The finite-dimensional sl2 quotient of a Verma module

Write $w_n=f^n{v}_m$. Induction from $[h,f]=-2f$, $[e,f]=h$, $e{w}_0=0$, and $h{w}_0=m{w}_0$ gives $h{w}_n=(m-2n)w_n$ and $e{w}_n=n(m-n+1)w_{n-1}$ for $n\ge 1$. In particular $e{w}_{m+1}=0$. The tail $W=\mathrm{span}\{w_n:n\ge m+1\}$ is stable under $e,f,h$ and equals $U({\mathfrak{s}\mathfrak{l}}_2)w_{m+1}$: it contains that generated submodule by invariance, while every tail vector is $f^k{w}_{m+1}$. Hence the quotient has the displayed $m+1$ surviving, distinct-weight PBW vectors.

Any nonzero submodule of the quotient contains a nonzero finite combination of its basis vectors. Their $h$-eigenvalues are distinct, so a finite interpolation polynomial in $h$ extracts some ${\bar{w}}_r$. Applying $e^r$ reaches a nonzero multiple of ${\bar{w}}_0$, because all coefficients $k(m-k+1)$ for $1\le k\le r\le m$ are nonzero. Applying $f$ then gives every displayed basis vector, so the quotient is simple. The unique-simple-quotient theorem identifies its kernel with $J(m)$ and the quotient with $L(m)$.