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 $\Gamma$ be a torsion-free finitely generated nilpotent group and let $G$ be its Malcev completion. For an irreducible unitary representation $\pi_l$ of $G$ attached to a rational Kirillov parameter $l\in\mathfrak{g}_{\mathbb Q}^{*}$, we prove an exact restriction theorem for $\pi_l|_\Gamma$. 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 $\Gamma$. 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.