and classified via group cohomology (Janssens et al., 2015). Classical examples include projective representations of abelian and Heisenberg groups, the Virasoro group (Diff $^+(S^1)$ ), and affine Kac–Moody algebras.

In sum, the theory of unitary representations of Lie groups—across finite and infinite dimensions—combines geometric, analytic, and algebraic methods to construct and classify representations, connect them to complex geometric and dynamical data, and analyze their structure via cohomology, invariants, and integrability criteria. The interplay between analytic vectors, induction via holomorphic or semigroup techniques, and explicit constructions in function spaces (Bergman, $L^2$ of orbits, holomorphic sections) is fundamental to the modern perspective. Classification results (Langlands, Dirac cohomology finiteness, evaluation representation classification) provide comprehensive frameworks, while dynamical systems and cocycle methods unify and generalize orbit- and functional-analytic constructions at both the group and algebra level.