Quaternionic Heisenberg Group

• Quaternionic Heisenberg group is a two-step nilpotent Lie group that extends the classical Heisenberg group with a noncommutative quaternionic structure and a natural sub-Riemannian geometry.
• Its algebraic and geometric framework supports advanced harmonic analysis, representation theory, and functional inequalities applicable in pluripotential theory and PDEs.
• The group underpins models in mathematical physics and automorphic forms, offering explicit transforms, control theoretic insights, and robust analytic tools.

The quaternionic Heisenberg group is a canonical example of a two-step nilpotent Lie group equipped with a quaternionic (noncommutative) structure. It appears as a central geometric and analytic object in sub-Riemannian geometry, harmonic and representation theory, pluripotential theory, automorphic forms, geometric group theory, and mathematical physics. Its algebraic, geometric, and analytic properties extend and generalize the classical (complex) Heisenberg group, incorporating the richer algebraic structure of the quaternions.

1. Algebraic and Geometric Structure

The quaternionic Heisenberg group of "rank" $n$—typically denoted by $\mathbb{H}_q$, $G(\mathbb{H})$, or $H$—is the 2-step stratified nilpotent Lie group with underlying manifold $\mathbb{H}^n \times \text{Im} \,\mathbb{H}$, i.e., $\mathbb{R}^{4n} \times \mathbb{R}^3$. Its group law is given by

$$(q',w') \cdot (q, w) = \left(q'+q,\, w'+w + 2\,\text{Im}(q'\overline{q}) \right),$$

where $q,q'\in \mathbb{H}^n$, $w,w'\in \text{Im}\,\mathbb{H} \cong \mathbb{R}^3$, and $\overline{q}$ denotes quaternionic conjugation. The center of the group is the subspace $\text{Im}\,\mathbb{H}$. The standard quaternionic Heisenberg group of dimension $7$ arises when $n=1$. The group admits a $\mathbb{Z}$-grading of its Lie algebra, with horizontal layer (first stratum) of dimension $4n$ and vertical (center) of dimension $3$. The Carnot group structure equips it with natural families of dilations and a rich sub-Riemannian geometry.

2. Sub-Riemannian, Contact, and Metric Structures

Sub-Riemannian structures on the quaternionic Heisenberg group are specified by a left-invariant distribution $H$ of dimension $4n$ (spanned by the first stratum), equipped with an inner product. For $n=1$, an explicit left-invariant frame is provided via vector fields:

$$\begin{aligned} e_1 &= \partial_{b_1} + b_2 \partial_{a_2} + b_3 \partial_{a_3} + b_4 \partial_{a_4},\ e_2 &= \partial_{b_2} - b_1 \partial_{a_2} + b_4 \partial_{a_3} - b_3 \partial_{a_4},\ e_3 &= \partial_{b_3} - b_4 \partial_{a_2} - b_1 \partial_{a_3} + b_2 \partial_{a_4},\ e_4 &= \partial_{b_4} + b_3 \partial_{a_2} - b_2 \partial_{a_3} - b_1 \partial_{a_4}, \end{aligned}$$

with non-holonomic bracket relations $[e_i, e_j] = c_{ij}e_k$, generating the center. This distribution is bracket generating of type $[4n, 4n+3]$. The horizontal distribution $H$ is endowed with a triple of $\text{SO}(3)$-related almost complex structures, providing a quaternionic (4n+3)-dimensional contact geometry. The fundamental 4-form

$$\Omega = \omega_1 \wedge \omega_1 + \omega_2 \wedge \omega_2 + \omega_3 \wedge \omega_3$$

reflects the quaternionic contact structure. There exists a canonical metric connection with skew torsion preserving the quaternionic contact structure and horizontal-vertical splitting (Agricola et al., 2015).

In dimension seven, the group admits a cocalibrated $G_2$ structure, and its canonical connection coincides with the characteristic connection of this $G_2$-structure, making it the unique 7-dimensional nilpotent model with an integrable left-invariant quaternionic contact structure (Conti et al., 2011).

3. Harmonic Analysis, Representation Theory, and Multiplier Theory

The group's noncommutative structure leads to a rich harmonic analysis. The convolution algebra of $K$-bi-invariant functions, for a compact subgroup $K$ of automorphisms, forms a commutative Gelfand pair $(K, \mathbb{H}_q)$. The spherical Fourier transform is constructed via a family of bounded spherical functions:

$$\phi_{\lambda,n}(u,t) = \frac{1}{n+1} e^{i \langle \lambda, t\rangle} e^{-|\lambda||u|^2} \mathscr{L}_n^{(1)} (2|\lambda||u|^2),$$

where $\lambda \in \text{Im}\, \mathbb{H}\setminus \{0\}$ and $n$ indexes the homogeneous polynomial degree (Ariyo et al., 3 Sep 2025). The spherical Fourier multiplier operator $T_{\mathfrak{m}}$ acts by

$$(\mathfrak{F}T_{\mathfrak{m}}f)(\lambda, n) = \mathfrak{m}(\phi_{\lambda, n}) \hat f(\lambda, n),$$

with boundedness on $L^p$ spaces ensured by Calderón-Zygmund kernel estimates. Similar analyses underpin the theory of wavelet transforms, radial function decompositions, and the Radon transform, including function space identifications (Semyanistyi–Lizorkin spaces) necessary for bijectivity of the Radon transform and for inversion formulas (He et al., 2011).

