Standard graph bimodules, support filtrations and characters

Graphs and supports. Keep $k=\mathbb{Q}$, $R=\mathbb{Q}[x_1,\dots ,x_n]$ with $\mathrm{deg}{x}_i=2$, and the affine scheme $V={\mathbb{A}}_k^n=\mathrm{Spec}R$, whose $k$-points are $V(k)=k^n$. Use the place-permuting action of $S_n$ from The standard type-A reflection realization and its polynomial ring. For $w\in S_n$ let $Gr(w):=\mathrm{Graph}(w:V\to V)\subseteq V{\times }_{k}V,Gr(w)(k)=\{(w\lambda ,\lambda ):\lambda \in k^n\},Gr(A):={\bigcup }_{w\in A}Gr(w)(A\subseteq S_n),$ so that $V{\times }_{k}V=\mathrm{Spec}(R{\otimes }_{k}R)$ and $Gr(w)$ is the graph of the map $\lambda \mapsto w\lambda$. A graded $(R,R)$-bimodule $M$ is the same thing as a graded module over $R{\otimes }_{k}R$, and the support of $m\in M$ is the zero set $\mathrm{supp}(m):=V(\mathrm{Ann}(m))\subseteq V\times V$ of its annihilator ideal in $R{\otimes }_{k}R$. For $A\subseteq S_n$ put ${\mathrm{\Gamma }}_{A}M:=\{m\in M:\mathrm{supp}(m)\subseteq Gr(A)\},$ a graded sub-bimodule of $M$. Since $Gr(A)$ is closed with radical ideal $I(Gr(A))={\bigcap }_{w\in A}I(Gr(w))$ (an intersection of prime ideals, since each $Gr(w)$ is a linear subspace of $V\times V$ complementing the second factor), an element $m$ lies in ${\mathrm{\Gamma }}_{A}M$ exactly when $I(Gr(A){)}^{N}m=0$ for some $N\ge 0$: this is the Nullstellensatz together with the Noetherianity of $R{\otimes }_{k}R$. The ideal itself need not annihilate such an element — in $(R{\otimes }_{k}R)\mathrm{/}I(Gr(e){)}^2$ the class of $1$ is supported on $Gr(e)$ but is not killed by $I(Gr(e))$ — which is why the criterion is stated with a power of the ideal. For $i\in \mathbb{Z}$ we write ${\mathrm{\Gamma }}_{\ge i}M:={\mathrm{\Gamma }}_{\{x:\mathrm{\ell }(x)\ge i\}}M,{\mathrm{\Gamma }}_{\le i}M:={\mathrm{\Gamma }}_{\{x:\mathrm{\ell }(x)\le i\}}M,$ using the length function of Type-A reduced words and the Coxeter presentation, while for $y\in S_n$ the $y$-indexed symbols denote the Bruhat principal-set cutoffs of Soergel's Notation 6.1, ${\mathrm{\Gamma }}_{\le y}M:={\mathrm{\Gamma }}_{\{x:x\le y\}}M,{\mathrm{\Gamma }}_{<y}M:={\mathrm{\Gamma }}_{\{x:x<y\}}M,{\mathrm{\Gamma }}_{\ge y}M:={\mathrm{\Gamma }}_{\{x:x\ge y\}}M,{\mathrm{\Gamma }}_{>y}M:={\mathrm{\Gamma }}_{\{x:x>y\}}M,$ with ${\mathrm{\Gamma }}_{y}M:={\mathrm{\Gamma }}_{\{y\}}M$ the elements concentrated on the single graph $Gr(y)$ and ${\mathrm{\Gamma }}_{\mathrm{\ne }y}M:={\mathrm{\Gamma }}_{S_n\setminus \{y\}}M$ the elements concentrated off $Gr(y)$. The integer-indexed symbols of the previous display are the length cutoffs used to define the flags below, and they must not be confused with the element-indexed Bruhat cutoffs: for $y=s_1$ in $S_4$ the submodule ${\mathrm{\Gamma }}_{\le 1}M$ contains $Gr(s_3)$ while ${\mathrm{\Gamma }}_{\le y}M={\mathrm{\Gamma }}_{\{e,s_1\}}M$ does not, and the two cutoffs agree at the extremes, ${\mathrm{\Gamma }}_{\le w_0}={\mathrm{\Gamma }}_{\le \mathrm{\ell }(w_0)}=M$ and ${\mathrm{\Gamma }}_{<e}={\mathrm{\Gamma }}_{\le -1}=0$. So ${\mathrm{\Gamma }}^{y}M:=M\mathrm{/}{\mathrm{\Gamma }}_{\mathrm{\ne }y}M$ is Soergel's $y$-layer quotient; following Soergel's Notation 6.1 we write the two $y$-layer quotients with an underline, ${\underset{‾}{\mathrm{\Gamma }}}_{\le y}M:={\mathrm{\Gamma }}_{\le y}M\mathrm{/}{\mathrm{\Gamma }}_{<y}M,{\underset{‾}{\mathrm{\Gamma }}}_{\ge y}M:={\mathrm{\Gamma }}_{\ge y}M\mathrm{/}{\mathrm{\Gamma }}_{>y}M,$ so that ${\underset{‾}{\mathrm{\Gamma }}}_{\le y}M$ is the quotient of the lower support submodule by ${\mathrm{\Gamma }}_{<y}M$ and ${\underset{‾}{\mathrm{\Gamma }}}_{\ge y}M$ the quotient of the upper support submodule by the part supported strictly above $Gr(y)$.

