A point-supported SL2 module gives the shifted Verma module

On the affine chart $U_0$, the module supported at $z=0$ is the Weyl algebra quotient $\mathbb{C}[z]⟨{\mathrm{\partial }}_z⟩\mathrm{/}\mathbb{C}[z]⟨{\mathrm{\partial }}_z⟩z$. Ordered monomials ${\mathrm{\partial }}_z^n{z}^m$ show that its global sections have basis $\delta ,{\mathrm{\partial }}_z\delta ,{\mathrm{\partial }}_z^2\delta ,\dots$; sections vanish off the support point. The relation $z\delta =0$ gives $z{\mathrm{\partial }}_z\delta =-\delta$. Therefore [F1] gives $e\delta =0,h\delta =(c+1-2)\delta =(c-1)\delta ,f\delta =-{\mathrm{\partial }}_z\delta \mathrm{.}$

The universal property in [F2] gives a surjection from the Verma module of highest weight $(c-1)\rho$ to $\mathrm{\Gamma }(X,{\delta }_{x_0})$, sending its highest vector to $\delta$. Its PBW basis is $f^{n}v$ for $n\ge 0$, and $f^n\delta =(-1{)}^n{\mathrm{\partial }}_z^n\delta$ is the basis in step 1.1. The map is therefore an isomorphism.

Since $e\delta =0$ and $h\delta =(c-1)\delta$, the Casimir acts by $ef\delta +fe\delta +\frac{1}{2}{h}^2\delta =((c-1)+\frac{1}{2}(c-1{)}^2)\delta =\frac{1}{2}(c^2-1)\delta \mathrm{.}$ In the chosen PBW order, $ef=fe+h$ and the Harish–Chandra projection of $C$ is $h+\frac{1}{2}{h}^2$. Evaluating it at $\kappa -\rho =(c-1)\rho$ gives the same scalar by [F4]. Thus the module has the exact central character stated.