Random walks on nilpotent groups driven by measures supported on powers of generators

We study the decay of convolution powers of a large family $\mu_{S,a}$ of measures on finitely generated nilpotent groups. Here, $S=(s_1,...,s_k)$ is a generating $k$-tuple of group elements and $a= (\alpha_1,...,\alpha_k)$ is a $k$-tuple of reals in the interval $(0,2)$. The symmetric measure $\mu_{S,a}$ is supported by $S^*=\{s_i^{m}, 1\le i\le k,\,m\in \mathbb Z\}$ and gives probability proportional to $$(1+m)^{-\alpha_i-1}$$ to $s_i^{\pm m}$, $i=1,...,k,$ $m\in \mathbb N$. We determine the behavior of the probability of return $\mu_{S,a}^{(n)}(e)$ as $n$ tends to infinity. This behavior depends in somewhat subtle ways on interactions between the $k$-tuple $a$ and the positions of the generators $s_i$ within the lower central series $G_{j}=[G_{j-1},G]$, $G_1=G$.