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Wall-crossing acts on standard modules by a two-step extension

Statement

Assume the Axiom of Choice, inherited from the published items. Let $θ_{s}$ be the wall-crossing functor of The wall-crossing functor and its unit and counit attached to the simple reflection $s$ in the regular integral block $O_{λ}$ . For every $w \in W$ the object $θ_{s} Δ (w \cdot λ)$ has a finite Verma flag with exactly two factors, namely $Δ (w \cdot λ)$ and $Δ (w s \cdot λ)$ , each with multiplicity one; in the Grothendieck group $[θ_{s} Δ (w \cdot λ)] = [Δ (w \cdot λ)] + [Δ (w s \cdot λ)] .$ Moreover $θ_{s}$ is exact and preserves Verma-filtered objects with finite Verma flags: if $X$ has a finite Verma flag with multiplicities $(X : Δ (v \cdot λ))$ , then $θ_{s} X$ has one with multiplicities $(θ_{s} X : Δ (u \cdot λ)) = (X : Δ (u \cdot λ)) + (X : Δ (u s \cdot λ))$ , and the same formula holds in $K_{0} (O_{λ})$ . The resulting flag is not in general split; splitting is neither asserted nor used.

Facts & Assumptions

Given: The regular integral block $O_{λ}$ , a simple reflection $s$ , the wall-crossing functor $θ_{s} = T_{o u t} T_{o n}$ , and the published single-wall translation datum for the wall of $s$ .

[F1]

The onto-wall translation satisfies $T_{o n} Δ (w \cdot λ) ≅ Δ (w \cdot μ)$ for every $w$ , and the out-of-wall translation satisfies $T_{o u t} Δ (w \cdot μ) ≅$ a finite Verma flag with exactly the two factors $Δ (w \cdot λ)$ and $Δ (w s \cdot λ)$ , each of multiplicity one, with class $[Δ (w \cdot λ)] + [Δ (w s \cdot λ)]$ (The wall-crossing functor and its unit and counit, Translation to and from a single wall on standard modules).

[F2]

$θ_{s}$ is exact; a finite Verma flag of $X$ pushes forward under an exact functor to a finite flag of $θ_{s} X$ ; Verma-flag multiplicities are independent of the flag; the class in $K_{0} (O_{λ})$ of a Verma-filtered object is the sum of its factor classes (The wall-crossing functor and its unit and counit, Finite Verma flags and their multiplicities, Verma-flag multiplicities are independent of the flag, The Grothendieck group and character of O).

[F3]

Direct summands of Verma-filtered objects are Verma-filtered, and tensoring by a finite-dimensional module preserves Verma flags (Direct summands of Verma-filtered objects are Verma-filtered, Finite-dimensional tensoring preserves Verma flags).

Proof

technique · direct

1.1F1F2

$θ_{s} Δ (w \cdot λ) ≅ T_{o u t} Δ (w \cdot μ)$ by the first clause of [F1], hence it has the two-factor finite Verma flag of the second clause of [F1] with factors $Δ (w \cdot λ)$ and $Δ (w s \cdot λ)$ , each once; its class in $K_{0} (O_{λ})$ is $[Δ (w \cdot λ)] + [Δ (w s \cdot λ)]$ by [F2].

2.1F2step 1.1

Let $X$ be Verma-filtered with finite flag $0 = X_{0} \subseteq X_{1} \subseteq \dots \subseteq X_{n} = X$ and factors $Δ (v_{i} \cdot λ)$ . Exactness of $θ_{s}$ by [F2] makes $0 = θ_{s} X_{0} \subseteq \dots \subseteq θ_{s} X_{n} = θ_{s} X$ an increasing chain of subobjects, and each quotient $θ_{s} X_{i} / θ_{s} X_{i - 1} ≅ θ_{s} (X_{i} / X_{i - 1}) ≅ θ_{s} Δ (v_{i} \cdot λ)$ carries the two-factor flag of step 1.1; concatenating these two-step flags through the chain yields a finite Verma flag of $θ_{s} X$ with factor multiplicities $(X : Δ (u \cdot λ)) + (X : Δ (u s \cdot λ))$ , where the independence of the flag [F2] makes the count well defined.

3.1F2F3step 2.1algebra∎

Summing the classes along the flag of step 2.1 gives $[θ_{s} X] = \sum_{i} ([Δ (v_{i} \cdot λ)] + [Δ (v_{i} s \cdot λ)]) = [X] + \sum_{i} [Δ (v_{i} s \cdot λ)]$ , with the multiplicities stated in step 2.1; no splitting of the two-step extensions is used. If $θ_{s} X = Y \oplus Z$ , [F3] makes both summands Verma-filtered, and their multiplicities add to those of $θ_{s} X$ ; the formula for the whole object does not prescribe either summand's individual multiplicities.

Depends on

Used by

Dependency tree · two levels

38 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Wolfgang Soergel, Kategorie O, perverse Garben und Moduln ueber den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990) 421-445 (MPIM preprint 1989/46 full text) (standard reference, not scraped)
Arthur Garnier, The classical theory of Soergel bimodules and the link with Lusztig's conjecture, Oberwolfach Arbeitsgemeinschaft note (2022) (standard reference, not scraped)