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

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.