Admissible positive-integral-level simple affine modules are semisimple

Let $D$ act as $q$ on $M(q)$ and extend the affine current action by $d=-D$. The grading shift in [F1] gives $[d,x_m]=m{x}_m$, so this is a module for the full untwisted affine algebra at level $\mathrm{\ell }$. For an ordered basis $a_1,\dots ,a_s$ from $\mathcal{B}$, PBW spans $U(\mathfrak{g})u$ by $a_1(0{)}^{r_1}\cdots a_s(0{)}^{r_s}u$. Local nilpotence [F2] makes the span of $a_s(0{)}^{r}u$ finite-dimensional. Recursively, if the suffix span is finite-dimensional, choose a finite spanning set; the powers of the next zero mode on each spanning vector have finite span by local nilpotence. Their finite sum is again finite-dimensional. After all $s$ steps this contains every PBW word, proving that $U(\mathfrak{g})u$ is finite-dimensional. Hence each horizontal orbit is completely reducible by [F2], and its Cartan acts diagonally.

Fix homogeneous $u\in M(q_0)$. The subspace $P=U(\mathfrak{g}[t{]}_{\ge 0})u$ is finite-dimensional. Indeed, PBW orders it as positive-loop words applied to the finite-dimensional zero-mode orbit $U(\mathfrak{g})u$; a positive mode lowers the nonnegative grade by its positive index, so a nonzero word has total index at most $q_0$, and there are finitely many such words in a finite basis of $\mathfrak{g}$. The full cyclic affine module $C=U(\widehat{\mathfrak{g}})u$ is spanned by negative-loop words applied to $P$. At each fixed admissible grade, only finitely many negative-mode index partitions and basis choices occur, so $C$ has finite-dimensional grade pieces. Each such piece is a finite-dimensional horizontal $\mathfrak{g}$-module, hence decomposes into Cartan weights by [F2]. The finite-dimensional positive-current space $P$ has a finite set of affine weights; negative affine root operators then show that the weight support of $C$ is contained in finitely many cones ${\lambda }_j-Q^{+}$. Thus $C$ lies in affine Kac–Moody category $\mathcal{O}$.

Every affine simple-root generator acts locally nilpotently on $C$, since [F2] supplies a current basis containing all finite simple-root and highest-root generators with nilpotent modes; thus $C$ is integrable. By [F5], the affine GCM $\widehat{A}$ is finite and symmetrizable, so the symmetrizable-GCM hypothesis of [F3] holds; [A1] supplies its AC hypothesis. Therefore $C$ is a direct sum of simple affine highest-weight modules. For any such summand $S$, current modes preserve $S$. If $a$ is a current state and $b$ is a state whose modes preserve $S$, module Jacobi gives

For fixed $u\in S$, each sum is finite by lower truncation of its rightmost mode; every term preserves $S$. Induction on PBW length in the current-generated VOA therefore makes $S$ a VOA submodule. In any such summand, let $v$ be an affine highest vector of weight $\lambda$, and put $H=U(\mathfrak{g}[0])v$. The finite root decomposition in [F5] gives finitely many negative root vectors; PBW [F7] orders them before the Cartan and positive root vectors. The positive root vectors kill $v$, Cartan vectors act by scalars, and each negative root zero mode is locally nilpotent by [F2]. The successive finite-span argument of step 1.1, applied to these ordered negative root modes, makes $H$ finite-dimensional. It is a completely reducible horizontal module by [F2]. Its top weight space is $\mathbb{C}v$ and all weights are at or below $\lambda {\mathrm{\mid }}_{\mathfrak{h}}$ by the PBW ordering; every nonzero projection of $v$ to a simple summand is therefore a highest vector of that same top weight. The one-dimensional top permits only one such summand, and $H=U(\mathfrak{g}[0])v$ has no summand with zero projection, so $H$ is irreducible. The quadratic definition of $L_0$ in [F4], applied to $v$ annihilated by positive loop modes, shows $L_{0}v$ is a linear combination of zero-mode products applied to $v$, hence lies in $H$. Since $[L_0,x_0]=0$ by [F4], $L_0$ preserves $H$; Schur's lemma [F7] makes it scalar there, say $h_S$. If $q_S$ is the admissible grade of $v$, then $dv=-q_{S}v$, while negative loop modes of total degree $n$ give $d$-eigenvalue $-q_S-n$ and $L_0$-eigenvalue $h_S+n$. These words applied to $H$ span $S$ by affine PBW, so $L_0+d=(h_S-q_S)I$ on $S$. Thus every VOA submodule of $S$, being $L_0$-stable, is also $d$-stable; full-affine simplicity now proves VOA simplicity. Each negative loop mode $x_{-m}$ raises the eigenvalue by $m>0$ by [F4]. The identity $L_0+d=(h_S-q_S)I$ identifies each $L_0$ eigenspace with a single admissible grade piece of $S$, which is finite-dimensional by step 2.1. The spectrum of each summand is a translate of ${\mathbb{Z}}_{\ge 0}$, so every summand is lower bounded and ordinary by [F6]. Hence $M$, being the sum of its homogeneous cyclic submodules $C$, is a sum of simple ordinary submodules.

