Proof. Start with the usual geometric root space \(V_0\) and its symmetric Tits form [4]. Let \(K\) be its radical and choose a complement \(V_1\) to \(K\). Adjoin a vector space \(K'\) dual to \(K\), pair \(K\) and \(K'\) nondegenerately, and take \(K'\) to be isotropic and orthogonal to \(V_1\). The resulting form on \[V=V_1\oplus K\oplus K'\] is nondegenerate and extends the original form. Its dimension is \(\dim V_0+\dim K\), the minimal possible dimension of such an extension. This is also minimal under the independent-root and independent-coroot conditions. Indeed, let \(G=((\alpha_s,\alpha_t))_{s,t\in S}\). In any realization \(\widetilde V\) with these pairings and independent simple roots and coroots, evaluation by the root covectors maps \(\widetilde V\) onto \(\mathbb R^S\). Its restriction to the coroot span has matrix \(G\), so its kernel there has dimension \(\operatorname{corank}G\). This subspace lies in the full kernel, so rank–nullity gives \[\dim\widetilde V\ge |S|+\operatorname{corank}G=\dim V_0+\dim K,\] which the construction attains. For distinct simple reflections with \(m_{st}<\infty\), their root plane is positive definite. The displayed reflections have product of order \(m_{st}\) on that plane and fix its orthogonal complement. Thus the Coxeter relations hold on all of \(V\); pairs with \(m_{st}=\infty\) impose no relation. The action restricts to the faithful geometric action on \(V_0\) [4], so it is faithful on \(V\) as well. The simple roots and their corresponding linear forms are independent by nondegeneracy.

For clarity, reflection faithfulness can also be seen directly. If an orthogonal transformation has rank-one difference from the identity, write it as \(g=1+u\otimes\varphi\). Orthogonality first implies \(u^{\flat}=c\varphi\) for some nonzero scalar \(c\), and then implies \(\varphi(u)=-2\). Thus \(g^2=1\) and \((u,u)\ne0\): \(g\) is an orthogonal reflection. If \(g\in W\), the finite-subgroup theorem for Coxeter groups [10] puts \(\langle g\rangle\), after conjugation, in a finite standard parabolic subgroup. Its root span is positive definite, and its orthogonal complement in \(V\) is fixed by that parabolic subgroup. The restriction of \(g\) to the root span still has rank-one difference from the identity, and finite Coxeter geometry identifies it as a Coxeter reflection. Conversely every Coxeter reflection has rank-one difference from the identity. This is the reflection-faithful realization used in [20].

Choose \(\rho\in V^*\) positive on every simple coroot. Such a choice exists by their independence. A real coroot has either nonnegative or nonpositive simple-coroot coefficients. Applying this observation to \(w^{-1}\alpha_s\) gives the displayed length-sign criterion. The same argument applies to roots under the symmetric identification.

The chosen \(\rho\) has trivial stabilizer. If \(w\ne1\), take a left descent \(s\) of \(w\); then \((w\rho)(\alpha_s)<0<\rho(\alpha_s)\), so \(w\rho\ne\rho\). The identification of \(V\) with its dual therefore supplies a regular orbit. The comparison constructions of [23] also use an orbit with stabilizer \(\{1,s\}\) for each simple reflection \(s\). Such an orbit exists here. Put \(H_s=V^s\). An element fixing \(H_s\) pointwise has rank at most one after subtracting the identity, so reflection faithfulness and uniqueness of the orthogonal reflection with hyperplane \(H_s\) force it to be \(1\) or \(s\). For every other element \(w\), \(H_s\cap V^w\) is therefore a proper subspace of \(H_s\). Since \(W\) is countable, a point outside their union exists over \(\mathbb R\) and has stabilizer exactly \(\{1,s\}\). ◻

Proof. We track the shifts explicitly. Temporarily double all degrees, so that linear forms have degree \(2\), and let \(z\) record the resulting generator degrees. Denote the regraded sheaf by \(\widehat B\), and write \(n=\ell(b)\) and \(m=\ell(x)\). With the shift convention \(M(k)^j=M^{j+k}\), put \[M=\Gamma(\widehat B)(n)\] and note that its top generator has degree \(-n\). This is the normalized self-dual indecomposable projective in [23].