Standard bimodules. For $w\in S_n$ let $R_w$ be the graded $(R,R)$-bimodule which equals $R$ as a graded $k$-vector space and has $f\cdot x=fx,x\cdot g=w(g)x(f,g,x\in R),$ where $w(g)(\lambda ):=g(w^{-1}\lambda )$; $R_w$ is generated by the element $1$ of degree $0$. Write $(k)$ for the external shift $M(k):=M\{-k\}$ of Associative graded algebras, bimodules, and internal shifts, so that $M(k{)}_d=M_{d+k}$; this is the shift convention of the type-A sources, and it is the one in which $B_i=R{\otimes }_{R^{s_i}}R(1)$ of The Soergel bimodule $B_i$ of a simple reflection has $1\otimes 1$ of degree $-1$. Put ${\mathrm{\Delta }}_w(d):=R_w(d-\mathrm{\ell }(w)),{\mathrm{\nabla }}_w(d):=R_w(d+\mathrm{\ell }(w))(d\in \mathbb{Z}),$ so that ${\mathrm{\Delta }}_w(d)$ is the standard bimodule $R_w$ with its generator in degree $\mathrm{\ell }(w)-d$, and ${\mathrm{\nabla }}_w(d)$ is $R_w$ with its generator in degree $-\mathrm{\ell }(w)-d$. Thus ${\mathrm{\Delta }}_w(0)=R_w\{\mathrm{\ell }(w)\}$, ${\mathrm{\nabla }}_w(0)=R_w\{-\mathrm{\ell }(w)\}$ and ${\mathrm{\Delta }}_e(d)={\mathrm{\nabla }}_e(d)=R(d)$ is generated in degree $-d$; in particular ${\mathrm{\Delta }}_e(0)={\mathrm{\nabla }}_e(0)=R_e=R$.

For graded free $R$-modules in this type-A page, a homogeneous free generator of degree $a$ contributes $v^a$ to graded rank. This is the $v\mapsto v^{-1}$ transform of Soergel's rank convention in his Notation 5.2; all rank formulas below use this convention.

