Invariant random subgroups of semidirect products

We study invariant random subgroups (IRSs) of semidirect products $G=A\rtimes \unicode[STIX]{x1D6E4}$. In particular, we characterize all IRSs of parabolic subgroups of $\text{SL}_{d}(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$ are either of the form $\mathbb{R}^{d}\rtimes K$ for some IRS of $\text{SL}_{d}(\mathbb{R})$, or are induced from IRSs of $\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$, where $\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$ is a lattice.