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 $\phi\sim \phi'$, where $\phi$, $\phi'\in \widehat{ \textbf{ k}^2}$ 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.
Categories: math.OA