We first compute the full upward-kernel rank. For \(w\le b\), let \(M^{[w]}\) be the lower-support layer \[M^{[w]}=\Gamma_{\le w}M/\Gamma_{<w}M,\] where \(\Gamma_JM\) means the submodule of sections supported on \(J\). Restriction to \(x\), together with flabbiness, identifies \(M^{[x]}\) with \(\widehat L_x(U_x)(n)\). Indeed a vector in the full upward kernel, placed at \(x\) and extended by zero outside \(\{y:y\le x\}\), is a section on the upward-closed set \([1,b]\setminus\{y:y<x\}\); flabbiness extends it to a global section supported on \(\{y:y\le x\}\).

The layer in [23] is described instead as the kernel of the map between the projections to the upper sets \(\{y:y\ge x\}\) and \(\{y:y>x\}\). Flabbiness identifies those projections with section modules; a tuple in the kernel has only its \(x\)-coordinate nonzero, with precisely the full upward vanishing conditions. Thus that layer and the lower-support quotient above are canonically identified through \(\widehat L_x(U_x)(n)\).

We now compare the character conventions in the cited sources. In the Hecke normalization \(H_s^2=1+(z^{-1}-z)H_s\), write \(\underline H_w\) for the Kazhdan–Lusztig basis element. The Hecke bar involution is determined by \(\overline z=z^{-1}\) and \(\overline{H_s}=H_s^{-1}\). Fiebig’s twisted graph indexed by \(w\) is \(\{(w^{-1}v,v):v\in V\}\) [23], whereas Elias–Williamson use \(\{(wv,v):v\in V\}\) [20]. Under Fiebig’s equivalence with Soergel bimodules [23], this index reversal and the top normalization identify the corresponding bimodule as the unshifted \(B_{b^{-1}}\). Let \(\mathfrak a\) be the \(\mathbb Z[z,z^{-1}]\)-linear anti-involution of the Hecke algebra defined by \(\mathfrak a(H_w)=H_{w^{-1}}\) [20].

Fiebig’s lower-support character of \(M\) is \(\mathfrak a(\operatorname{ch}_{\nabla}(B_{b^{-1}}))\) in Elias–Williamson’s notation. The coefficient convention can be checked on one generator. Their standard lower-support module \(\nabla_{x^{-1}}\) has its generator in degree \(-m\). A generator of \(M^{[x]}\) in degree \(j\) therefore corresponds to \(\nabla_{x^{-1}}(-m-j)\). Its shift contributes \(z^{-m-j}\) before the coefficient conjugation in \(\operatorname{ch}_{\nabla}\) changes this to \(z^{m+j}\). Thus the coefficient after applying \(\mathfrak a\) is \(z^m\operatorname{grk}_z(M^{[x]})\), as in [23].

Let \(\operatorname{ch}\) denote the standard character in [20]. For a Soergel bimodule, the characters satisfy \(\operatorname{ch}_{\nabla}(B)=\overline{\operatorname{ch}(B)}\) [20]. The character theorem, together with bar invariance of the Kazhdan–Lusztig basis, therefore gives \(\operatorname{ch}_{\nabla}(B_{b^{-1}})=\underline H_{b^{-1}}\). The anti-involution \(\mathfrak a\) commutes with bar and inversion preserves Bruhat order, so the triangular characterization gives \(\mathfrak a(\underline H_{b^{-1}})=\underline H_b\). Using the Kazhdan–Lusztig expansion in [20], the full character identity is \[\begin{align*} \sum_{w\le b}z^{\ell(w)}\operatorname{grk}_z(M^{[w]})H_w &=\mathfrak a(\operatorname{ch}_{\nabla}(B_{b^{-1}})) =\underline H_b\\ &=\sum_{w\le b}z^{n-\ell(w)}P_{w,b}(z^{-2})H_w. \end{align*}\] Comparison of the coefficient of \(H_x\) gives \[ z^m\operatorname{grk}_z(M^{[x]}) =z^{n-m}P_{x,b}(z^{-2}). \tag{5}\] Since \(M^{[x]}\simeq\widehat L_x(U_x)(n)\), this gives \[ \operatorname{grk}_z\widehat L_x(U_x) =z^{2(n-m)}P_{x,b}(z^{-2}). \tag{6}\]