Support filtrations. A $\mathrm{\Delta }$-flag of a graded $R$-bimodule $M$ is a finite chain $0=M_0\subseteq M_1\subseteq \cdots \subseteq M_k=M$ of graded sub-bimodules which refines the length filtration $M={\mathrm{\Gamma }}_{\ge 0}M\supseteq {\mathrm{\Gamma }}_{\ge 1}M\supseteq \cdots \supseteq {\mathrm{\Gamma }}_{\ge \mathrm{\ell }(w_0)+1}M=0$ of $M$ (here ${\mathrm{\Gamma }}_{\ge 0}M=M$ for every bimodule supported on $Gr(S_n)={\bigcup }_{w}Gr(w)$, which all bimodules of this page are, and ${\mathrm{\Gamma }}_{\ge i}M=0$ for $i>\mathrm{\ell }(w_0)$ because no graph $Gr(x)$ with $\mathrm{\ell }(x)>i$ remains) and whose successive quotients are standard bimodules $M_j\mathrm{/}{M}_{j-1}\cong R_{x_j}\{a_j\}$. We require each length layer to be a finite direct sum of shifted graph bimodules of that length, as in Soergel's Definition 5.4. Thus the quotients carry a numbering with $\mathrm{\ell }(x_1)\ge \cdots \ge \mathrm{\ell }(x_k)$ and each $R_{x_j}\{a_j\}$ is one graph, so that $M$ has a "support flag" in the sense of Soergel's $F_{\mathrm{\Delta }}$. A $\mathrm{\nabla }$-flag has the same direct-sum condition on its length layers, with the reverse length filtration ${\mathrm{\Gamma }}_{\le i}M$ and $\mathrm{\ell }(x_1)\le \cdots \le \mathrm{\ell }(x_k)$. The refinement condition is a condition on supports, not an ordering of incomparable quotients of equal length; only the lengths are ordered. For $i=\mathrm{\ell }(x)$ the graded multiplicity is $(M:{\mathrm{\Delta }}_x(d)):=\text{multiplicity of }{\mathrm{\Delta }}_x(d)\text{ in }{\mathrm{\Gamma }}_{\ge i}M\mathrm{/}{\mathrm{\Gamma }}_{\ge i+1}M,$ and, for a $\mathrm{\nabla }$-flag, $(M:{\mathrm{\nabla }}_x(d))$ is the multiplicity of ${\mathrm{\nabla }}_x(d)$ in ${\mathrm{\Gamma }}_{\le i}M\mathrm{/}{\mathrm{\Gamma }}_{\le i-1}M$; a priori both symbols refer to a chosen compatible enumeration (the next items on this page prove that they do not depend on it). With $T_x$ the Hecke basis of The standard basis of the type-A Hecke algebra and its multiplication rule and ${\tilde{T}}_x:=v^{\mathrm{\ell }(x)}{T}_x$, the character sums attached to a flag are $h_{\mathrm{\Delta }}(M):={\sum }_{x,d}(M:{\mathrm{\Delta }}_x(d))v^d{\tilde{T}}_x,h_{\mathrm{\nabla }}(M):={\sum }_{x,d}(M:{\mathrm{\nabla }}_x(d))v^{-d}{\tilde{T}}_x,$ and they are used only through the items that prove these sums intrinsic.

Small $n$. For $n\le 1$ there is a single element $e$ and every object is a finite direct sum of shifts ${⨁}_{j}R(d_j)$. Its one-graph flags have these summands as quotients, so $h_{\mathrm{\Delta }}={\sum }_j{v}^{d_j}{\tilde{T}}_e$ and $h_{\mathrm{\nabla }}={\sum }_j{v}^{-d_j}{\tilde{T}}_e$; in particular both characters send the unit $R$ to $1$.

(a) Maps between standard bimodules. For $v,w\in S_n$ and $a,b\in \mathbb{Z}$, ${\mathrm{Hom}}_{R\text{-}R}(R_v(a),R_w(b))\cong \{\begin{array}{ll}R(b-a),& v=w,\\ 0,& v\mathrm{\ne }w,\end{array}$ where $R(c)$ denotes the free graded $R$-module on one generator of degree $-c$. Indeed a bimodule map is determined by $\phi (1)=h\in R$, and $f\cdot 1\cdot g\mapsto fhw(g)$ must equal $v(g)fh$; since $R$ is a domain this forces $v(g)h=w(g)h$ for all $g$, hence $h=0$ or $v=w$. The generator $1$ of $R_v(a)$ has degree $-a$ and maps to $h$, which has degree $\mathrm{deg}h-b$ as an element of $R_w(b)$, so the homogeneous map ${\phi }_h$ has degree $\mathrm{deg}h+a-b$ and $\mathrm{Hom}(R_v(a),R_w(b))\cong R(b-a)$ in the case $v=w$.

