The annihilator of the trivial sl(2)-module

Given: The Axiom of Choice, $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$ with basis $e,f,h$, the trivial module $\mathbb{C}$, and the ideal ${\mathrm{Ann}}_{U(\mathfrak{g})}\mathbb{C}$ of operators acting by $0$ on $\mathbb{C}$.

The trivial module is the unital module on which $e,f,h$ act by $0$; the corresponding algebra homomorphism is the augmentation $\epsilon \text{\hspace{0.17em}⁣}:U(\mathfrak{g})\to \mathbb{C}$ with $\epsilon (\mathfrak{g})=0$, so its kernel is the two-sided ideal generated by $\mathfrak{g}$: after killing $e,f,h$, every nonempty tensor word is zero and the quotient is spanned by $1$; $\epsilon (1)=1$ makes the remaining scalar quotient exactly $\mathbb{C}$ (The annihilator of a module over an enveloping algebra, The special linear Lie algebra sl_2).

Under AC the center is $\mathbb{C}[\mathrm{\Omega }]$, so $\ker {\chi }_0=(\mathrm{\Omega })$ and $U(\mathfrak{g})\ker {\chi }_0=(\mathrm{\Omega })U(\mathfrak{g})$; the trivial module is simple and nonzero, so its annihilator is primitive and the center acts on it through the central character ${\chi }_0$ with ${\chi }_0(\mathrm{\Omega })=0$, the Casimir eigenvalue of the trivial module (The central reduction of U(sl2) is simple away from the finite-dimensional central characters, The center of the enveloping algebra is polynomial on rank-many generators, The Axiom of Choice, Primitive ideals of an enveloping algebra, A primitive ideal determines a central character, Central character of a Lie algebra module, The quadratic Casimir eigenvalue on a highest-weight module is $(\lambda ,\lambda +2\rho )$).

$M(-2)$ is a simple Verma module by the Verma irreducibility criterion, since $-2+1=-1\mathrm{\notin }{\mathbb{Z}}_{>0}$; its highest vector $v_{-2}$ satisfies $h{v}_{-2}=-2v_{-2}$, and its central character is ${\chi }_{-2}$, which equals ${\chi }_0$ because the normalized Casimir $\mathrm{\Omega }=ef+fe+\frac{1}{2}{h}^2$ acts on both by $0$, by the eigenvalue formula $\mathrm{\Omega }\mapsto \lambda (\lambda +2)\mathrm{/}2$ and $-2\cdot 0\mathrm{/}2=0$ (Verma modules, The Verma irreducibility criterion from Shapovalov determinants, Highest-weight vectors and cyclic highest-weight modules, The central reduction of U(sl2) is simple away from the finite-dimensional central characters).

$U(\mathfrak{g})\ker {\chi }_{\lambda }\subseteq {\mathrm{Ann}}_{U(\mathfrak{g})}M(\lambda )$ for every $\lambda$ (The Verma annihilator contains the central-character ideal).