Let
- $X_{j}=\frac{\partial}{\partial x_{j}}-\frac{1}{2}y_{j}\frac{\partial}{\partial t}$, $j=1,2,\dots,n$
- $Y_{j}=\frac{\partial}{\partial y_{j}}+\frac{1}{2}x_{j}\frac{\partial}{\partial t},j=1,2,\dots,n$.
The Schrõdinger representations $\pi_\lambda, \lambda\in\Bbb R^*$ are realised on $\phi\in L^2(\Bbb R^n)$ and given explicitly by $$\pi_\lambda(x,y,t)\phi(\xi)=e^{i\lambda t} e^{i\lambda(x.\xi+\frac{1}{2}x.y)}\phi(\xi+y).$$ with $(x,y)\in\Bbb R^n\times\Bbb R^n$ and $t\in\Bbb R$
Put
$$
\pi^*(X_j) u=\left.\frac{d}{d s}\right|_{s=0} \pi(\exp s X_j) u
$$
and
$$
\pi^*(Y_j) u=\left.\frac{d}{d s}\right|_{s=0} \pi(\exp s Y_j) u
$$
where $u\in H$ for some Hilbert space.
Question. Why
$$
\pi_\lambda^*\left(X_j\right)=i \lambda \xi_j, \quad \pi_\lambda^*\left(Y_j\right)=\frac{\partial}{\partial \xi_j} \;?
$$
