Every nonzero weak positive-integral-level simple affine module contains a simple ordinary submodule

For $u\in M\setminus \{0\}$, put $E(u)=\mathrm{span}\{a_{n}u:a\in \mathcal{B},n>0\}$. Lower truncation and finiteness of $\mathcal{B}$ make $E(u)$ finite-dimensional. Choose nonzero $u$ minimizing $d(u)=\dim E(u)$. If $d(u)=0$, then $u$ is already killed by all positive currents. Otherwise let $k>0$ be the largest index for which $a_{k}u\mathrm{\ne }0$ for some $a\in \mathcal{B}$, choose such $a$, and let $r\ge 1$ be maximal with $a_k^{r}u\mathrm{\ne }0$. Set $v=a_k^{r}u$.

In the remaining case $d(u)>0$, for every $b\in \mathcal{B}$ and $n>0$, expand $b_n{a}_k^{r}u$ by repeatedly commuting $b_n$ past $a_k$. The term with no commutator is $a_k^r{b}_{n}u$. Every other term contains a current mode of index $n+jk>k$ for some $j\ge 1$. Move that mode right past the remaining $a_k$ factors; each new commutator has still larger positive index, so induction on the number of remaining factors reduces every term to one with a mode of index $>k$ acting directly on $u$, which vanishes by maximality of $k$. Thus $b_{n}v=a_k^r{b}_{n}u$ and $E(v)\subseteq a_k^{r}E(u)$. But $a_{k}u\mathrm{\ne }0$ lies in $E(u)$ and is killed by $a_k^r$, so $\dim E(v)<\dim E(u)$, contradicting minimality. Therefore the minimum is zero and $\mathrm{\Omega }=\{w\in M:a_{n}w=0\text{ for every }a\in \mathfrak{g},n>0\}$ is nonzero.

The subspace $\mathrm{\Omega }$ is stable under the zero-mode action of $\mathfrak{g}$: for $b\in \mathfrak{g}$, $a_n{b}_{0}w=b_0{a}_{n}w+[a_n,b_0]w=0$, since $[a_n,b_0]=[a,b{]}_n$ and $n>0$. For nonzero $w\in \mathrm{\Omega }$, Order the finite basis $\mathcal{B}$ as $a_1,\dots ,a_s$. PBW [F7] spans $U(\mathfrak{g})w$ by ordered zero-mode words. The powers of $a_s(0)$ on $w$ span a finite-dimensional space by local nilpotence [F1]. Recursively take powers of $a_i(0)$ on a finite spanning set for the suffix space: each string is finite, and their finite sum is finite-dimensional. After the $s$ basis elements this contains every ordered PBW word, so $U(\mathfrak{g})w$ is finite-dimensional. It is a nonzero finite-dimensional $\mathfrak{g}$-module inside $\mathrm{\Omega }$. By [F3], it contains a nonzero finite-dimensional irreducible summand; choose a highest vector $v$ there and put $H=U(\mathfrak{g})v$. Then $H$ is that irreducible summand, and $v$ is killed by all positive loop modes and all positive finite-root spaces.

Positive loop modes kill $v$, so the quadratic Sugawara formula in [F5] shows that $L_{0}v$ is a linear combination of zero-mode products applied to $v$, hence belongs to $H$. The commutator $[L_0,x_0]=0$ then makes $H$ invariant under $L_0$; since $H$ is finite-dimensional irreducible, Schur's lemma [F7] gives $L_0{\mathrm{\mid }}_H=hI$ for some $h\in \mathbb{C}$. On $R=U(\widehat{\mathfrak{g}})v$, define the affine derivation $d=-L_0$; [F5] gives $[d,x_n]=n{x}_n$, and $v$ is a weight vector for the full affine Cartan. By The affine simple root alpha zero is delta minus the highest root, let $\theta$ be the highest finite root. For the untwisted affine matrix $\widehat{A}$, write $a_{ij}={\alpha }_j(h_i)$, with node-zero data ${\alpha }_0=\delta -\theta$, $h_0=c-{\theta }^{\vee }$, and finite simple pairs ${\alpha }_i,h_i={\alpha }_i^{\vee }$ [F5], with $\delta {\mathrm{\mid }}_{\mathfrak{h}\oplus \mathbb{C}c}=0$, $\delta (d)=1$, and ${\alpha }_i(c)=0$. Under the finite invariant-form identification of the Cartan dual with its root span, ${\alpha }_i^{\vee }=2{\alpha }_i\mathrm{/}({\alpha }_i,{\alpha }_i)$ and ${\theta }^{\vee }=2\theta \mathrm{/}(\theta ,\theta )$. Normalize the finite root inner product so $(\theta ,\theta )=2$, put ${\beta }_0=-\theta$, ${\beta }_i={\alpha }_i$, ${\tau }_0=1$, and ${\tau }_i=({\alpha }_i,{\alpha }_i)\mathrm{/}2>0$. For finite indices, ${\tau }_i{a}_{ij}=({\alpha }_i,{\alpha }_j)={\tau }_j{a}_{ji}$. For $j>0$, $a_{0j}={\alpha }_j(h_0)=-({\alpha }_j,\theta )$ and $a_{j0}={\alpha }_0(h_j)=-2(\theta ,{\alpha }_j)\mathrm{/}({\alpha }_j,{\alpha }_j)$, so ${\tau }_0{a}_{0j}={\tau }_j{a}_{j0}$; and $a_{00}={\alpha }_0(h_0)=2$. Hence $\widehat{A}$ is a finite symmetrizable GCM, and the matrix hypothesis of [F4] holds. The module $R$ is integrable because every affine simple raising or lowering generator is one of the root-current modes covered by [F1]. It is generated by the affine highest vector $v$ of weight $\lambda$, so Verma PBW gives finite-dimensional weight spaces and support in $\lambda -Q^{+}$; hence $R$ lies in affine category $\mathcal{O}$. By [F4] and [A1], $R$ is a direct sum of simple highest-weight modules. Its weight-$\lambda$ space is the line $\mathbb{C}v$. Each projection of $v$ to a simple summand is singular; in a simple highest-weight module every singular vector is in its highest line, since a singular vector generates the whole simple module and PBW bounds the generated weights below its weight. Therefore exactly one summand has a nonzero projection of $v$; because $v$ generates $R$, that summand is all of $R$. To see $R$ is a VOA submodule, note it is stable under every current mode by its definition. For a current state $a$ and a state $b$ whose modes preserve $R$, module Jacobi gives

On fixed $u\in R$, lower truncation of the rightmost factor makes both sums finite, and each term preserves $R$. Induction over current-generated states shows every state mode preserves $R$. Affine simplicity then implies VOA simplicity.

Affine PBW orders $R$ as negative-loop words followed by zero modes on $v$; positive loops kill $v$. Equivalently it is spanned by negative-loop words applied to the finite-dimensional space $H$. A word of total loop degree $q$ raises the $L_0$ eigenvalue from $h$ by $q$ by [F5]. There are finitely many ordered negative-loop monomials of fixed total degree (finitely many mode indices summing to $q$ and finitely many basis vectors of $\mathfrak{g}$), each applied to finite-dimensional $H$. Thus the $L_0$ eigenspaces are finite-dimensional and the spectrum is contained in $h+{\mathbb{Z}}_{\ge 0}$. PBW spanning by $L_0$ eigenvectors makes $L_0$ semisimple, and this spectrum is real-part lower bounded; [F6] makes the simple weak submodule an ordinary module.