Equidistribution of joinings under off-diagonal polynomial flows of nilpotent Lie groups

Let $G$ be a connected nilpotent Lie group. Given probability-preserving $G$-actions $(X_i,\Sigma_i,\mu_i,u_i)$, $i=0,1,...,k$, and also polynomial maps $\phi_i:\mathbb{R}\to G$, $i=1,...,k$, we consider the trajectory of a joining $\lambda$ of the systems $(X_i,\Sigma_i,\mu_i,u_i)$ under the `off-diagonal' flow $$(t,(x_0,x_1,x_2,...,x_k))\mapsto (x_0,u_1^{\phi_1(t)}x_1,u_2^{\phi_2(t)}x_2,...,u_k^{\phi_k(t)}x_k).$$ It is proved that any joining $\lambda$ is equidistributed under this flow with respect to some limit joining $\lambda'$. This is deduced from the stronger fact of norm convergence for a system of multiple ergodic averages, related to those arising in Furstenberg's approach to the study of multiple recurrence. It is also shown that the limit joining $\lambda'$ is invariant under the subgroup of $G^{k+1}$ generated by the image of the off-diagonal flow, in addition to the diagonal subgroup.