Full divided powers force root-of-unity weight reflection symmetry

Fix one root color, and first work in a single $P\mathrm{/}\mathbb{Z}{\alpha }_i$ coset. Take a finite-dimensional simple graded module for the rank-one algebra, and choose a nonzero vector $v$ at its highest integer coroot label $n$. It is killed by every positive divided power. Reordering shows that the cyclic module is spanned by the distinct weight vectors $F^{(r)}v$, $r\ge 0$, and simplicity makes this the whole module. If $n=-b<0$, put $r=2\mathrm{\ell }k$. The negative-binomial identity gives ${\left[\begin{array}{c}-b\\ r\end{array}\right]}_{\zeta }=(-1{)}^r{\left[\begin{array}{c}b+r-1\\ r\end{array}\right]}_{\zeta }$. To see this is nonzero for arbitrarily large $k$, write $b-1=a\mathrm{\ell }+s$, $0\le s<\mathrm{\ell }$, and use ${\prod }_{j=0}^{N-1}(1+{\zeta }^{N-1-2j}z)={\sum }_j{\left[\begin{array}{c}N\\ j\end{array}\right]}_{\zeta }{z}^j$ with $N=b+r-1$. Every block of length $\mathrm{\ell }$ equals $1+{\zeta }^{\mathrm{\ell }(N-1)}{z}^{\mathrm{\ell }}$, because ${\zeta }^{-2}$ has odd order $\mathrm{\ell }$; the residual block has degree $s<\mathrm{\ell }$. The coefficient of $z^{2\mathrm{\ell }k}$ is therefore $\left(\genfrac{}{}{0px}{}{a+2k}{2k}\right)$: the sign from ${\zeta }^{\mathrm{\ell }(N-1)}=\pm 1$ is raised to the even power $2k$. It is nonzero. But finite support makes $F^{(r)}v=0$ for large $r$, contradicting $E^{(r)}{F}^{(r)}v={\left[\begin{array}{c}n\\ r\end{array}\right]}_{\zeta }v$. Hence $n\ge 0$.

For this simple cyclic module the span of $F^{(r)}v$, $r>n$, is stable. Positive divided powers landing at index $r-s\le n$ have coefficient ${\left[\begin{array}{c}n-r+s\\ s\end{array}\right]}_{\zeta }=0$, since the upper index is then a nonnegative integer smaller than $s$; all other terms remain in that span. Negative divided powers increase the index and toral operators act diagonally. This is a proper submodule, as it avoids the top weight, so it is zero. The module is thus a quotient of the specialized free rank-one string $W(n)$ with basis $u_0,\dots ,u_n$ and the actions in [F1]. On $W(n)$ the diagonal bilinear form $(u_r,u_r)={\left[\begin{array}{c}n\\ r\end{array}\right]}_{\zeta }$, $(u_r,u_s)=0$ for $r\mathrm{\ne }s$, is contravariant for $E^{(a)}\leftrightarrow F^{(a)}$ and fixes the torus. Indeed the required identity is ${\left[\begin{array}{c}n\\ r\end{array}\right]}_q{\left[\begin{array}{c}r\\ a\end{array}\right]}_q={\left[\begin{array}{c}n\\ r-a\end{array}\right]}_q{\left[\begin{array}{c}n-r+a\\ a\end{array}\right]}_q$, valid as a Laurent-polynomial identity before specialization. Its radical is consequently stable. Every surviving weight line of the quotient by this radical can be raised to $u_0$ using $E^{(r)}$, since its coefficient is nonzero; the quotient is simple. Every proper graded submodule of $W(n)$ lies in the radical, since it cannot contain $u_0$ and the same raising test applies. Therefore the original simple module is this quotient. Its surviving indices are exactly those $r$ with ${\left[\begin{array}{c}n\\ r\end{array}\right]}_{\zeta }\mathrm{\ne }0$, a set invariant under $r\mapsto n-r$ by Gaussian symmetry. Its character is reflected about coroot label zero.

Every finite-dimensional graded rank-one module has a composition series in the graded module category: choose proper graded submodules of largest dimension and continue, so the process terminates. Its simple quotients have the symmetric characters proved in steps 1.1 and 2.1; character additivity proves symmetry for the original module. Coset by coset, the reflection sends $\mu$ to $\mu -⟨\mu ,h_i⟩{\alpha }_i$, so this is precisely $s_i$-invariance of $\mathrm{ch}M$. Applying this for each color gives Weyl invariance.

Now let $v$ be any nonzero rank-one highest vector, without simplicity. Its finite cyclic module is spanned by $F^{(r)}v$ and has support at $n-2r$, $r\ge 0$. By step 3.1 this support is invariant under negation, forcing $n\ge 0$. Its span of indices $r>n$ is a submodule by the calculation in step 2.1, and all its coroot labels are strictly negative. Applying step 3.1 to this finite submodule forces it to be zero. Thus every higher divided power $F^{(r)}v$ with $r>n$ vanishes. This proves the stated highest-vector assertion for every color.