(b) Composition. The $R$-balanced assignment $R_w{\otimes }_R{R}_v⟶R_{wv},a\otimes b⟼aw(b),$ where $w(b)(\lambda )=b(w^{-1}\lambda )$ and the product is taken in $R$, is a well-defined degree-zero isomorphism of graded $(R,R)$-bimodules. It is well defined because for $f\in R$ the relation $(a\cdot f)\otimes b=a\otimes (fb)$ reads $w(f)a\otimes b=a\otimes fb$ and maps to $w(f)aw(b)=aw(f)w(b)$ on both sides; it is left $R$-linear, and it respects the right actions because $w(bh)=w(b)w(h)$ for $h\in R$. It sends the $1$-tensor $1\otimes 1$ of degree $0$ to $1$ and is therefore surjective, and both sides are free of rank one as left $R$-modules, so it is an isomorphism. In particular $R_w{\otimes }_R{R}_v\cong R_{wv}$ (and $R_w{\otimes }_R{R}_v\cong R_w$ only when $v=e$), while tensoring with $w=e$ shows that the assignment is compatible with the unit. The single assignment $1\otimes 1\mapsto 1$ does extend uniquely to a bimodule map, and the map it extends to is exactly the twisted one above; what fails is the untwisted formula $a\otimes b\mapsto ab$, which is not balanced when $w\mathrm{\ne }e$: the equality $(1\cdot f)\otimes 1=1\otimes f$ in $R_w{\otimes }_R{R}_v$ would map to $w(f)$ on the left and $f$ on the right, and these differ for a polynomial moved by $w$.

(c) The rank-one calculus. Let $s=s_i$, $R^s=R^{s_i}$, $\alpha ={\alpha }_i$, $\delta :=\alpha \mathrm{/}2$, and $B_s=R{\otimes }_{R^s}R(1)$ with the images $u:=1\otimes 1$ of degree $-1$ and $w_0:=1\otimes \delta$ of degree $1$ (The Soergel bimodule $B_i$ of a simple reflection); these two elements form a graded left $R$-basis of $B_s$, and every $g\in R$ has a unique decomposition $g=g^{+}+\delta g^{-}$ with $g^{\pm }\in R^s$, given by $g^{+}=\frac{1}{2}(g+s(g))$. The right action on the basis is $u\cdot g=g^{+}u+g^{-}{w}_0,w_0\cdot g={\delta }^2{g}^{-}u+g^{+}{w}_0,$ as follows from $1\otimes g=1\otimes (g^{+}+\delta g^{-})=g^{+}u+g^{-}{w}_0$ and $\delta g=\delta g^{+}+{\delta }^2{g}^{-}$ with ${\delta }^2={\alpha }^2\mathrm{/}4\in R^s$. Hence $(\delta u+w_0)\cdot g=g(\delta u+w_0),(\delta u-w_0)\cdot g=s(g)(\delta u-w_0),$ so $R(\delta u+w_0)\cong R(-1)$ and $R(\delta u-w_0)\cong R_s(-1)$ are sub-bimodules of $B_s$, generated in degree $1$. Moreover for $f,h\in R$ one has $(fu+h{w}_0)\mapsto f+h\delta$ under $f\otimes g\mapsto fg$ and $(fu+h{w}_0)\mapsto f-h\delta$ under $f\otimes g\mapsto fs(g)$, because $1\otimes g=g^{+}u+g^{-}{w}_0$ and $s(g)=g^{+}-\delta g^{-}$; both maps are surjective, their kernels are the two displayed sub-bimodules, and all four maps of $0\to R(-1)\stackrel{1\mapsto \delta u+w_0}{\to }{B}_s\stackrel{f\otimes g\mapsto fs(g)}{\to }{R}_s(1)\to 0,$ $0\to R_s(-1)\stackrel{1\mapsto \delta u-w_0}{\to }{B}_s\stackrel{f\otimes g\mapsto fg}{\to }R(1)\to 0$ are degree zero: sources and targets are generated in degree whose shifts match the degree-$1$ elements $\delta u\pm w_0\in B_s$. Finally the two quotients are the image of $u$, a free rank-one $R$-module with right action $u\cdot g\equiv gu$ modulo $\delta u-w_0$ and $u\cdot g\equiv s(g)u$ modulo $\delta u+w_0$ respectively, that is $R(1)$ and $R_s(1)$.

(d) Imported statements of Soergel, recorded for the later items. The later items of this page cite the following results of Soergel's Kazhdan–Lusztig-Polynome und unzerlegbare Bimoduln through this definition; we record their content here, in the notation fixed above, so that every later citation has a textual anchor in this item. Let $\mathcal{B}$ be Soergel's category of special bimodules (Definition 5.11). The concrete criterion of Lemma 5.13 is that $B\in \mathcal{B}$ precisely when $B\oplus C\cong D$ for finite sums $C,D$ of shifted Bott–Samelson products. It does not say that $B$ itself is such a sum. Let $\mathrm{add}\mathcal{B}$ be the category of direct summands of special bimodules (Bemerkung 6.12). Since every special bimodule is a summand of such a $D$, and every word is special, this is exactly the idempotent completion defining ${\mathrm{S}\mathrm{B}\mathrm{i}\mathrm{m}}_n$.