For the stalk rank, write \(M^x\) for the image of the projection of \(M\) to \(\widehat B^x(n)\). Generation by global sections gives \(M^x=\widehat B^x(n)\). Fiebig’s integral graded-module statement [23] says, in the shift convention above, that \[M^x\simeq\bigoplus_i A(k_i) \quad\Longrightarrow\quad M^{[x]}\simeq\bigoplus_i A(2m-k_i).\] Both sides are the actual graded free modules over \(A\). In our positive-generator rank convention, this relation becomes \[\operatorname{grk}_z(M^x) =z^{-2m}\operatorname{grk}_z(M^{[x]})\big|_{z\mapsto z^{-1}}.\] Using \(M^x=\widehat B^x(n)\) and (5), we obtain \[z^{-n}\operatorname{grk}_z(\widehat B^x) =z^{-2m}\left.z^{n-2m}P_{x,b}(z^{-2})\right|_{z\mapsto z^{-1}} =z^{-n}P_{x,b}(z^2).\] Thus \(\operatorname{grk}_z\widehat B^x=P_{x,b}(z^2)\). Finally set \(q=z^2\) in this formula and in (6). This gives (4). ◻

Proof. We use the reflection-subgroup theorem: a reflection subgroup has a canonical Coxeter system whose simple roots are positive ambient roots with positive squared lengths and pairwise nonpositive pairings; see [12, 16, 17, 18]. Every reflection with root in \(\Pi\) preserves \(\Pi\) and has the image of its difference from the identity contained in \(\Pi\). These properties persist under products. If an ambient reflection belongs to \(D\), its root line, being the image of its difference from the identity, therefore lies in \(\Pi\).

The canonical simple roots of \(D\) consequently lie in \(\Pi\). There cannot be three of them: in the pointed intersection of \(\Pi\) with the ambient positive cone, one of three distinct rays lies strictly between the other two. Its root \(\gamma\) is a positive linear combination of the outside roots \(\alpha,\beta\). Pairing this combination with \(\gamma\) would give \[(\gamma,\gamma) =c(\gamma,\alpha)+d(\gamma,\beta)\le0, \qquad c,d>0,\] contrary to its positive squared length. Since \(D\) contains two distinct reflections with independent root lines, its canonical rank is exactly two.

Choose an element \(a_0\) of least ambient length in \(D a\). For \(\gamma\in\Phi_D^+\), minimality and the reflection length-sign criterion (2) give \(a_0^{-1}\gamma>0\). Thus \(a_0^{-1}\) preserves the signs of all roots of \(D\). For \(w\in D\), the signs of \((wa_0)^{-1}\gamma\) and \(w^{-1}\gamma\) agree. This proves the inversion-count formula and identifies the orientations and labels of all relative reflection edges. Conversely, an ambient reflection carrying one element of \(D a\) to another belongs to \(D\), so there are no further ambient edges on this coset. The relative graph has a unique minimum, hence \(a_0\) is also the unique ambient-length minimum. Relative directed paths are ambient directed paths, which proves the asserted implication between orders. In the finite setting, this is the coset graph property of [17].

For the final assertion, use the nondegenerate ambient bilinear space, even when its restriction to \(\Pi\) is degenerate. For independent roots \(\alpha,\beta\), set \[a(v)=\frac{2(v,\alpha)}{(\alpha,\alpha)}, \qquad b(v)=\frac{2(v,\beta)}{(\beta,\beta)}.\] The normal forms \(a,b\) are independent, and the reflection formula gives \[(r_\alpha r_\beta-1)v =\alpha\bigl(-a(v)+a(\beta)b(v)\bigr)-\beta b(v).\] The two coefficient forms are independent as well. Therefore \[\operatorname{im}(r_\alpha r_\beta-1) =\operatorname{span}(\alpha,\beta).\] This image depends only on the product. It gives the common plane for any other two-reflection factorization, and part (1) supplies the maximal dihedral subgroup attached to that plane. The labels of a directed two-edge path are distinct, since repeating a reflection would return to its starting vertex. ◻

Proof. Reduced words alternate between the two canonical generators. In the infinite group there are two elements of every positive length. In \(I_2(m)\) there are two elements of each length strictly between \(0\) and \(m\), and one element at lengths \(0\) and \(m\). The identity is below every element. At positive shorter length, either alternating word occurs as a subword of every longer alternating word: choose the run beginning at its first or second letter. The subword criterion [4] gives comparability in length order. At equal length, comparability forces equality. These facts give the stated interval shape, including a possible finite longest element as the singleton top.

