The Crowley-Nordstr\"om invariants of $G_2$-Structures on Aloff--Wallach Spaces

We study homogeneous $G_2$-structures on the Aloff-Wallach spaces $N_{k,l}=SU(3)/S^1_{k,l},$ and compute the Crowley--Nordström invariants associated to these structures. In the case where the first Pontryagin class is rationally trivial, we derive a new intrinsic formula for the $\xi$ invariant. Using Goette's expressions for the $\eta$ invariants of homogeneous spaces, we obtain explicit formulas for the invariants of homogeneous $G_2$-structures on $N_{k,l}$. In particular, we prove that \[ \bar{\nu}(\varphi)=0, \qquad \xi(\varphi)=\frac{3}{2}kl(k+l). \]
As a consequence, we obtain examples of manifolds carrying nearly-parallel $G_2$-structures that lie in distinct connected components of the moduli space of $G_2$-structures.