Bader–Muchnik irreducibility conjecture for quasi-regular representations

Let $G$ be a locally compact group, let

be a spread-out probability measure on $G$, and let the associated

-boundary be a measurable $G$-space equipped with its boundary measure. The corresponding quasi-regular representation of $G$ on the associated $L^2$-space is the unitary representation induced by the boundary action.

Bader–Muchnik irreducibility conjecture. For every locally compact group $G$ and every spread-out probability measure

on $G$, the quasi-regular representation associated with a

-boundary of $G$ is irreducible.

This conjecture connects the irreducibility of boundary quasi-regular representations with ergodic properties of random walks and boundary actions. The supplied source attributes it to Bader and Muchnik; no resolution is indicated here.