Choose a maximal family of simple ordinary submodules of $M$ whose sum is direct; such a family exists by [A1], since unions of chains of independent families remain independent. If its sum were not $M$, the sum-of-simple-submodules statement in step 3.1 supplies a simple ordinary submodule $S$ not contained in it. The intersection of $S$ with the chosen sum is a submodule of simple $S$, so it is either $0$ or $S$; the latter is excluded, and the former would enlarge the independent family. Therefore the sum is all of $M$, proving complete reducibility.

It remains to count the possible simple ordinary isomorphism classes. Let $N$ be any simple ordinary module. For each $L_0$ eigenvalue coset in $\mathbb{C}\mathrm{/}\mathbb{Z}$, the sum of its eigenspaces is stable under every current mode by [F4], hence under all state modes by the current-generation argument of step 3.1. Simplicity leaves one nonzero coset. Its nonempty lower-bounded spectrum therefore has a lowest eigenvalue $h$, whose eigenspace is finite-dimensional. Positive current modes lower $L_0$ and hence kill this eigenspace. It is stable under zero modes, so choose a finite-dimensional irreducible horizontal $\mathfrak{g}$-summand and a highest vector $w$ in it. This is killed by every affine simple raising generator. Every affine simple lowering generator acts locally nilpotently by [F2]; together with the raising operator killing $w$, the rank-one brackets make the generated ${\mathfrak{s}\mathfrak{l}}_2$ string finite-dimensional, so [F2] makes each affine Dynkin label $m_i=w(h_i)$ a nonnegative integer. The finite-root labels determine the horizontal irreducible uniquely by [F7]; the node-zero label is included, and the level is $\mathrm{\ell }$. For this classification, extend the current action on $N$ to the full affine algebra by $d=-L_0$. This is a valid extension because [F4] gives $[-L_0,x_m]=m{x}_m$, and $L_0$ commutes with the scalar central action. It is a new choice of extension on $N$, independent of $d=-D$ used for $M$ in step 1.1; no equality of the admissible grading and conformal weights is assumed. Thus $w$ is an affine highest vector of a dominant integral weight $\lambda$ at level $\mathrm{\ell }$, with $\lambda (d)=-h$. Any current-invariant subspace of $N$ is stable under all state modes by step 3.1's finite Jacobi formula, so VOA simplicity makes $N$ simple for the current algebra and hence for this full affine action. In particular $w$ generates $N$. Affine Verma universality in [F3] gives a surjection from the Verma module of weight $\lambda$, and [F3] identifies its unique simple quotient.

Write $m_i=\lambda (h_i)\in {\mathbb{Z}}_{\ge 0}$ for the affine Dynkin labels. The coroot ${\theta }^{\vee }$ is a positive coroot, so in the finite simple-coroot basis it has ${\theta }^{\vee }={\sum }_{i=1}^r{b}_i{h}_i$ with positive integers $b_i$: it lies in the positive coroot lattice, and its support is full because the highest root has full support by [F5]. Since $h_0=c-{\theta }^{\vee }$, the level identity gives $\mathrm{\ell }=\lambda (c)=m_0+{\sum }_i{b}_i{m}_i$. Every coefficient on the right is positive, so this equation has only finitely many nonnegative integer solutions. On a top vector $w$, the positive-mode terms in the quadratic Sugawara formula [F4] vanish, so $2(\mathrm{\ell }+h^{\vee })L_{0}w=Cw$, where $C={\sum }_a{x}_a(0)x^a(0)$ for $B$-dual bases. Invariance of $B$ makes ${\sum }_a{x}_a\otimes x^a$ invariant under the diagonal adjoint action (it corresponds to the identity map of $\mathfrak{g}$); hence $[z,C]=0$ for every $z\in \mathfrak{g}$. Schur's lemma [F7] makes $C$ scalar on the irreducible horizontal top, and its scalar is determined by that representation, which [F7] uniquely fixes from the finite highest-weight labels. Thus the top conformal weight $h$ is fixed by those labels. Consequently only finitely many affine highest weights, including their derivation values, occur; [F3]'s unique simple highest-weight quotient fixes the full affine module for the extension $d=-L_0$. An affine intertwiner commutes with all current modes, and the same finite Jacobi induction as in step 3.1 makes it commute with all state modes, so it is a VOA intertwiner. Thus these finitely many weights fix the simple ordinary VOA modules up to isomorphism. This proves finiteness.