Strong affine linkage for root Weyl modules and tilting sections

Put $R=u^{-}$ and $H=R⋊\mathbb{C}[T]$. This is an actual finite Hopf algebra: $\mathrm{\Delta }(F_i)=F_i\otimes 1+K_i\otimes F_i$, the torus is group-like, and the antipode preserves this Borel. The odd-order type-1 toral quotient is a Hopf quotient because the $K_i^{\mathrm{\ell }}-1$ generate a Hopf ideal. The positive-height ideal $R_{+}$ is nilpotent, and $J=R_{+}H=H{R}_{+}$ is nilpotent with quotient $\mathbb{C}[T]$. This quotient is semisimple: its commuting toral operators diagonalize simultaneously, since their finite-order minimal polynomials divide separable polynomials $X^d-1$. A nilpotent ideal annihilates every simple module, and a semisimple quotient has zero radical; hence $J=\mathrm{rad}H$ and $R_{+}=\mathrm{rad}R$.

In rank one of highest coordinate $m=(\kappa ,\alpha )$, take $v_j=F^{(j)}{v}_0$ for $0\le j<\mathrm{\ell }$; in the opposite module of lowest coordinate $m^{\mathrm{\prime }}=m-2(\mathrm{\ell }-1)$ take $v_j^{\mathrm{\prime }}=E^{(j)}{v}_0^{\mathrm{\prime }}$. Their actions are $F{v}_j=[j+1]v_{j+1}$, $E{v}_j=[m-j+1]v_{j-1}$, $E{v}_j^{\mathrm{\prime }}=[j+1]v_{j+1}^{\mathrm{\prime }}$ and $F{v}_j^{\mathrm{\prime }}=-[m^{\mathrm{\prime }}+j-1]v_{j-1}^{\mathrm{\prime }}$, with out-of-range vectors zero. The map $v_j\mapsto c_j{v}_{\mathrm{\ell }-1-j}^{\mathrm{\prime }}$, where $c_0=1$ and $c_{j+1}=-[m-\mathrm{\ell }-j]c_j\mathrm{/}[j+1]$, intertwines these actions. For $E$ compatibility use $[\mathrm{\ell }-j]=-s[j]$ and $[a-\mathrm{\ell }]=s[a]$, where $s={\zeta }^{\mathrm{\ell }}\in \{1,-1\}$; $F$ compatibility is its defining recurrence, and the endpoint and toral relations follow from the displayed actions and weights. All denominators are nonzero. If $1\le n\le \mathrm{\ell }$ is congruent to $m+1$ modulo $\mathrm{\ell }$, then $c_j\mathrm{\ne }0$ exactly for $j<n$. The kernel is zero for $n=\mathrm{\ell }$ and otherwise is spanned by $v_n,\dots ,v_{\mathrm{\ell }-1}$, generated by the singular vector $F^{(n)}{v}_0$. It is an image of the baby Verma of label $\kappa -n\alpha$: the $i$-th positive generator kills that vector because $[m-n+1]=0$, and the other simple positive generators commute with $F_i$.

By [F2], choose a Frobenius functional $f$ on $H$. Then $b\mapsto [a\mapsto f(ab)]$ identifies the left regular module $H$ with $H^{\ast }$ under $(h\varphi )(a)=\varphi (ah)$. The socle of $H^{\ast }$ is the annihilator of $J$, of dimension $\dim (H\mathrm{/}J)=\mathrm{\mid }T\mathrm{\mid }$. On the other hand ${\mathrm{soc}}_{H}H=\{h:Jh=0\}=\{h:R_{+}h=0\}={\mathrm{soc}}_{R}H$, since $J=H{R}_{+}$. As a left $R$-module, $H={⨁}_{t\in T}R{K}_t$, so $\mathrm{\mid }T\mathrm{\mid }\dim {\mathrm{soc}}_{R}R=\mathrm{\mid }T\mathrm{\mid }$. Thus $R$ has one-dimensional regular socle, spanned by its nonzero maximum-height PBW monomial; every positive-height factor annihilates it. This transports to braid-transformed halves and to $u^{+}$ by algebra isomorphisms; transformed Borels need not be Hopf subalgebras for the original coproduct.

