Finite A1 specialization of Weyl Kac

The rank-one presentation in F2 has the three generators $e,h,f$ with $[h,e]=2e$, $[h,f]=-2f$, $[e,f]=h$, so it is ${\mathfrak{s}\mathfrak{l}}_2$. There is one positive root $\alpha =2\omega$, and its reflection sends $\omega$ to $-\omega$, giving $W=\{1,s\}$ and $\rho =\omega$. Inserting these in F1 gives the displayed quotient.

Set $x=e^{-2\omega }$. After cancelling a monomial, the quotient in step 1.1 is $e^{m\omega }(1-x^{m+1})\mathrm{/}(1-x)$. The polynomial identity $(1-x){\sum }_{j=0}^m{x}^j=1-x^{m+1}$ proves the finite expansion, since $1-x$ is a formal unit. Distinct $j$ give distinct weights, each with coefficient one, so the total dimension is $m+1$. At $m=0$ the sum is $1$, and at $m=1$ it is $e^{\omega }+e^{-\omega }$. No numerical division at $x=1$ is used; the dimension is read from the already finite polynomial.