I'm studying the Schrodinger representation for the Heisenberg Group in Foundations of Harmonic Analysis by Rottensteiner (https://services.phaidra.univie.ac.at/api/object/o:1265111/get)
Studying this document, I have not been able to understand two facts regarding the representations. I attach the images.
For simplicity, I will consider $n=1$.
\begin{align} (t,p,q)\mapsto i(tI+qx+hD) \end{align} where $D=-i\partial_x$
Question 1. Why are antisymmetric operators on the Schwartz space considered in this representation? Is it because if $f\in \mathcal{S}(\mathbb{R})$ then $xf(x),\, Df(x),\, \in\mathcal{S}(\mathbb{R})$ ?
Question 2. Why is the space of bounded linear operators on $L^2(R)$ considered? The exponential $\mathrm{e}^{(i(htI+qx+hpD)}$ is bounded on $L^2(\mathbb{R})$? Thank you.