I am studying the quantitative decay of matrix cofficients from the book Howe-Tan. I have the following doubt;
They analyze unitary representations of $SL(2,\mathbb{R})\rtimes \mathbb{R}^2$ with no $\mathbb{R}^2$ fixed vectors and showed that their matrix coefficients restricted to $SL(2,\mathbb{R})$ are in $L^{2+\epsilon}$ for all $c >0 $. I have gone through the rough sketch of the proof but it's bit unclear to me why particularly the $SL(2,\mathbb{R})\rtimes\mathbb{R}^2$ has been chosen, I mean what could go wrong if someone works with some other embedded subgroup in $SL(n,R)$ where $n\geq 3$ then restricting the representation to $SL(2,\mathbb{R})$. Which step would crucially fail if one does so? Any insight is very much appreciated.