Weyl quantum dimensions and affine Weyl signs

Evaluate the finite character identity in [F1] at $e^{\mu }=q^{(2\rho ,\mu )}$. Symmetry of the invariant form and changing $w$ to $w^{-1}$ show that its alternating numerator equals ${\sum }_w\det (w)q^{(2w\rho ,\lambda +\rho )}$. Evaluate the Weyl denominator identity instead at $e^{\mu }=q^{2(\mu ,\lambda +\rho )}$; this gives the numerator product ${\prod }_{\alpha >0}(q^{(\lambda +\rho ,\alpha )}-q^{-(\lambda +\rho ,\alpha )})$, since ${\sum }_{\alpha >0}\alpha =2\rho$. The denominator is the same product with $\lambda =0$. Thus the character specialization is the quotient of these two products as a rational function, or equivalently their cross-multiplied Laurent-polynomial identity. Replacing $q$ by $q^{-1}$ changes neither the quotient nor the quantum-integer factors, so it is also the character evaluated at $e^{\mu }=q^{-(2\rho ,\mu )}$.

Every positive-root height $⟨\rho ,{\alpha }^{\vee }⟩$ lies between one and $h-1$, strictly below $\mathrm{\ell }$. Since $\mathrm{ord}({\zeta }^2)=\mathrm{\ell }$, a quantum integer $[n{]}_{\zeta }$ vanishes exactly when $\mathrm{\ell }$ divides $n$; also $\zeta -{\zeta }^{-1}\mathrm{\ne }0$. The denominator in step 1.1 is therefore nonzero after specializing, so ordinary scalar division is now justified. By [F2] its value is the quantum dimension, proving the formula with no singular evaluation of a formal quotient.

A finite simple reflection permutes positive roots except for reversing its own simple root; invariance of the form shows that the numerator product changes sign under the dot action of that reflection. These reflections generate $W$, yielding finite Weyl alternation. A translation $\lambda \mapsto \lambda +\mathrm{\ell }\beta$ with $\beta \in Q$ multiplies each numerator factor by ${\zeta }^{\mathrm{\ell }(\beta ,\alpha )}$, because ${\zeta }^{\mathrm{\ell }}$ is one or minus one. The full multiplier is ${\zeta }^{\mathrm{\ell }(\beta ,2\rho )}=1$: write $\beta =\sum n_i{\alpha }_i$ and use $({\alpha }_i,\rho )=1$ to get $(\beta ,2\rho )=2\sum n_i$. Therefore the product is $\mathrm{\ell }Q$-periodic. Together these give the affine sign relation. A numerator factor vanishes exactly when $⟨\lambda +\rho ,{\alpha }^{\vee }⟩\in \mathrm{\ell }\mathbb{Z}$, namely an affine reflection wall. In the open bottom alcove every such integer lies strictly between zero and $\mathrm{\ell }$, so none vanishes.

Trace is additive over an invariant-subspace exact sequence: choose a weight-adapted basis and use the block triangular action of $g$. Hence quantum dimension adds along any Weyl filtration. If every section label is affine singular, step 3.1 makes each of its quantum dimensions zero, so their sum is zero. These conclusions are all that has been asserted about tilting dimensions here; cancellation for nonsingular labels outside the alcove requires a separate argument.