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

We prove that $L(SL_2(textbf{k}))$ is a maximal Haagerup von Neumann subalgebra in $L(textbf{k}^2rtimes SL_2(textbf{k}))$ 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 $phisim phi$, where $phi$, $phiin widehat{textbf{k}^2}$ and $phi(x, y):=phi(-x, -y)$ 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.