On $L^1$-$L^2$ dichotomy for flat symmetric spaces

For rank 1 flat symmetric spaces, continuous orbital measures admit absolutely continuous convolution squares, except for Cartan type AI. Hence $L^1$-$L^2$ dichotomy for these spaces holds true in parallel to the compact and non-compact rank 1 symmetric spaces. We also study $L^1$-$L^2$ dichotomy for flat symmetric spaces of ranks $p=2,3$ of type AIII, i.e.\ associated with $SU(p,q)/S(U(p)\times U(q))$ where $q\geq p$. For continuous orbital measures given by regular points $L^1$-$L^2$ dichotomy holds. We study such measures given by certain singular points when $p=2$, and show that $L^1$-$L^2$ dichotomy fails. This is the first time such results are observed for any type of symmetric spaces of rank 2.