An odd alternating reduced word is a palindrome and hence is conjugate to its middle generator, while every reflection has odd sign. Thus the reflections of \(D\) are exactly its elements of odd Coxeter length. For \(p<_D q\), the sign of \(qp^{-1}\) is \((-1)^{\ell_D(q)-\ell_D(p)}\), which proves the left-reflection edge criterion. A two-edge path must therefore have an even total rank difference. If \(n=2h>0\), its intermediate vertex may occupy precisely the \(h\) odd offsets \(1,3,\ldots,2h-1\). Each of those internal ranks has two vertices, giving \(2h=n\) paths. ◻

Proof. For a finite Bruhat interval \(Q\), let \(\mathcal E_Q\) be the least relation containing its cover pairs and closed under the following rule. Arrange six distinct vertices in four layers \[\{a\}\mid\{p,q\}\mid\{r,s\}\mid\{c\}.\] If all eight ordered pairs from one layer to the next are in \(\mathcal E_Q\), add \((a,c)\). This rule depends only on the poset.

First suppose that the eight pairs in an application of the rule are actual directed reflection edges. The two paths from \(a\) to \(r\) through \(p,q\) have a common root plane by Lemma 7(3). The two labels leaving \(a\) are distinct and span that plane. Applying the same argument to the paths from \(a\) to \(s\) puts all four middle edges in this plane. The labels of \(p\to r\) and \(p\to s\) are again distinct, so the two paths from \(p\) to \(c\) put the last two labels there as well. Thus all six vertices lie in one maximal plane coset. By Lemmas 7 and 8, the three edges from \(a\) through \(p,r\) to \(c\) have positive odd relative gaps; their sum is positive and odd. Lemma 8 gives a relative reflection edge \(a\to c\), which is an ambient edge by Lemma 7. Hence the rule adds only actual reflection edges.

To prove the converse, we first refine every non-cover reflection edge \(a\xrightarrow{t}c=ta\) to a directed three-edge path in one maximal plane coset. Its gap is odd, so write \(d=d(a,c)\ge3\). We use a consequence of simple-reflection lifting: for a directed edge \(z\xrightarrow{\tau}\tau z\) and a simple reflection \(v\), left multiplication gives a directed edge \(vz\to v\tau z\) with label \(v\tau v\), unless \(\tau=v\). For a gap at least three, the new gap is at least the old gap minus two and is positive. For a cover, a reversal would require \(vz>z\) and \(v(\tau z)<\tau z\). The lifting property [4] then gives \(z\le v(\tau z)\); their lengths are equal, so \(z=v(\tau z)\) and \(\tau=v\). When \(\tau=v\), left multiplication indeed swaps the endpoints.

We prove the refinement by induction on \(\ell(c)\) in the ambient group. Choose a simple left descent \(v\) of \(c\). If \(va>a\), then \[a\longrightarrow va\longrightarrow vc\longrightarrow c\] is directed: the middle gap is \(d-2\ge1\). Its labels are \(v,vtv,v\) and lie in the plane subgroup spanned by the roots of \(v\) and \(t\). These roots are independent, since \(v=t\) would make the original gap one.

Suppose instead that \(va<a\). The edge \(va\to vc\) has the same gap \(d\) and an upper vertex of length \(\ell(c)-1\). By induction it has a directed path \[w_0=va\longrightarrow w_1\longrightarrow w_2\longrightarrow w_3=vc\] whose labels \(r_1,r_2,r_3\) lie in a maximal plane subgroup \(D\). Its last label cannot be \(v\): that would give \(w_2=vw_3=c>w_3\). Nor can both \(r_1,r_2\) equal \(v\), since two consecutive uses of \(v\) return to the same vertex. If neither equals \(v\), left multiplication by \(v\) preserves the whole directed path, and its labels lie in the conjugate plane subgroup \(vDv\). If \(r_1=v\) or \(r_2=v\), use instead \[a=vw_0\longrightarrow w_2\longrightarrow w_3\longrightarrow vw_3=c.\] Its first edge is respectively the original second edge, because \(vw_0=w_1\), or the \(v\)-translate of the original first edge, because \(vw_1=w_2\). In the latter case \(r_1\ne v\), so that translate is directed. The last two edges are the original last edge and the upward \(v\)-edge. All labels lie in \(D\), since \(v\in D\) in these cases. This proves the refinement. Its final path is directed with endpoints \(a,c\), so it lies inside \([a,c]\), even though the descent induction need not stay there.

