= Root-binomial divisibility of reflection anti-invariants
{title2=$s_\alpha u=-u\ \Longrightarrow\ u/(1-e(\alpha))\in\mathbb Z[P]$}
For a <root reflection> on the <weight lattice>, $s_\alpha\lambda=\lambda-m\alpha$ with $m=\lambda(H_\alpha)\in\mathbb Z$. A finite anti-invariant element of the <group ring of a weight lattice> is a sum of terms $a_\lambda[e(\lambda)-e(s_\alpha\lambda)]$; fixed weights have zero coefficient. Each pair is divisible by $1-e(\alpha)$, because $1-t^m$ is divisible by $1-t$ in $\mathbb Z[t,t^{-1}]$ for every integer $m$. Thus the quotient has finite support.
The formal expression $(\sum_{n\geq0}e(n\alpha))u$ is interpreted coefficientwise. Along each coset of $\mathbb Z\alpha$, only finitely many coefficients of $u$ are nonzero and their total is zero by reflection anti-invariance. Partial coefficient sums therefore vanish before and after that finite interval. The result has finite support, even though the individual summands $e(n\alpha)u$ do not eventually vanish.
// Target: lie-algebra.bigb
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