Free right induction from the minimal parabolic in [F1] transports this exact rank-one kernel calculation. For adjacent Borels $w,w{s}_i$, $\beta =w{\alpha }_i$ and $w$-highest label $\kappa$, it gives a homomorphism $Z^w(\kappa )\to Z^{w{s}_i}(\kappa -(\mathrm{\ell }-1)\beta )$. If $1\le n\le \mathrm{\ell }$ is congruent to $(\kappa +w\rho ,\beta )$ modulo $\mathrm{\ell }$, its kernel is zero for $n=\mathrm{\ell }$ and otherwise is an image of $Z^w(\kappa -n\beta )$. This is exact induction of an actual module map; it is not deduced from character equality.

Each baby Verma is regular over its transformed negative half. Every nonzero submodule therefore contains its lowest PBW line by step 2.1, and that line generates its simple socle. Also $Z(\kappa )$ has a unique simple head $L_u(\kappa )$: the sum of all submodules missing its one-dimensional highest line still misses that line and is the unique maximal submodule. Every finite graded simple has such a highest label, by taking a maximal label in its finite support. Their formal characters are linearly independent, since a maximal highest label in a finite relation occurs with coefficient one only in its own simple character.

Choose a reduced expression $w_0=s_{i_1}\cdots s_{i_N}$, write $w_j=s_{i_1}\cdots s_{i_{j-1}}$ and ${\beta }_j=w_j{\alpha }_{i_j}>0$, and put $\kappa (w)=\kappa +(\mathrm{\ell }-1)(w\rho -\rho )$. Compose step 2.2 along these successive Borels from $Z(\kappa )$ to $Z^{w_0}(\kappa -2(\mathrm{\ell }-1)\rho )$. Every module in this chain has the character of $Z(\kappa )$: the transformed root-product character changes by $e^{(\mathrm{\ell }-1)(\rho -w\rho )}$, canceled by its label shift. Each map is nonzero on the unique label-$\kappa$ line. Indeed in its source that line has exponent $\mathrm{\ell }-1$ along already inverted roots and zero along not-yet-inverted roots. The current positive root ${\beta }_j$ is not yet inverted, so its rank-one exponent is zero and its image coefficient is $c_0=1$; the complementary free parabolic basis makes the resulting vector nonzero.

The composite image is generated by its label-$\kappa$ line. In the opposite-Borel target this is the maximum-height line of its regular $u^{+}$ representation, so step 2.1 makes its $u$-span the simple socle. Thus the composite image is the simple head $L_u(\kappa )$ of its source. Every other composition factor $L_u(\eta )$ of $Z(\kappa )$ consequently occurs in its kernel. The successive-image filtration of the kernel of a finite composite places it among the composition factors of one of the individual kernels. Step 2.2 makes that kernel an image of $Z^{w_j}(\kappa (w_j)-n_j{\beta }_j)$ with $1\le n_j<\mathrm{\ell }$ and $n_j\equiv (\kappa +\rho ,{\beta }_j)(\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{\ell })$, because $\kappa (w_j)+w_j\rho -(\kappa +\rho )=\mathrm{\ell }(w_j\rho -\rho )$. This transformed baby Verma has the character of $Z(\kappa -n_j{\beta }_j)$; independence of simple characters in step 3.1 transfers the composition factor to that ordinary baby Verma. Repeat: the nonnegative root height of the current highest label minus $\eta$ strictly decreases, so the procedure terminates. Each change $\kappa \mapsto \kappa -n_j{\beta }_j$ is a strict downward affine reflection, in the level $k\mathrm{\ell }=(\kappa +\rho ,{\beta }_j)-n_j$. Hence every baby-Verma composition label satisfies $\eta {⪯}_{\mathrm{\ell }}\kappa$.

Let $p(\beta )$ denote the Kostant partition numbers, defined by ${\sum }_{\beta \in Q_{+}}p(\beta )e^{-\beta }={\prod }_{\alpha >0}(1-e^{-\alpha }{)}^{-1}$. The baby PBW character and the actual Weyl formula in [F3] give $\mathrm{ch}Z(\kappa )=e^{\kappa }{\prod }_{\alpha >0}\frac{1-e^{-\mathrm{\ell }\alpha }}{1-e^{-\alpha }},\mathrm{ch}{\mathrm{\Delta }}_{\zeta }(\lambda )={\sum }_{w\in W,\beta \in Q_{+}}\det (w)p(\beta )\mathrm{ch}Z(w\cdot \lambda -\mathrm{\ell }\beta )\mathrm{.}$ The second identity follows by multiplying the finite Weyl numerator by ${\prod }_{\alpha >0}(1-e^{-\mathrm{\ell }\alpha }{)}^{-1}$. It holds in the completion on $\lambda -Q_{+}$: $w\cdot \lambda \le \lambda$, and only finitely many $\beta$ contribute to a fixed label. Simple graded characters are a unitriangular basis in that completion: recursively subtract their highest coefficients by increasing root height; every bounded-height interval has finitely many lattice labels, so the recursion is well defined and unique. In particular the coefficient of a fixed simple character on the right is a finite signed sum, since its contributing highest labels must satisfy $w\cdot \lambda -\mathrm{\ell }\beta \ge \eta$.

