Maximal Haagerup subalgebras in $L(\mathbb{Z}^2\rtimes SL_2(\mathbb{Z}))$

We prove that $L(SL_2(\textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(\textbf{k}^2\rtimes SL_2(\textbf{k}))$ for $\textbf{k}=\mathbb{Q}$. Then we show how to modify the proof to handle $\textbf{k}=\mathbb{Z}$. The key step for the proof is a complete description of all intermediate von Neumann subalgebras between $L(SL_2(\textbf{k}))$ and $L^{\infty}(Y)\rtimes SL_2(\textbf{k})$, where $SL_2(\textbf{k})\curvearrowright Y$ denotes the quotient of the algebraic action $SL_2(\textbf{k})\curvearrowright \widehat{\textbf{k}^2}$ by modding out the relation $φ\sim φ'$, where $φ$, $φ'\in \widehat{\textbf{k}^2}$ and $φ'(x, y):=φ(-x, -y)$ for all $(x, y)\in \textbf{k}^2$. As a by-product, we show $L(PSL_2(\mathbb{Q}))$ is a maximal von Neumann subalgebra in $L^{\infty}(Y)\rtimes PSL_2(\mathbb{Q})$; in particular, $PSL_2(\mathbb{Q})\curvearrowright Y$ is a prime action, i.e. it admits no non-trivial quotient actions.