Nilpotent Floquet–Bloch prime orbit asymptotic conjecture

Let $\varphi_t$ be a smooth, transitive and weakly mixing Anosov flow on a compact manifold $N$. Let $\Gamma$ be a finitely generated torsion-free nilpotent group, let $\Phi:\pi_1(N)\to\Gamma$ be surjective, and let $\alpha$ be a conjugacy class of a central element of $\Gamma$. Set $H=\Gamma/[\Gamma,\Gamma]$, let $b$ be the rank of $H$, and let $H^\dagger$ be the dual of $H$. Let $G$ be the Malcev completion of $\Gamma$, and let $d$ be the polynomial growth order of $\Gamma$. Nilpotent Floquet–Bloch asymptotic conjecture. If the winding cycle $\Psi$ vanishes on $H^\dagger$, then

$$\pi(x,\Phi,\alpha)\sim\frac{Ce^{hx}}{x^{1+d/2}},$$

where

$$C=\frac{1}{(2\pi)^{b/2}h\operatorname{vol}(\widehat{H})}\zeta_H(d/2),$$

and $\zeta_H(d/2)$ is the special value at $s=d/2$ of the spectral zeta function of a hypo-elliptic operator associated with some irreducible unitary representations of $G$. The group $G$ is the simply connected nilpotent Lie group containing $\Gamma$ as a lattice subgroup. This conjecture proposes the nilpotent analogue of the abelian asymptotic for closed orbits, with the polynomial growth order determining the power of $x$; its status is not resolved in the supplied source.