Now induct on the ambient gap of a reflection edge \(a\to c\) inside \(Q\). For gap one it is a cover and belongs to \(\mathcal E_Q\). Otherwise choose the three-edge path just constructed in a maximal plane coset. In \([a,c]_D\), take both vertices at each of the two internal relative ranks occupied by this path. Lemma 8 supplies these vertices and all eight reflection edges between the four selected layers, since the three relative gaps are odd. These edges are ambient edges in \([a,c]\subseteq Q\) by Lemma 7. Each meets an internal layer, so its ambient gap is smaller than \(d(a,c)\). By induction all eight are in \(\mathcal E_Q\), and the closure rule adds \((a,c)\).

Thus \(\mathcal E_Q\) is exactly the directed Bruhat graph of \(Q\). A poset isomorphism preserves covers and the closure rule, hence preserves all its directed reflection edges. The rule recovers their presence and direction, without identifying their root labels. ◻

Proof. Write \(n=\ell_D(c)-\ell_D(x)\). By Lemma 8, the interval has one vertex at each end, two vertices in every internal rank, and all possible directed edges between consecutive ranks. Its vertices lie in \([u,b]\), by Lemma 7.

The two arrows from \(x\) to its relative atoms have distinct labels after applying \(\iota\). Let \(\Pi'\) be the plane of those two root labels. For each vertex in the next relative rank, the two paths through the atoms have the same plane by Lemma 7(3). Their remaining arrows therefore have labels in \(\Pi'\). Repeat this argument through the ranks. More explicitly, if the arrows into a two-vertex rank have labels in \(\Pi'\), choose a vertex of the preceding rank and form the two paths through these vertices to each vertex of the following rank. Their common plane is \(\Pi'\), so the next arrows also lie in that plane. This proves that all image vertices lie in one coset of \(D'=D_{\Pi'}\).

Consider an original pair of vertices with relative rank difference two. By Remark 9, exactly two ambient directed two-edge paths join them. Every intermediate vertex of such a path belongs to \([u,b]\), so graph preservation implies that their images also have exactly two ambient two-edge paths.

In the target dihedral coset, the two-edge paths imply a positive even relative rank difference \(r\). Lemma 8 gives exactly \(r\) relative two-edge paths. All are ambient paths and lie in \([u',b']\); conversely, the common-plane property puts every ambient two-edge path for these endpoints in this coset. Thus \(r=2\). The two edges of any such path each have positive relative rank difference, so both have difference one.

Every consecutive-rank edge of \([x,c]_D\) lies in a consecutive two-edge path, because \(n\ge2\). Hence every relative cover maps to a relative cover. The image endpoints have relative rank difference \(n\), and the image contains two vertices in each internal relative rank. These exhaust the full target dihedral interval. The comparability and odd-gap criteria in Lemma 8 now show that the restricted map preserves and reflects the relative order and all reflection edges. ◻

Proof. All ambient real roots have squared length \(2\), since the action preserves the form and the simple roots have that length. Define \[p_0=0,\qquad p_1=1,\qquad p_{n+1}=2\kappa p_n-p_{n-1}\quad(n\ge1).\] For an alternating word of length \(n+1\), let \(w_n\) be its prefix of length \(n\) and \(\sigma_n\) its next letter, with \(\alpha_s=\alpha\) and \(\alpha_t=\beta\). The reflection formulas give, by induction, the two roots \(w_n\alpha_{\sigma_n}\) \[\delta_n=p_{n+1}\alpha+p_n\beta,\qquad \epsilon_n=p_n\alpha+p_{n+1}\beta\] for the words beginning with \(s\) and \(t\), respectively. Their left labels \(w_n\sigma_nw_n^{-1}\) are the alternating palindromes with parameters \(2n+1\) and \(-(2n+1)\).

