Noncommutative Floquet--Bloch Theory for Nilpotent Groups:\ Representation-Theoretic Foundations

Abstract

Classical Floquet--Bloch theory decomposes abelian periodic problems over the character torus of the lattice. For nonabelian nilpotent lattices, the non--type~I obstruction rules out a comparable parametrization of the full unitary dual. We do not attempt to remove this obstruction. Instead, we construct an exact Bloch-type replacement on the representation-theoretic part of the theory which is visible from rational Kirillov data and from finite-dimensional rational fibers. Let \(Γ\) be a torsion-free finitely generated nilpotent group and let \(G\) be its Malcev completion. For an irreducible unitary representation \(π_l\) of \(G\) attached to a rational Kirillov parameter \(l\in\mathfrak g_{\mathbb Q}^{*}\), we prove an exact restriction theorem for \(π_l|_Γ\). The branching is first described by induced representations attached to rational polarizations. On the rational odd locus relevant to finite-dimensional representations, it further decomposes into finite-dimensional irreducible representations of \(Γ\). On these finite-dimensional rational fibers we construct a positive finitely additive Plancherel measure. It gives Fourier inversion and normalized trace identities for nilpotent lattices, recovering Pytlik's formula in the discrete Heisenberg case.