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Let $\Gamma$ be a lattice in a simply connected nilpotent Lie group $G$.Given an infinite measure preserving action $T$ of $\Gamma$ and a "direction"in $G$ (i.e. an element $\theta$ of the projective space $P(\goth g)$ of theLie algebra $\goth g$ of $G$), some notions of recurrence and rigidity for $T$along $\theta$ are introduced. It is shown that the set of recurrent directions$\Cal R(T)$ and the set of rigid directions for $T$ are both $G_\delta$. In thecase where $G=\Bbb R^d$ and $\Gamma=\Bbb Z^d$, we prove that (a) for each$G_\delta$-subset $\Delta$ of $P(\goth g)$ and a countable subset$D\subset\Delta$, there is a rank-one action $T$ such that $D\subset\CalR(T)\subset\Delta$ and (b) $\Cal R(T)=P(\goth g)$ for a generic infinitemeasure preserving action $T$ of $\Gamma$. This answers partly a question froma recent paper by A.~Johnson and A.~{\c S}ahin. Some applications to thedirectional entropy of Poisson actions are discussed. In the case where $G$ isthe Heisenberg group $H_3(\Bbb R)$ and $\Gamma=H_3(\Bbb Z)$, a rank-one$\Gamma$-action $T$ is constructed for which $\Cal R(T)$ is not invariant underthe natural "adjoint" $G$-action.

Directional recurrence and directional rigidity for infinite measure preserving actions of nilpotent lattices