1. Closure under tensoring with a generator. (Proposition 5.7(1) and Proposition 5.9(1).) If $M\in F_{\mathrm{\Delta }}$ then $R{\otimes }_{R^s}M\in F_{\mathrm{\Delta }}$, and if $M\in F_{\mathrm{\nabla }}$ then $R{\otimes }_{R^s}M\in F_{\mathrm{\nabla }}$; in both cases the quotients of the resulting flag are again standard bimodules, indexed by the quotients of the flag of $M$. Moreover (the proofs of the same two propositions) the multiplicities of the new flag are given, for $\mathrm{\ell }(x)>\mathrm{\ell }(sx)$, by the recursions $(B_s{\otimes }_{R}M:{\mathrm{\Delta }}_x(d))=(M:{\mathrm{\Delta }}_x(d+1))+(M:{\mathrm{\Delta }}_{sx}(d)),(B_s{\otimes }_{R}M:{\mathrm{\Delta }}_{sx}(d))=(M:{\mathrm{\Delta }}_x(d))+(M:{\mathrm{\Delta }}_{sx}(d-1)),$ and dually $(B_s{\otimes }_{R}M:{\mathrm{\nabla }}_x(d))=(M:{\mathrm{\nabla }}_x(d-1))+(M:{\mathrm{\nabla }}_{sx}(d)),(B_s{\otimes }_{R}M:{\mathrm{\nabla }}_{sx}(d))=(M:{\mathrm{\nabla }}_x(d))+(M:{\mathrm{\nabla }}_{sx}(d+1))\mathrm{.}$ In Notation 5.6 the source uses the shifted reflection functor ${\theta }_s=R[1]{\otimes }_{R^s}(-)=B_s{\otimes }_R-$, with $[1]=(1)=\{-1\}$. Read in the other direction, the two recursions say that a flag quotient of $M$ with graph $y$ and index $e$ produces the two quotients with graphs $sy$ and $y$ of $B_s{\otimes }_{R}M$: the quotient with graph $sy$ carries the same index $e$, while the index of the quotient with graph $y$ changes by $-1$ in the $\mathrm{\Delta }$-chart when $\mathrm{\ell }(y)>\mathrm{\ell }(sy)$ and by $+1$ when $\mathrm{\ell }(sy)>\mathrm{\ell }(y)$, the two signs being exchanged in the $\mathrm{\nabla }$-chart. The two pairs match the two Hecke expansions of $({\tilde{T}}_s+v)h_{\mathrm{\Delta }}(M)$ and $({\tilde{T}}_s+v)h_{\mathrm{\nabla }}(M)$ in the basis $\{{\tilde{T}}_x\}$, which is the source's own interpretation of the recursions.

2. Stability of the flag categories. (Bemerkung 5.5.) The layers ${\mathrm{\Gamma }}_{\ge i}M\mathrm{/}{\mathrm{\Gamma }}_{\ge i+1}M$ are additive in $M$, so $F_{\mathrm{\Delta }}$ is stable under finite direct sums and, by the graded Krull–Schmidt theorem, under direct summands as well; the same holds for $F_{\mathrm{\nabla }}$ with the filtration ${\mathrm{\Gamma }}_{\le i}$.

3. Localization at a reflection. (Notation 6.9 and Lemma 6.10.) For a reflection $t$ let $R_{(t)}$ be the localization of $R$ at the homogeneous functions that do not vanish identically on the reflection hyperplane $V^t$. For every special bimodule $B$ its localization $B{\otimes }_R{R}_{(t)}$ is a direct summand of a finite direct sum of graded shifts of $R_y{\otimes }_R{R}_{(t)}$ and $R_{y,yt}{\otimes }_R{R}_{(t)}$, for $y<yt$. Here $R_{y,yt}:=(R{\otimes }_{k}R)\mathrm{/}(I(Gr(y))\cap I(Gr(yt)))$ is the coordinate bimodule of the union of the two graphs. The two-graph module generally does not split into its two graph modules over $R_{(t)}$: their common reflection equation has not been inverted. After additionally inverting that equation, the two graphs separate and the two-graph piece splits.