For \(0\le n<m\) in finite type and every \(n\ge0\) in infinite type, the word of length \(n+1\) is reduced. Its last relative cover is an ambient upward edge by Lemma 7. The sign criterion (2) then identifies \(\delta_n,\epsilon_n\) as the positive ambient roots of their labels. Applying \(h\) to both and adding gives \[p_{n+1}+p_n>0\] throughout those ranges.

In finite type, the parameter \(2m-1\) equals \(-1\) modulo \(2m\), so \(\delta_{m-1}\) lies on the \(\beta\) line and \(p_m=0\). If \(\kappa\ge1\), the recurrence instead gives \(p_{n+1}>p_n\ge0\) by induction. Thus finite type has \(\kappa<1\). Write \(\kappa=\cos\theta\) with \(0<\theta\le\pi/2\). Then \[p_n=\frac{\sin(n\theta)}{\sin\theta}, \qquad p_{n+1}+p_n>0 \quad\Longleftrightarrow\quad \sin\bigl((n+\tfrac12)\theta\bigr)>0.\] The equation \(p_m=0\) gives \(m\theta=j\pi\) for a positive integer \(j\). If \(j\ge2\), the first \(n\) with \((n+\tfrac12)\theta\ge\pi\) is less than \(m\), and its angle lies in \([\pi,\pi+\theta)\subseteq[\pi,3\pi/2]\). Its sine is nonpositive, a contradiction. Hence \(j=1\) and \(\kappa=\cos(\pi/m)\). In infinite type, \(\kappa<1\) would give the same first sign crossing for some \(n\), contradicting positivity for every \(n\). Thus \(\kappa\ge1\) there.

The recurrence also gives \(p_{n+1}^2-p_np_{n+2}=1\) by induction. Therefore the slopes \(r_n=p_n/p_{n+1}\) of the \(\delta_n\) satisfy \[r_{n+1}-r_n=\frac{1}{p_{n+1}p_{n+2}}>0\] whenever the denominators are positive. In finite type, the sine formula makes these slopes increase from \(0\) to \(+\infty\), ending at \(\beta\); their parameters are all the odd residues \(1,3,\ldots,2m-1\). In infinite type, \(p_{n+1}>p_n\ge0\), so the slopes of the \(\delta_n\) increase below \(1\), while the reciprocal slopes of the \(\epsilon_n\) decrease from \(+\infty\) above \(1\). The two families exhaust the reflections. Hence no other root intervenes, whether their limiting rays coincide or are distinct, and their slope order is the displayed positive string followed by the reversed negative string.

Since \(h(\alpha),h(\beta)>0\), the normalized point \((\alpha+r\beta)/h(\alpha+r\beta)\) moves monotonically along the affine line \(h=1\) as the slope \(r\) increases. The functional \(f\) in the geometric construction separates the endpoint rays, so \(f/h\) gives this order or its reverse. ◻

Proof. If \([x,c]_D\) has relative rank one, then \(c=y\), and both sides vanish. Otherwise apply \(\iota\) using Proposition 11, then identify the target coset with its dihedral group by Lemma 7. The transported schedule is now the target label order, and the relative gaps are unchanged. By the definition of reflection order, this label order is one of the two angular orders in Lemma 12. Rename the resulting group and vertices \(D,x,y,c\), and use the integer indexing above. Lemma 8 says that \(i\to j\) is an arrow exactly when \[|i|<|j|\quad\text{and}\quad i-j\ \text{is odd}.\] We first use the order starting with \(s\); the reversed order will follow from the count below.

Write \(i,j\) for the indices of \(x,y\), and put \[L=|i|,\qquad M=|j|,\qquad C=\ell_D(c), \qquad k=\frac{M-L-1}{2}.\] We use the following midpoint comparison, which we verify after deriving the count. Every row concerns only an arrow that exists, so its endpoint has the required parity and greater absolute length. In the finite group \(-m\) is omitted because \(m\) already represents the longest element.

Suppose first that \(j=-M\). The table leaves precisely the negative endpoints in the eligible ranks strictly between \(L\) and \(M\). By Lemma 8, those vertices belong to \([x,c]_D\). Their ranks form the possibly empty list \[L+1,L+3,\ldots,M-2,\] which has \(k\) entries. The table gives no later outgoing arrow at \(j\), so \(h_x-h_y=k\).

