Real and imaginary factors in the affine sl2 denominator

Here $\mathrm{\ell }=1$ and ${\mathrm{\Phi }}_0^{+}=\{\alpha \}$. The roots $n\delta$ give $q^n$ for $n\ge 1$; $\alpha +n\delta$ give $z{q}^n$ for $n\ge 0$; $-\alpha +n\delta$ give $z^{-1}{q}^n$ for $n\ge 1$. F1 says all these factors have exponent one in this rank. Substitution gives the stated product.

In particular its degree-zero factor is $1-z$. To first degree in $q$, the remaining factors are $(1-q)(1-zq)(1-z^{-1}q)$; factors with index at least two contribute only at degree at least two. Thus $P=(1-z)(1-(1+z+z^{-1})q+O(q^2))\mathrm{.}$ This calculation checks both index endpoints: omitting $n=0$ from the positive family loses $1-z$, while including it in the negative family adds the nonexistent root $-\alpha$. The grouping is coefficientwise formal as in F1, with no analytic Jacobi identity asserted.