4. Sections have graph support, and the four layers. (Bemerkung 6.2.) Let a graded bimodule carry a finite filtration by sub-bimodules whose successive quotients are single graphs $Gr(x)$. For any section $m$, filter its cyclic $(R{\otimes }_{k}R)$-submodule $C=(R{\otimes }_{k}R)m$ by intersecting with the given filtration. Each successive quotient of this induced filtration embeds in a single-graph module. Every nonzero submodule of a single-graph module has that same support, since its coordinate ring is the domain $R$; a zero quotient has empty support. Support is the union of the supports of the successive quotients in a finite filtration, so $\mathrm{supp}(m)=\mathrm{supp}(C)$ is a union of graphs $Gr(x)$ with $x\in S_n$. Consequently ${\mathrm{\Gamma }}_{y}B\cap {\mathrm{\Gamma }}_{\mathrm{\ne }y}B=0$, so the evident maps ${\mathrm{\Gamma }}_{y}B\to {\underset{‾}{\mathrm{\Gamma }}}_{\le y}B$ and ${\underset{‾}{\mathrm{\Gamma }}}_{\ge y}B\to {\mathrm{\Gamma }}^{y}B$ (the second is the composite of the quotient map onto ${\mathrm{\Gamma }}_{\ge y}B\mathrm{/}{\mathrm{\Gamma }}_{>y}B$ with the inclusion of that quotient in $B\mathrm{/}{\mathrm{\Gamma }}_{\mathrm{\ne }y}B$, which is defined because ${\mathrm{\Gamma }}_{>y}B\subseteq {\mathrm{\Gamma }}_{\mathrm{\ne }y}B$) are injective, and ${\mathrm{\Gamma }}_{y}B={\mathrm{\Gamma }}_{\le y}B\cap {\mathrm{\Gamma }}_{\ge y}B$. Moreover right multiplication by a nonzero $p\in R$ does not change supports, $({\mathrm{\Gamma }}_{A}B)p={\mathrm{\Gamma }}_A(Bp)$, so the localizations below may be formed layer by layer.

5. Top-layer localization. (Notation 6.5 and Satz 6.6, extended in Lemma 6.13.) For $y\in S_n$ put $p_y:={\prod }_{t:yt<y}{c}_t$, the product over the finitely many reflections $t=(ab)$ with $a<b$, using the fixed hyperplane equations $c_t=x_a-x_b$ of The standard type-A reflection realization and its polynomial ring. For $B\in \mathcal{B}$ the natural morphisms induce isomorphisms ${\mathrm{\Gamma }}_{y}B\to ({\underset{‾}{\mathrm{\Gamma }}}_{\le y}B)p_y$ and ${\underset{‾}{\mathrm{\Gamma }}}_{\ge y}B\to ({\mathrm{\Gamma }}^{y}B)p_y$, where ${\mathrm{\Gamma }}^{y}B=B\mathrm{/}{\mathrm{\Gamma }}_{\mathrm{\ne }y}B$ is the $y$-layer quotient of Notation 6.1; the first paragraph of the proof of Lemma 6.13 observes that the isomorphism ${\mathrm{\Gamma }}_{y}N\cong ({\underset{‾}{\mathrm{\Gamma }}}_{\le y}N)p_y$ of Satz 6.6 holds for all $N\in \mathrm{add}\mathcal{B}$ as well, and the same holds for the second isomorphism, so both are available for every $N\in \mathrm{add}\mathcal{B}$.

