Let $\Pi_{\lambda}$ be the Schrodinger representations on the Heisenberg group $H^n=\Bbb C^n\times\Bbb R$. For $\phi\in L^2(\Bbb R^n)$, we have $$\Pi_{\lambda} (x,y,t)\phi(\xi)=e^{i\lambda t} e^{i\lambda(x.\xi+\frac{1}{2}x.y)}\phi(\xi+y). $$ with $x.\xi=\sum^n_{i=1}x_i\xi_i$ and $x.y=\sum^n_{i=1}x_iy_i$ and $(x,y)\in\Bbb R^{n}\times\Bbb R^n$ with $t,\lambda\in\Bbb R^*$.
Put $\Pi_{\lambda}(X)\phi=\frac{d}{dt}\Big|_{t=0}\Big(\Pi_\lambda\big((e^{tX}\big)\phi \Big).\ $ and $\ X_j=\frac{\partial}{\partial x_j}-\frac{1}{2}y_j\frac{\partial}{\partial t}$. Why $$\Pi_{\lambda}(X_j)\phi(\xi)=-i\lambda \xi_j\phi(\xi)\quad? $$ Thanks in advance