Further, the Weyl transform and uncertainty principles extend to the quaternionic setting: any function supported on a set of finite measure whose Weyl transform is finite rank must vanish identically (Ghosh et al., 2019).

4. Analysis, Pluripotential Theory, and PDEs

The quaternionic Heisenberg group serves as the model space for subelliptic and pluripotential analysis beyond both Euclidean and classical Heisenberg settings. Sharp forms of the Folland–Stein Sobolev inequality and Hardy–Littlewood–Sobolev inequality hold, with explicit best constants, extremal functions, and characterization of equality achieved up to group symmetries (Ivanov et al., 2010, Christ et al., 2014). The qc Yamabe problem is completely solved on both the (compact) quaternionic sphere and the (noncompact) Heisenberg group, including a full classification of extremals and uniqueness theorems akin to Obata’s result (Ivanov et al., 2015).

In quaternionic pluripotential theory, the group admits a well-behaved quaternionic Monge–Ampère operator:

$\det(Q_i Q_j u + 8 \delta_{ij} \partial_t u),$

where $Q_i$ are tangential Cauchy–Fueter operators. Associated first-order differential operators $d_0$, $d_1$ satisfy $d_0 d_1 = -d_1 d_0$, in contrast to the bad behavior of the complex tangential Cauchy–Riemann operator on the Heisenberg group. This allows for the definition of closed positive currents, a robust Chern-Levine-Nirenberg estimate, the existence of Monge-Ampère measures for continuous plurisubharmonic functions, and a minimum principle (Wang, 2019). These analytic structures are critical for the paper of sub-Riemannian potential theory and boundary value problems.

5. Automorphic Forms, Arithmetic Structures, and Homogeneous Dynamics

The quaternionic Heisenberg group realizes the boundary geometry of quaternionic hyperbolic spaces and arises as the unipotent radical of Heisenberg parabolic subgroups of higher-rank groups (e.g., quaternionic $E_8$). In automorphic representation theory, explicit Fourier–Jacobi expansions are constructed for automorphic forms generating quaternionic discrete series, where nonconstant terms are entirely described via generalized Whittaker models associated to the Heisenberg parabolic (Narita, 12 Jan 2025, Pollack, 2018). The Fourier coefficients are indexed by characters of the unipotent radical, and the spectral contributions of the Jacobi group can be sharply identified.

Sharp counting and equidistribution theorems for "rational" points, arithmetic chains, and orbits under arithmetic subgroups are established in these nilpotent groups, generalizing and extending classical results such as Mertens's formula and Neville equidistribution to the quaternionic context (Parkkonen et al., 2019, Parkkonen et al., 2020). These results rely on integrating harmonic analysis, invariant geometry (e.g., Cygan metrics), ergodic theory of flows, and arithmetic of quaternion algebras.

6. PDE, Geometric Control, and Mathematical Physics

The quaternionic Heisenberg group underlies highly symmetric models for geometric analysis and physics. In dimension 7, its algebraic structure can be realized explicitly via matrices in $\mathfrak{sp}(2,1)$. Local control and sub-Riemannian geodesics are described by a left-invariant frame, and the associated control system and Hamiltonian dynamics are integrable with explicit cut times determined by group symmetries (Eisner et al., 13 Apr 2024). This has a direct impact on the geometry of the space of curves and on optimal control theory.

In mathematical physics, the partial Fourier transform of the sub-Laplacian (the Landau operator) of the quaternionic Heisenberg group models the Hamiltonian of two harmonic oscillators in a uniform magnetic field. These Landau operators provide the spectral framework for quantum Hall effect models, noncommutative geometry, and quantum gauge theory (Zinoun et al., 2010).

7. Further Directions, Function Theory, and Open Problems

Sophisticated constructions—such as explicit Cauchy–Szegö kernels for Hardy spaces, Calderón–Zygmund theory for singular integrals, descriptions of BMO and VMO spaces, and harmonic analysis associated to Gelfand pairs—continue to expand the analytic theory on the quaternionic Heisenberg group (Chang et al., 2019, Ariyo et al., 3 Sep 2025). Fundamental results on function extension (Hartogs phenomenon for $k$-Cauchy–Fueter complexes), minimum principles for Monge–Ampère equations, and sharp filling volume inequalities (Gruber, 2017) underscore the group's centrality in both analysis and geometry.

The quaternionic Heisenberg group thus plays the role of the flat, noncommutative, and non-Riemannian model space for quaternionic and sub-Riemannian geometry, harmonic analysis, representation theory, automorphic forms, arithmetic geometry, and certain domains in mathematical physics. Its structure and the evolving analytic machinery built upon it drive ongoing research in several branches of mathematics.