6. Hom formula for the additive closure. (Theorem 5.15, Lemma 6.13 and Satz 6.14(4).) For $M\in F_{\mathrm{\Delta }}$ and $N\in \mathrm{add}\mathcal{B}$, and likewise for $M\in \mathrm{add}\mathcal{B}$ and $N\in F_{\mathrm{\nabla }}$, the graded module ${\mathrm{Hom}}_{R\text{-}R}(M,N)$ is free with $\mathrm{rk}{\mathrm{Hom}}_{R\text{-}R}(M,N)={\sum }_{x,d,e}(M:{\mathrm{\Delta }}_x(d))(N:{\mathrm{\nabla }}_x(e))v^{d-e},$ and $\mathrm{add}\mathcal{B}=\mathcal{B}$ by Satz 6.14(4), where $\mathcal{B}$ has the stable-sum meaning just specified. For the first step of Lemma 6.13, use item 5 and item 7: the canonical lower layer of $N$ is ${⨁}_{\mu }{\mathrm{\nabla }}_y(\mu {)}^{\oplus (N:{\mathrm{\nabla }}_y(\mu ))}$, and its submodule obtained by multiplication by $p_y$ is ${\mathrm{\Gamma }}_{y}N$. Since $\mathrm{deg}{p}_y=2\mathrm{\ell }(y)$, its graded rank in our convention is ${\sum }_{\mu }(N:{\mathrm{\nabla }}_y(\mu ))v^{\mathrm{\ell }(y)-\mu }$. Item 8 identifies this with $\mathrm{Hom}(R_y,N)$; replacing $R_y$ by ${\mathrm{\Delta }}_y=R_y(-\mathrm{\ell }(y))$ subtracts $\mathrm{\ell }(y)$ from map degrees, giving ${\sum }_{\mu }(N:{\mathrm{\nabla }}_y(\mu ))v^{-\mu }$, as required. In general one takes $x$ of maximal length with $(M:{\mathrm{\Delta }}_x(d))\mathrm{\ne }0$ for some $d$; then ${\mathrm{\Gamma }}_{x}M\hookrightarrow M\twoheadrightarrow \mathrm{coker}$ is a short exact sequence with both outer terms in $F_{\mathrm{\Delta }}$ and $(M:{\mathrm{\Delta }}_y(d))=({\mathrm{\Gamma }}_{x}M:{\mathrm{\Delta }}_y(d))+(\mathrm{coker}:{\mathrm{\Delta }}_y(d))$, and Theorem 5.15 forces the sequence $\mathrm{Hom}(\mathrm{coker},N)\hookrightarrow \mathrm{Hom}(M,N)\twoheadrightarrow \mathrm{Hom}({\mathrm{\Gamma }}_{x}M,N)$ to be exact for every $N\in \mathcal{B}$; being additive in $N$, it stays exact for every $N\in \mathrm{add}\mathcal{B}$, which closes the induction over the length of a $\mathrm{\Delta }$-flag of $M$. The second case is analogous, using the dual maximality argument over a $\mathrm{\nabla }$-flag of $N$.

7. Bruhat-refining enumerations. (Lemma 6.3 and its proof.) Let $B\in F_{\mathrm{\nabla }}$ and let $y_0,y_1,\dots$ be an enumeration of $S_n$ in which Bruhat-larger elements have larger index; put $C(k)=\{y_0,\dots ,y_k\}$ and $y_k=y$. Then the evident map ${\mathrm{\Gamma }}_{\le y}B\mathrm{/}{\mathrm{\Gamma }}_{<y}B\to {\mathrm{\Gamma }}_{C(k)}B\mathrm{/}{\mathrm{\Gamma }}_{C(k-1)}B$ is an isomorphism, both sides are finite direct sums of objects of the form ${\mathrm{\nabla }}_y(\mu )$, and ${\mathrm{\nabla }}_y(\mu )$ occurs in this quotient exactly $(B:{\mathrm{\nabla }}_y(\mu ))$ times as a direct summand; the analogous statement holds for $F_{\mathrm{\Delta }}$ after reversing the enumeration and replacing the lower layer by ${\mathrm{\Gamma }}_{\ge y}B\mathrm{/}{\mathrm{\Gamma }}_{>y}B$: for a descending Bruhat enumeration $z_0,z_1,\dots$, $C^{\mathrm{\prime }}(k)=\{z_0,\dots ,z_k\}$ and $z_k=y$, this upper layer maps isomorphically to ${\mathrm{\Gamma }}_{C^{\mathrm{\prime }}(k)}B\mathrm{/}{\mathrm{\Gamma }}_{C^{\mathrm{\prime }}(k-1)}B$ and is a direct sum of ${\mathrm{\Delta }}_y(\mu )$ with multiplicities $(B:{\mathrm{\Delta }}_y(\mu ))$. The proof compares two such enumerations through the finitely many steps that swap two adjacent incomparable elements: incomparable elements differ by no reflection, so by the extension-vanishing input there is no ${\mathrm{E}\mathrm{x}\mathrm{t}}^1$ between the corresponding subquotients, and the two filtrations have the same subquotients up to order.

