The semisimple Lie algebra $\mathfrak{g}:=sl(2,\mathbb{R})$ has the Casimir operator $$\Omega:=(1/2)h^2+ef+fe$$ with respect to the standard basis $(h,e,f)$. For an irreducible unitary representation $\pi$ of $G:=SL(2,\mathbb{R})$ the global character $\Theta_\pi$ satisfies the differential equation $$\Omega.\Theta_\pi=\chi(\Omega) \cdot \Theta_\pi$$ when $\Omega \in Z(U\mathfrak{g})$ is considered a left-invariant fundamental vector field on $G$. On the regular points of $G$ the distribution $\Theta_\pi$ is given by an analytic function, see the explicit form in "Knapp: Representation Theory of Semisimple Groups, Prop. 10.14".
My question:
Which explicit form has the differential equation above for the analytic function $\Theta_\pi$?
Which differential operators in explicit form define $\Omega$ on the two Cartan subgroups and their conjugates?