The $\bar{\nu}$-Invariant of $G_2$-Structures on Aloff-Wallach Spaces

We compute the $\bar{\nu}$-invariant of homogeneous nearly-parallel $G_2$-structures on Aloff--Wallach spaces $N_{k,l} = SU(3)/S^1_{k,l}$. Using Goette's formulas for the $\eta$-invariants of homogeneous spaces, we derive an explicit expression for $\bar{\nu}$ in terms of representation-theoretic data and show that for the two homogeneous nearly-parallel structures $\varphi^\pm$ on $N_{k,l}$ one has $$\bar{\nu}(\varphi^\pm) = \mp 41.$$ Additionally, we compare the $\bar{\nu}$-invariants of the nearly-parallel $G_2$-structures arising from the 3-Sasakian structure.