If $L_{\zeta }(\eta )$ is a composition factor of ${\mathrm{\Delta }}_{\zeta }(\lambda )$, restriction to $u$ is exact and its highest vector generates a baby-Verma quotient with simple head $L_u(\eta )$. Thus $L_u(\eta )$ has positive multiplicity in the actual restricted Weyl module. The finite signed character coefficient in step 4.2 must therefore have some nonzero baby multiplicity, giving $\eta {⪯}_{\mathrm{\ell }}w\cdot \lambda -\mathrm{\ell }\beta$ by step 4.1. This uses virtual characters only to locate a composition factor, not to assert a module filtration.

For every $x$ and positive root $\alpha$, the translation $x\mapsto x-\mathrm{\ell }\alpha$ is a downward chain. Write $(x,\alpha )=k\mathrm{\ell }+r$, $0\le r<\mathrm{\ell }$. If $r>0$, reflect in levels $k\mathrm{\ell }$ and $(k-1)\mathrm{\ell }$, subtracting respectively $r\alpha$ and $(\mathrm{\ell }-r)\alpha$; if $r=0$, reflect directly in level $(k-1)\mathrm{\ell }$, subtracting $\mathrm{\ell }\alpha$. For dominant $\lambda$, $w\cdot \lambda {⪯}_{\mathrm{\ell }}\lambda$ as well: build a reduced word by adjoining simple reflections on the left. At a length-increasing step $s_{i}v$, $v^{-1}{\alpha }_i$ is positive, so $(v(\lambda +\rho ),{\alpha }_i)>0$ and the level-zero reflection is downward. Apply the negative translations to the positive-root summands of $\beta$. Thus step 5.1 proves $\eta {⪯}_{\mathrm{\ell }}\lambda$ for Weyl composition factors. The isometry $-w_0$ preserves positive roots and these downward chains; exact duality in [F3] proves the same assertion for ${\mathrm{\nabla }}_{\zeta }(\lambda )$.

Work in the finite dominance ideal $\mathrm{\Gamma }=\{\mu \in P^{+}:\mu \le \lambda \}$ and put $S=\{\mu \in \mathrm{\Gamma }:\mu {⪯}_{\mathrm{\ell }}\lambda \}$. Reciprocity in [F4] and step 6.1 show that every Weyl section ${\mathrm{\Delta }}_{\zeta }(\nu )$ of the projective kernel $K_{\mu }$ has $\mu {\prec }_{\mathrm{\ell }}\nu$. Suppose $X$ has all composition labels in $S$ and $\mu \mathrm{\notin }S$. Such a section label $\nu$ cannot belong to $S$, since transitivity would give $\mu {⪯}_{\mathrm{\ell }}\lambda$. Hence $\mathrm{Hom}({\mathrm{\Delta }}_{\zeta }(\nu ),X)=0$: a nonzero image is a highest quotient with simple head $L_{\zeta }(\nu )$, absent from $X$. Induction along the kernel flag gives $\mathrm{Hom}(K_{\mu },X)=0$. The projective sequence consequently gives ${\mathrm{Ext}}^1({\mathrm{\Delta }}_{\zeta }(\mu ),X)=0$.

The actual finite universal-extension construction in [F4] starts with $X={\mathrm{\Delta }}_{\zeta }(\lambda )$, whose composition labels lie in $S$ by step 6.1. Step 7.1 shows that it never adds a Weyl quotient with label outside $S$. Adding a section with label in $S$ preserves this composition support by step 6.1 and transitivity. The final tilting and its indecomposable highest-label-$\lambda$ summand therefore have all Weyl labels in $S$, using summand closure in [F4]. Their composition labels also lie in $S$ by the Weyl result already proved. This proves every assertion, for both possible orders of $\zeta$ and without primality of $\mathrm{\ell }$; the only scalar zeros used were $[a]=0$ exactly when $\mathrm{\ell }\mid a$.