Invariant random subgroups of semidirect products

We study invariant random subgroups (IRSs) of semidirect products $G = A \rtimes Γ$. In particular, we characterize all IRSs of parabolic subgroups of $\mathrm{SL}_d(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^d \rtimes \mathrm{SL}_d(\mathbb{R})$ are either of the form $\mathbb{R}^d \rtimes K$ for some IRS of $\mathrm{SL}_d(\mathbb{R})$, or are induced from IRSs of $Λ\rtimes \mathrm{SL}(Λ)$, where $Λ< \mathbb{R}^d$ is a lattice.

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