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Type-A top support layers are controlled by reflection localization

Statement

For a reflection $t = (a b)$ with $a < b$ , fix the hyperplane equation $c_{t} : = β_{a b} = x_{a} - x_{b}$ of The standard type-A reflection realization and its polynomial ring. For $y \in S_{n}$ put $p_{y} : = \prod_{t : y t < y} c_{t} \in R$ . Let $M$ be a Bott–Samelson bimodule or a direct summand of one, with its support submodules $Γ_{y} M \subseteq Γ_{\leq y} M \subseteq M$ and $Γ_{y} M \subseteq Γ_{\geq y} M \subseteq M$ and the $y$ -layer quotients ${\underline{Γ}}_{\leq y} M = Γ_{\leq y} M / Γ_{< y} M$ and ${\underline{Γ}}_{\geq y} M = Γ_{\geq y} M / Γ_{> y} M$ of Standard graph bimodules, support filtrations and characters, and write $Γ^{y} M : = M / Γ_{\neq y} M$ for Soergel's $y$ -layer quotient. Then:

the natural morphism $Γ_{y} M \to {\underline{Γ}}_{\leq y} M$ , the inclusion of the $y$ -layer into the lower support submodule followed by the quotient map, is injective with image the submodule $p_{y} \cdot {\underline{Γ}}_{\leq y} M$ obtained as the image of right multiplication by $p_{y}$ , so it becomes an isomorphism after inverting $p_{y}$ , that is $Γ_{y} M ≅ ({\underline{Γ}}_{\leq y} M) p_{y}$ , and the $y$ -layer is generated by the lower $y$ -layer after inverting the hyperplane equations of the reflections $t$ with $y t < y$ ;
dually, the natural morphism ${\underline{Γ}}_{\geq y} M \to Γ^{y} M = M / Γ_{\neq y} M$ is injective with image the image $p_{y} \cdot Γ^{y} M$ of right multiplication by $p_{y}$ , so it becomes an isomorphism after the same inversion, that is ${\underline{Γ}}_{\geq y} M ≅ (Γ^{y} M) p_{y}$ and the upper $y$ -layer is determined by the $y$ -layer quotient after inverting $p_{y}$ ;
in particular the graded ranks satisfy $rk Γ_{y} M = \sum_{ν} (M : \nabla_{y} (ν)) v^{ℓ (y) - ν}$ and $rk Γ^{y} M = \sum_{ν} (M : Δ_{y} (ν)) v^{- ℓ (y) - ν}$ , so that each graded rank determines the corresponding $y$ -multiplicities of its support flag.

The inclusion chains displayed above are two separate chains: the lower and the upper support submodule are not nested in general, and because the cutoffs of Standard graph bimodules, support filtrations and characters are the Bruhat principal sets ${x : x \leq y}$ and ${x : x \geq y}$ of Soergel's Notation 6.1, their intersection is exactly $Γ_{y} M = Γ_{\leq y} M \cap Γ_{\geq y} M$ . This is Soergel's Satz 6.6 in the notation of his Notation 6.1; the localization input is his Lemma 6.10, quoted in the Facts block, and the rank identities are the computation of Satz 6.6's proof.

Facts & Assumptions

Given: An element $y \in S_{n}$ , the polynomial ring $R$ with the reflections of $S_{n}$ and their fixed hyperplanes in $V = Spec R$ , a Bott–Samelson bimodule $M$ or a direct summand of one, and the elements $p_{y} = \prod_{y t < y} c_{t}$ .

[F1]

$Γ_{y} M$ is the support sub-bimodule on the single graph $G r (y)$ ; $Γ_{\leq y} M = Γ_{{x : x \leq y}} M$ and $Γ_{\geq y} M = Γ_{{x : x \geq y}} M$ are the Bruhat lower and upper support submodules, so that both contain $Γ_{y} M$ and their intersection is $Γ_{y} M$ ; ${\underline{Γ}}_{\leq y} M = Γ_{\leq y} M / Γ_{< y} M$ and ${\underline{Γ}}_{\geq y} M = Γ_{\geq y} M / Γ_{> y} M$ are the two $y$ -layer quotients and $Γ^{y} M = M / Γ_{\neq y} M$ is the $y$ -layer quotient of Notation 6.1; the filtration multiplicity $(M : Δ_{y} (d))$ counts $Δ_{y} (d)$ in the upper $y$ -layer and $(M : \nabla_{y} (d))$ counts $\nabla_{y} (d)$ in the lower $y$ -layer (Standard graph bimodules, support filtrations and characters).

[F2]

$M$ has finitely many support layers, all of the form $G r (x)$ for $x \in S_{n}$ , and each layer is a finite direct sum of standard bimodules, so every $Γ$ -submodule of $M$ is finitely generated and the multiplicities of [F1] are defined and intrinsic (The type-A support filtration multiplicities are intrinsic, Bott–Samelson bimodules carry delta and nabla support filtrations).

[F3]

The two $y$ -layer quotients are finite direct sums of shifted standards, ${\underline{Γ}}_{\leq y} M ≅ ⨁_{ν} \nabla_{y} (ν)^{(M : \nabla_{y} (ν))}$ and ${\underline{Γ}}_{\geq y} M ≅ ⨁_{ν} Δ_{y} (ν)^{(M : Δ_{y} (ν))}$ , with the multiplicities of [F1] (Bruhat-refining enumeration of Soergel's Lemma 6.3, recorded in the definition) (Standard graph bimodules, support filtrations and characters).

