Exponential multiple mixing for commuting automorphisms of a nilmanifold

Let $l\in \mathbb {N}_{\ge 1}$ and $\alpha : \mathbb {Z}<^>l\rightarrow \text {Aut}(\mathscr {N})$ be an action of $\mathbb {Z}<^>l$ by automorphisms on a compact nilmanifold $\mathscr{N}$. We assume the action of every $\alpha (z)$ is ergodic for $z\in \mathbb {Z}<^>l\smallsetminus \{0\}$ and show that $\alpha $ satisfies exponential n-mixing for any integer $n\geq 2$. This extends the results of Gorodnik and Spatzier [Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), 127-159].