I have been reading Armand Borel's book, Automorphic Forms on $SL_{2}(\mathbb{R})$. In his second chapter entitled "Review of $SL_{2}(\mathbb{R})$, Differential Operators and Convolution", Borel presents a standard bases for Lie(G) = { $M \in M_{2}(\mathbb{R}) \: | \: \text{trace}\: M = 0 $ }. Namley $$ H = \left[ {\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array} } \right], E = \left[ {\begin{array}{cc} 0 & 1 \\ 0 & 0\\ \end{array} } \right] , F = \left[ {\begin{array}{cc} 0 & 0 \\ 1 & 0 \\ \end{array} } \right]$$ He continues to say that if we consider the upper half complex plane $\mathbb{H}$ and if we denote $V^*$, the Vector Field on $\mathbb{H}$ defined by $V \in $ Lie(G), and since $G = SL_{2}(\mathbb{R})$ operates on the left on $\mathbb{H} = \frac{ SL_{2}(\mathbb{R})}{SO_2}$ , then the map $V \mapsto V^*$ extends to an anti-isomorphism of $\: \mathcal{U}(\mathbb{g})$ (the universal enveloping Algebra) onto an algebra of differential operators on $ \mathbb{H}$.
He then states that we may wirte $e^{tV}(z) = x(t) + i y(t) \: \: \: (z \in \mathbb{H})$, and the value of $V^*_z$ of $V^*$ is $$\frac{dx(t)}{dt} \bigg{|}_{t =0} \: \frac{\partial}{\partial x} + \frac{dy(t)}{dt} \bigg{|}_{t =0} \: \frac{\partial}{\partial y}.$$ From all this Borel claims that it is easy to check that $$E^* = \frac{\partial}{\partial x},\\ F^* = (y^2 - x^2) \frac{\partial}{\partial x} - 2xy \frac{\partial}{\partial y},\\ H^* = 2x \frac{\partial}{\partial x} + 2y \frac{\partial}{\partial y}$$ My question is, how would i even begin to verify what $E*, F^* $ and $H^*$ are? Any and all help would be much appreciated.