8. Freeness over the invariant ring of a graph. (Proposition 6.4.) For $B\in \mathcal{B}$ and $y\in S_n$ the graded modules ${\underset{‾}{\mathrm{\Gamma }}}_{\le y}B$, ${\underset{‾}{\mathrm{\Gamma }}}_{\ge y}B$, ${\mathrm{\Gamma }}^{y}B$ and ${\mathrm{\Gamma }}_{y}B$ are free graded right $R$-modules on which the action of $R{\otimes }_{k}R$ factors through the graph ring $R_y$; in particular evaluation at the generator gives an isomorphism ${\mathrm{Hom}}_{R\text{-}R}(R_y,B)\cong {\mathrm{\Gamma }}_{y}B$, and $B{\otimes }_R{R}_x\in F_{\mathrm{\Delta }}$ for every $x\in S_n$.

(e) Opposite bimodules. For a graded $(R,R)$-bimodule $M$ let $M^{\mathrm{o}\mathrm{p}}$ be the graded $(R,R)$-bimodule with the same underlying graded $k$-module and operations $r{\cdot }^{\mathrm{o}\mathrm{p}}m{\cdot }^{\mathrm{o}\mathrm{p}}s=smr$; since $R$ is commutative this is again a graded $(R,R)$-bimodule, and the following four remarks are used by the later support-filtration items.

1. The flip $m\otimes n\mapsto n\otimes m$ is a well-defined degree-zero bijection $(M{\otimes }_{R}N{)}^{\mathrm{o}\mathrm{p}}\to N^{\mathrm{o}\mathrm{p}}{\otimes }_R{M}^{\mathrm{o}\mathrm{p}}$, because $mr\otimes n=m\otimes rn$ maps to $n\otimes mr=(n{\cdot }^{\mathrm{o}\mathrm{p}}r)\otimes m=rn\otimes m$ in $N^{\mathrm{o}\mathrm{p}}{\otimes }_R{M}^{\mathrm{o}\mathrm{p}}$.
2. $R_x^{\mathrm{o}\mathrm{p}}\cong R_{x^{-1}}$ through $m\mapsto x^{-1}(m)$, and $B_s^{\mathrm{o}\mathrm{p}}\cong B_s$ through the flip $f\otimes g\mapsto g\otimes f$; both are degree-zero isomorphisms of graded $(R,R)$-bimodules.
3. Write $\tau$ for the flip of the two factors of $V\times V$. A nonzero element $m$ of a standard bimodule has support exactly the single graph of its standard, since the annihilator of the generator of $R_w$ is the prime ideal $I(Gr(w))$; for the same reason ${\mathrm{Ann}}_{M^{\mathrm{o}\mathrm{p}}}(m)=\sigma ({\mathrm{Ann}}_M(m))$ under the flip $\sigma (f\otimes g)=g\otimes f$, so ${\mathrm{supp}}_{M^{\mathrm{o}\mathrm{p}}}(m)=\tau ({\mathrm{supp}}_M(m))$. Since $\tau (Gr(x))=Gr(x^{-1})$ and $\mathrm{\ell }(x^{-1})=\mathrm{\ell }(x)$, the length filtrations satisfy ${\mathrm{\Gamma }}_{\ge i}(M^{\mathrm{o}\mathrm{p}})={\mathrm{\Gamma }}_{\ge i}M$ and ${\mathrm{\Gamma }}_{\le i}(M^{\mathrm{o}\mathrm{p}})={\mathrm{\Gamma }}_{\le i}M$ as $k$-submodules.
4. Consequently a $\mathrm{\Delta }$-flag of $M$ with quotients $R_{x_j}\{a_j\}$ is a $\mathrm{\Delta }$-flag of $M^{\mathrm{o}\mathrm{p}}$ with quotients $R_{x_j^{-1}}\{a_j\}$ in the same order, and a $\mathrm{\nabla }$-flag of $M$ is a $\mathrm{\nabla }$-flag of $M^{\mathrm{o}\mathrm{p}}$ in the same order: the graph index inverts, the length is unchanged by item 3, and the refined length filtration is preserved by the opposite functor, so this functor preserves each of the two flag categories, $M\in F_{\mathrm{\Delta }}⟺M^{\mathrm{o}\mathrm{p}}\in F_{\mathrm{\Delta }}$ and $M\in F_{\mathrm{\nabla }}⟺M^{\mathrm{o}\mathrm{p}}\in F_{\mathrm{\nabla }}$.