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, $S=(s_1,\mathrm{.}\mathrm{.}\mathrm{.},s_k)$ is a generating $k$-tuple of group elements and $a=({\alpha }_1,\mathrm{.}\mathrm{.}\mathrm{.},{\alpha }_k)$ is a $k$-tuple of reals in the interval $(0,2)$. The symmetric measure ${\mu }_{S,a}$ is supported by $S^{\ast }=\{s_i^m,1\le i\le k,m\in \mathbb{Z}\}$ and gives probability proportional to $$(1+m{)}^{-{\alpha }_i-1}$$ to $s_i^{\pm m}$, $i=1,\mathrm{.}\mathrm{.}\mathrm{.},k,$ $m\in \mathbb{N}$. 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 $G_j=[G_{j-1},G]$, $G_1=G$.