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 $N_{k,l}=SU(3)\mathrm{/}{S}_{k,l}^1$. 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 ${\phi }^{\pm }$ on $N_{k,l}$ one has $$\bar{\nu }({\phi }^{\pm })=\mp 41.$$ Additionally, we compare the $\bar{\nu }$-invariants of the nearly-parallel $G_2$-structures arising from the 3-Sasakian structure.