Simplicity of Weyl modules in the closed bottom alcove

Define the folding map $p$ on an open alcove $w{C}^{\circ }$ by $p(x)=w^{-1}x$. It extends continuously across facets: the two inverse maps on adjacent alcoves differ by the corresponding fundamental-facet reflection, which fixes their common facet pointwise. At a point on several walls, use a small ball meeting only the finitely many incident walls; its local chambers are connected by adjacent local chambers, so the same agreement proves that all limiting inverse maps coincide there. This yields a continuous map on the whole Euclidean space, identity on $C$, and $p(wx)=p(x)$ for every affine Weyl element. On a line segment crossing walls generically, local finiteness divides it into finitely many pieces on each of which $p$ is an isometry, so the distance of the images of its endpoints is at most its length. General endpoints follow by arbitrarily small perturbation of the endpoints to avoid the finitely many codimension-two wall intersections and passage to continuity. Hence $p$ is one-Lipschitz.

The origin belongs to $C$, so $p(0)=0$. Thus for $x\in C$ and any affine Weyl element $w$, step 1.1 gives $\mathrm{\parallel }x\mathrm{\parallel }=\mathrm{\parallel }p(wx)-p(0)\mathrm{\parallel }\le \mathrm{\parallel }wx\mathrm{\parallel }$. Put $x=\lambda +\rho$. If a dominant $\mu <\lambda$ belonged to its dot orbit, then $\mu +\rho =wx$ while $\mathrm{\parallel }\lambda +\rho {\mathrm{\parallel }}^2-\mathrm{\parallel }\mu +\rho {\mathrm{\parallel }}^2=(\lambda -\mu ,\lambda +\mu +2\rho )>0$: expand the nonzero positive-root-lattice vector $\lambda -\mu$ in simple roots and use strict positivity of every pairing with $\lambda +\mu +2\rho$. This contradicts the norm inequality. By [F2] there is no proper composition factor of ${\mathrm{\Delta }}_{\zeta }(\lambda )$, and it is simple.

The dual Weyl module has a unique simple socle at the same label and the same character and dimension, so it too equals that simple module. This object is consequently tilting, and uniqueness identifies it with $T_{\zeta }(\lambda )$. By [F3] its quantum dimension is nonzero in the strict bottom alcove and zero on the upper wall, proving all assertions.