[F4]

For a Bott–Samelson bimodule $B$ , evaluation at the generator identifies ${Hom}_{R - R} (R_{y}, B)$ with $Γ_{y} B$ , and the four $y$ -layers ${\underline{Γ}}_{\leq y} B$ , ${\underline{Γ}}_{\geq y} B$ , $Γ^{y} B$ and $Γ_{y} B$ are free graded right $R$ -modules on which $R \otimes_{k} R$ acts through $R_{y}$ (Soergel Proposition 6.4, recorded in the definition). These properties pass to a direct summand $M = e B$ : the support functors, their quotients, and the evaluation map commute with the idempotent $e$ , while a graded summand of a finite free module over the positively graded polynomial ring $R$ is graded free by graded Nakayama. Thus the same evaluation and freeness statements hold for every $M$ in the statement (Standard graph bimodules, support filtrations and characters).

[F5]

Imported (Soergel Satz 6.6, with the observation at the start of the proof of Lemma 6.13 that the isomorphism $Γ_{y} N ≅ ({\underline{Γ}}_{\leq y} N) p_{y}$ of Satz 6.6 holds for every $N$ in the additive closure of the Bott–Samelson bimodules as well): for $M$ a Bott–Samelson bimodule or a direct summand of one the natural morphisms $Γ_{y} M \to {\underline{Γ}}_{\leq y} M$ and ${\underline{Γ}}_{\geq y} M \to Γ^{y} M$ are injective with images the images $p_{y} \cdot {\underline{Γ}}_{\leq y} M$ and $p_{y} \cdot Γ^{y} M$ of right multiplication by $p_{y}$ , and they induce the isomorphisms $Γ_{y} M ≅ ({\underline{Γ}}_{\leq y} M) p_{y}$ and ${\underline{Γ}}_{\geq y} M ≅ (Γ^{y} M) p_{y}$ (Standard graph bimodules, support filtrations and characters).

Proof

1.1

Injectivity and image: [F5] provides both natural morphisms with their injectivity and their images, and the two localisation statements are equivalent to these: localising the inclusion $p_{y} {\underline{Γ}}_{\leq y} M \subseteq {\underline{Γ}}_{\leq y} M$ at $p_{y}$ gives an equality because every element $n$ of the localised target is $(1 / p_{y}) (p_{y} n)$ with $p_{y} n$ in the image, and conversely a morphism which is injective with $p_{y}$ -divisible image has localisation an isomorphism; the same argument applies to ${\underline{Γ}}_{\geq y} M \to Γ^{y} M$ . [F5] 1.2 Ranks of the layer quotients: by [F1] and [F3] the two $y$ -layer quotients are direct sums of copies of $\nabla_{y} (ν) = R_{y} {- ℓ (y) - ν}$ and of $Δ_{y} (ν) = R_{y} {ℓ (y) - ν}$ , so their graded ranks are $rk {\underline{Γ}}_{\leq y} M = \sum_{ν} (M : \nabla_{y} (ν)) v^{- ℓ (y) - ν}$ and $rk {\underline{Γ}}_{\geq y} M = \sum_{ν} (M : Δ_{y} (ν)) v^{ℓ (y) - ν}$ ; the element $p_{y}$ is homogeneous of degree $2 ℓ (y)$ , because each $c_{t}$ is a linear form of degree $2$ and the number of reflections $t$ with $y t < y$ equals $ℓ (y)$ , and multiplication by a nonzero homogeneous element of a free graded right $R$ -module is injective and raises degrees by its own degree; the sums are finite and their coefficients are the intrinsic multiplicities of [F2]. [F1, F2, F3, F4] 2.1 The two rank identities: by step 1.1 the modules $Γ_{y} M$ and $p_{y} {\underline{Γ}}_{\leq y} M$ are isomorphic, so by step 1.2 $rk Γ_{y} M = v^{2 ℓ (y)} rk {\underline{Γ}}_{\leq y} M = \sum_{ν} (M : \nabla_{y} (ν)) v^{ℓ (y) - ν}$ ; dually $Γ^{y} M$ is isomorphic to the quotient, inverse to the inclusion, of $p_{y} Γ^{y} M$ , so $rk Γ^{y} M = v^{- 2 ℓ (y)} rk {\underline{Γ}}_{\geq y} M = \sum_{ν} (M : Δ_{y} (ν)) v^{- ℓ (y) - ν}$ , both computations using that the four modules of [F4] are free and hence $p_{y}$ -torsion-free. [F4, step 1.1, step 1.2] 3.1 Conclusion: claims (1) and (2) are steps 1.1 and 2.1 read together with [F5], and claim (3) is the pair of rank identities of step 2.1; the result is stated for Bott–Samelson bimodules and their direct summands because [F5] is, and no statement about the full Hom formula for arbitrary summands is used. ∎

step 1.1step 2.1

Depends on

Used by

The type-A Soergel Hom formula Theorem

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Sources

Soergel, Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln, Notation 6.5, Lemma 6.10, Lemma 6.13 and Satz 6.6, PDF pp.19–21 (standard reference, not scraped)
Elias–Williamson, Soergel Calculus, §3.6 (standard reference, not scraped)