Root specialization of integral Weyl lattices and their characters

In finite type ${\chi }_{\lambda }$ has finite support and finite multiplicities, so [F1] makes $V_A(\lambda )$ finite free in total rank. Tensor its homogeneous basis with $\mathbb{C}$; the resulting vectors remain a basis, each weight multiplicity is unchanged, and in particular its highest vector survives. All action identities on this stable lattice are integral identities, hence remain valid after specialization, including the full toral-binomial character and divided-power label shifts. Finite support makes all sufficiently high divided powers act by zero. Thus the root module is finite dimensional, integrable, and type 1, with character ${\chi }_{\lambda }$. This is free base change and not base change of a generic direct-sum decomposition.

For a highest vector $m$ of label $\lambda$ in a finite module, [F2] gives every singular divided-power relation $F_i^{(r)}m=0$ with $r>\lambda (h_i)$. The negative-half map $u\mapsto um$ therefore factors through the specialized quotient in [F1], since tensoring an exact quotient presentation is right exact. Mixed reordering, the specified highest toral character and positive-generator annihilation make this factorization a homomorphism for the full root algebra. It sends the highest vector to $m$ and is unique because that vector generates the source. This proves the asserted universal property.

Every proper submodule misses the one-dimensional highest space, because any nonzero vector on that line generates the full module. Label preservation means their sum also misses that space. Hence the sum of all proper submodules is a proper submodule containing every other one, and its quotient is simple. In any finite dimensional simple module choose a maximal label from its finite support. Its vector is killed by all positive divided powers. By [F2] its label is dominant, and it generates the module by simplicity, so step 2.1 presents it as the unique simple head at that label. A simple module's highest label is recovered from its support; different labels therefore give nonisomorphic simples.

Exact ordinary duality turns the simple head of ${\mathrm{\Delta }}_{\zeta }(-w_0\lambda )$ into the unique simple socle of its dual. Its character is ${\chi }_{-w_0\lambda }(e^{-\mu })={\chi }_{\lambda }$, so its highest label is $\lambda$ and all its weights are at most $\lambda$. The simple socle has highest label $\lambda$: the weights of the simple quotient of a Weyl module are Weyl invariant by the finite rank-one reflection symmetry in [F2], so its dual's highest label is the negative longest-Weyl image of the original highest label. Step 3.1 identifies this socle with $L_{\zeta }(\lambda )$. This proves all claims, with Choice confined to [F3].