Weyl flag summands, standard-costandard orthogonality and tilting uniqueness

If a finite module $V$ has no label strictly greater than $\lambda$, every surjection $V\to {\mathrm{\Delta }}_{\zeta }(\lambda )$ splits: lift its highest vector to a label-$\lambda$ vector $v\in V$. All positive divided powers kill $v$ because they would increase its label strictly. The universal property in [F1] extends the highest-vector lift to a module map from ${\mathrm{\Delta }}_{\zeta }(\lambda )$, and its composite with the surjection fixes the highest generator, hence is the identity. This argument permits labels incomparable with $\lambda$ in $V$.

A nonzero map ${\mathrm{\Delta }}_{\zeta }(\lambda )\to {\mathrm{\nabla }}_{\zeta }(\mu )$ has image containing the unique simple socle $L_{\zeta }(\mu )$: every nonzero finite-length submodule contains a simple submodule, and this socle is unique. Its weights are at most $\lambda$, so $\mu \le \lambda$. On the other hand the highest generator has label $\lambda$ in the target, so $\lambda \le \mu$. Thus $\lambda =\mu$. At equal labels the image of the highest line is an arbitrary scalar multiple of the unique highest line, which lies in the simple socle; its generated image is precisely that socle. This gives a one-dimensional Hom space, and the map is the simple-head quotient followed by socle inclusion.

If $M$ is Weyl-filtered and $\lambda$ is a maximal support label, the canonical action map ${\mathrm{\Delta }}_{\zeta }(\lambda ){\otimes }_{\mathbb{C}}{M}_{\lambda }\to M$ is injective and its quotient is Weyl-filtered with no section of label $\lambda$. To prove this, induct along a chosen Weyl filtration. If its final section has label $\lambda$, the short exact sequence splits by step 1.1; the trace acquires exactly this additional copy. If its final section has different label $\gamma$, it has no $\lambda$ weight, since otherwise $\lambda \le \gamma$ would contradict maximality and distinctness. Thus $M_{\lambda }$ and its generated trace are contained in the preceding flag subobject, where induction applies, and the quotient extends by ${\mathrm{\Delta }}_{\zeta }(\gamma )$. This proves the claim in both cases; it also identifies the submodule canonically, independently of the chosen filtration.

Consider an extension $0\to {\mathrm{\nabla }}_{\zeta }(\mu )\to V\to {\mathrm{\Delta }}_{\zeta }(\lambda )\to 0$. If $\lambda ≮\mu$, no weight of $V$ is strictly greater than $\lambda$, since such a weight would lie at most $\mu$ and force $\lambda <\mu$. Thus it splits by step 1.1. If $\lambda <\mu$, dualizing produces an extension with Weyl quotient of label ${\mu }^{\ast }$ and dual Weyl kernel of label ${\lambda }^{\ast }$. The involution $-w_0$ preserves the dominance order, so ${\lambda }^{\ast }<{\mu }^{\ast }$, and the already settled case splits this dual extension. Exact involutive duality then splits the original extension. Therefore all standard-costandard degree-one extension groups vanish. Inducting on a Weyl filtration in the first variable and a dual Weyl filtration in the second variable, using the Hom-extension exact sequences (equivalently pullback and pushout of short exact sequences), proves that every extension of a Weyl-filtered module by a dual-Weyl-filtered module splits.

Suppose $M=Q\oplus R$ is Weyl-filtered. Its canonical maximal trace from step 2.1 is stable under the two projection idempotents, since it is generated by $M_{\lambda }$; the restrictions to this trace are the identity on the Weyl factor tensored with the corresponding projections of $M_{\lambda }$. Hence its intersections with $Q,R$ are respectively ${\mathrm{\Delta }}_{\zeta }(\lambda )\otimes Q_{\lambda }$ and ${\mathrm{\Delta }}_{\zeta }(\lambda )\otimes R_{\lambda }$. The quotient decomposes as the direct sum of $Q$ and $R$ modulo these intersections, and is Weyl-filtered of smaller total dimension. Induction on total dimension therefore gives Weyl filtrations of these two quotients, and adjoining the respective Weyl traces gives Weyl filtrations of $Q,R$. The base $M=0$ is immediate. Exact duality proves the same for dual Weyl flags, hence for tiltings. Any finite dimensional module splits into indecomposables by repeatedly decomposing a decomposable summand, since positive dimensions decrease; the established summand closure makes each resulting tilting summand tilting.

For a sufficiently high power of any endomorphism $a$ of a finite dimensional module, kernel and image stabilize and give the module decomposition $T=\ker a^n\oplus \mathrm{im}{a}^n$. For an indecomposable $T$, every endomorphism is therefore nilpotent or invertible. The maximal-trace proof in step 2.1 also works simultaneously for the finite set $\mathrm{\Gamma }$ of all maximal support labels: the corresponding Weyl sections split past one another by step 1.1, and their sum is the injected submodule $S={⨁}_{\lambda \in \mathrm{\Gamma }}{\mathrm{\Delta }}_{\zeta }(\lambda )\otimes T_{\lambda }$ with Weyl-filtered quotient. If the total dimension of ${⨁}_{\lambda \in \mathrm{\Gamma }}{T}_{\lambda }$ exceeded one, define a homomorphism $S\to T$ fixing one selected highest basis line and killing another, by acting on the displayed multiplicity spaces. step 2.2 extends this map to an endomorphism of $T$ because its cokernel is Weyl-filtered. That endomorphism fixes a nonzero vector and kills another, so it is neither nilpotent nor invertible, a contradiction. Thus an indecomposable tilting has a unique maximal label and a one-dimensional highest space.

Let $T,T^{\mathrm{\prime }}$ be indecomposable tiltings and $\lambda$ their common maximal label. step 2.1 supplies inclusions of a copy of ${\mathrm{\Delta }}_{\zeta }(\lambda )$ into each, with Weyl-filtered cokernels. step 2.2 extends the common highest-line identification to homomorphisms $f:T\to T^{\mathrm{\prime }}$ and $g:T^{\mathrm{\prime }}\to T$, whose composites fix that line. By step 3.2 these composites are not nilpotent and hence are invertible. Therefore $f$ is an isomorphism. This proves the claimed uniqueness of indecomposable tiltings at each label where one exists, without asserting existence at the remaining labels.