An extension of the Floquet-Bloch theory to nilpotent groups and its applications

TL;DR

This work extends Floquet-Bloch theory from abelian covers to torsion-free discrete nilpotent groups by leveraging Malcev completions and a novel branching framework that links finite-dimensional lattice representations to infinite-dimensional unitary representations of associated nilpotent Lie groups. The authors develop a Plancherel-type inversion supported on finite-dimensional representations, enabling rigorous perturbative analysis around the trivial representation and practical approximation via the Malcev completion; this underpins spectral decompositions of twisted Laplacians and leads to unified asymptotics for heat kernels and prime geodesics on nilpotent covers. The framework yields a semi-classical expansion for the Harper operator and connects spectral data to hypoelliptic operators tied to nilpotent Lie structures, with concrete results for the Heisenberg and Engel groups and systematic pathways for general nilpotent groups. The paper also outlines future directions, including extensions to broader hyperbolic flows, noncompact geometries, Riemann-Hilbert/opers connections, and potential links to knot invariants and infinite-number theory phenomena, highlighting the broad applicability of noncommutative Floquet-Bloch ideas in geometric analysis and spectral theory.

Abstract

We develop a generalized Floquet-Bloch theory for discrete torsion-free nilpotent groups by exploiting their Malcev completions. Our main result is a branching formula that relates finite-dimensional representations of a discrete nilpotent lattice to infinite-dimensional unitary representations of its simply connected nilpotent Lie group. This generalization enables to extend the following two classical asymptotic problems (i) a Chebotarev density analogue for prime closed geodesics on compact negatively curved manifolds with nilpotent covers, and (ii) long time asymptotic expansions of heat kernels on such coverings to the nilpotent setting. As a by-product, we derive a semi-classical expansion for the Harper operator, presenting an alternative to mathematical justification of Wilkinson's formula by Helffer-Sjöstrand. We conclude by proposing several avenues for future work: extensions to general hyperbolic flows and noncompact manifolds (in particular knot complements and related quasi-morphisms), connections to modified Riemann-Hilbert problems and opers. Furthermore, we give a brief comment on asymptotic behavior of knot invariants and infinite extensions in number theory.

Figures (5)

• Figure 1: Hofstadter's butterfly (created by Hisashi Naito). The vertical axis represents the value of $\theta$ corresponding to the strength of a magnetic field or the magnetic flux density in the interval $[0,2\pi]$, and the horizontal axis represents the range of the spectrum of $H_\theta$, which is a subset of the closed interval $[-4,4]$.
• Figure 2: The Cayley graph $X$ of $({\rm Heis}_3(\mathbb{Z}), \{u,v\})$ created by Satoshi Ishiwata: This is a regular graph of degree four embedded in $\mathbb{R}^3$. Solid lines express the edges of the graph, but the dotted line represents the $z$-axis of the orthogonal coordinates of $\mathbb{R}^3$, which does not mean the edges.
• Figure 3: Relation between twisted Laplacian $\Delta_{\rho_{\tiny{{\rm fin}}, x}}$ and Harper operator $H_{\theta}$ in terms of covering of graphs.
• Figure 4: Double well potential
• Figure 5: Joku-ikou no Genri

Theorems & Definitions (5)

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• proof : Proof of Theorem \ref{['chenpi1derham']}