Integrability criterion for simple highest weight kac moody modules

If $L_A(\lambda )$ is integrable, its nonzero highest vector is supplied by F3, so F1 gives $\lambda \in P^{+}$. This implication excludes no boundary label: zero labels are permitted in $P^{+}$.

Conversely let $\lambda \in P^{+}$. F2 makes $N$ proper, so $N\subseteq N_{\lambda }$ by F3. Consequently the quotient map factors as a surjection $M_A(\lambda )\mathrm{/}N\to L_A(\lambda )$. If a simple generator has a power killing a vector upstairs, the same power kills its image. Every vector downstairs has a preimage, so both $e_i$ and $f_i$ act locally nilpotently on $L_A(\lambda )$.

The quotient also retains a weight decomposition. To see this directly, a submodule of a weight module contains every weight component of each of its elements: for finitely many distinct weights choose a Cartan element separating them and apply its Lagrange interpolation polynomials. Existence follows because finitely many nonzero linear functionals cannot vanish everywhere over the infinite field $\mathbb{C}$; explicitly their evaluations on ${\sum }_{a=1}^d{t}^{a-1}{h}_a$ in a finite Cartan basis exclude only finitely many $t$. The quotient is therefore the direct sum of quotient weight spaces. Applied to the kernel in 1.2, this supplies the remaining integrability condition. With both nilpotences established there, sufficiency follows. All arguments permit $N=0$, zero labels and one-dimensional quotients. The simple quotient itself is never zero by F3. The empty interpolation product is $1$ when only one weight occurs. No choice principle or symmetrizer is used in either direction.