2026/07/13 by Atsushi Katsuda
Mathematics · #math.RT #math.GR #math.OA
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 πl of G attached to a rational Kirillov parameter l∈\mathfrakg\mathbb Q*, we prove an exact restriction theorem for πl|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.