Boundaries and equivariant maps for ergodic groupoids

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman \cite{BF14} for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu)$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman \cite{BF:Unpub}, we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu)$. In this way, given any measurable representation $\rho:\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm=\mathbf{L}_\pm(\kappa)$ for some $\kappa$-subgroups $\mathbf{L}_\pm<\mathbf{H}$. In the particular case when $\kappa=\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.