Finite-Weyl canonical basic sets from Lusztig properties

Suppose $d_{E,M}>0$. Choose $w$ with $a(w)=a_M$ and $D_{w}M\mathrm{\ne }0$. Since $M$ is a composition factor of $\ast V_E$ by [F4], $D_w$ cannot act as zero on that whole module. Thus $\varphi (D_w)$ acts nontrivially on $V_E$. In the formula of [F1], some surviving summand has $h_{w,d,z}\mathrm{\ne }0$ and $t_z{V}_E\mathrm{\ne }0$. The latter implies $a(z)={\alpha }_E$ by [F2], while P4 gives $a_M=a(w)\le a(z)$. Hence every positive entry satisfies $a_M\le {\alpha }_E$.

By [F3] and [F4], $d_{E(M),M}=1$ and ${\alpha }_{E(M)}=a_M$. Together with step 1.1 this proves that every column is nonzero and its least $\alpha$ is exactly $a_M$. The assignment $M\mapsto E(M)$ is already injective. It remains to prove uniqueness of a generic label at that minimum.

Fix $M$ and a projective cover $P=H_{k}e$ as in [F4]. Put $b=\min \{i:B_{i}e\mathrm{\ne }0\}$. The finite filtration starts with $I_{0}e=P$ and ends with zero, so this minimum exists, and the vanishing earlier layers implies $P=I_{b}e$. The quotient $P^b=B_{b}e=P\mathrm{/}{I}_{b+1}e$ is nonzero. The kernel $K$ of the projective-cover map $P\to M$ is maximal, and superfluity makes it the unique maximal submodule: any proper submodule not contained in $K$ would sum with $K$ to $P$. Hence $I_{b+1}e\subseteq K$, so the cover map factors through a surjection $\bar{\pi }:P^b\to M$. Its kernel $\bar{K}=K\mathrm{/}{I}_{b+1}e$ is superfluous in $P^b$, since any submodule whose sum with $\bar{K}$ is $P^b$ lifts to a submodule whose sum with $K$ is $P$. The kernel $\bar{K}$ is also maximal because the quotient is simple. Therefore it is the unique maximal submodule and $P^b$ has simple head $M$. This makes $P^b$ indecomposable: a direct sum of two nonzero finite-dimensional modules has at least two simple summands in its head, even if isomorphic. But $P^b$ is semisimple as a $J_k$-module and each $J_k$-summand is an $H_k$-submodule, since the action restricts through $\varphi$. Consequently it is exactly one $V_{E_1}$, with multiplicity one. Reciprocity gives ${\alpha }_{E_1}=b$ and $d_{E_1,M}>0$. Step 1.1 yields $a_M\le b$. Conversely, [F3] and [F4] give $d_{E(M),M}=1$ and ${\alpha }_{E(M)}=a_M$, so reciprocity makes $B_{a_M}e$ nonzero and $b\le a_M$. Hence $b=a_M$. Reciprocity then puts $V_{E(M)}$ in the single $J_k$-simple summand of $B_{b}e$, so $E_1=E(M)$ and $d_{E_1,M}=1$.

If $d_{E,M}>0$ and ${\alpha }_E=a_M$, reciprocity puts $V_E$ in the nonzero layer $B_{{\alpha }_E}e$, so ${\alpha }_E\ge b$. Since $a_M\le b$ by step 2.2, equality holds throughout. The unique summand in $B_{b}e$ is $V_{E_1}$, so $E=E_1$. This applies to $E(M)$ by step 2.1, giving $E_1=E(M)$. Thus a positive entry at the least $\alpha$ has precisely the label $E(M)$; every other positive entry has strictly larger $\alpha$.

The image of the injection in [F3] now satisfies all axioms in [F5], proving existence. Those axioms also make the generic label the unique minimum in each column, proving uniqueness of the set and bijection. The parameter identities follow from $S_s=v^{L(s)}{T}_s$, which changes $(T_s-v^{L(s)})(T_s+v^{-L(s)})=0$ into $(S_s-v^{2L(s)})(S_s+1)=0$; applying $\theta$ gives the claimed specializations. Only $f_E$ was inverted in [F2] and [F4], and step 1.1 used the explicitly assumed Lusztig properties. Hence the proof imposes neither full-Schur nonvanishing nor any unstated general Coxeter assertion.