Now suppose that \(j=M\). The same \(k\) negative endpoints below rank \(M\) contribute at \(i\). If \(c=y\), there are no further contributions. If \(C>M\), the opposite endpoint \(-j\) is an internal vertex and contributes one more later arrow at \(i\). In the finite case \(C>M\) implies \(M<m\), so this endpoint is represented. Put \(H=C-M\). Above rank \(M\), all outgoing arrows from either \(i\) or \(j\) are later. At an internal rank \(M+t\), the parity condition permits arrows from \(i\) when \(t\) is even and from \(j\) when \(t\) is odd. The two vertices there therefore contribute \(2(-1)^t\) to \(h_x-h_y\), while the singleton top contributes \((-1)^H\). The total contribution above rank \(M\) is \[2\sum_{t=1}^{H-1}(-1)^t+(-1)^H=-1.\] It cancels the extra arrow to \(-j\), leaving \(h_x-h_y=k\).

For the reversed angular order, Lemma 8 shows that the number of outgoing edges at \(a\) in \([a,c]_D\) equals \(\ell_D(c)-\ell_D(a)\): each eligible internal rank contributes two vertices, and the singleton top contributes when this difference is odd. The outgoing edges at \(y\) in \([x,c]_D\) are exactly those in \([y,c]_D\), so the total outgoing-edge counts at \(x\) and \(y\) differ by \(2k+1\). The edge \(e\) contributes one at \(x\) and none at \(y\), and no other edge incident to \(x\) or \(y\) has its label. Removing it and the later arrows leaves an earlier-arrow difference of \((2k+1)-1-k=k\). Those arrows are later in the reversed order.

It remains to verify the midpoint table. In the infinite group, \(r>|i|\) implies \(i+r>0\) and \(i-r<0\), while the sign of \(i+j\) is the sign of \(j\). The positive midpoint block precedes the negative block, and each block is ordered by the ordinary integer order. Comparing \(i+r\) and \(i-r\) with \(i+j\) gives the two rows starting at \(i\). For arrows from \(j\), use \(r>M>|i|\). If \(j=M\), the positive midpoint \(M+r\) exceeds \(i+M\) and the negative midpoint lies in the later block. If \(j=-M\), the positive midpoint lies in the earlier block and the negative midpoint \(-M-r\) is smaller than \(i-M\). This gives the other two rows.

In the finite group, a negative midpoint \(a\) is represented by \(2m+a\). Comparisons inside one block are unchanged. For a cross-block arrow from \(i\) when \(j=M\), the negative endpoint \(-r\) is later because \[2m+i-r>i+M\] is equivalent to \(r+M<2m\). This is strict since \(r\le m-1\) for a represented negative endpoint and \(M\le m\). When \(j=-M\), a positive endpoint could be later only if \(r+M>2m\), which is impossible since \(r\le m\) and \(M\le m-1\).

For arrows starting at \(j=M\), the two possible midpoint residues \(M+r\) and \(2m+M-r\) exceed \(i+M\): the first inequality follows from \(r>|i|\), and the second from \(2m-r>i\), using \(r\le m\) and \(i<m\). For arrows starting at \(j=-M\), the residues \(r-M\) and \(2m-M-r\) are smaller than \(2m+i-M\): these are respectively \(r<2m+i\), using \(r\le m\) and \(i>-m\), and \(-r<i\), using \(r>|i|\). These inequalities include the possible longest endpoint, so the table has no wrap-around exception. This completes its verification and the proof. ◻

Proof. Let \(\alpha\) be the positive root of \(e\), so that \(y=r_\alpha x\). For every positive root \(\gamma\), set \[\varepsilon(\gamma) =\mathbf{1}_{\{y^{-1}\gamma<0\}} -\mathbf{1}_{\{x^{-1}\gamma<0\}}.\] The two ambient inversion sets are finite, so \(\varepsilon\) has finite support and \[\sum_{\gamma\in\Phi^+}\varepsilon(\gamma) =\ell(y)-\ell(x)=d(x,y).\] The direction of the arrow gives \(x^{-1}\alpha>0\) and \(y^{-1}\alpha<0\), hence \(\varepsilon(\alpha)=1\).

Partition the positive roots other than \(\alpha\) according to their planes \(\Pi=\operatorname{span}(\alpha,\gamma)\). By the relative inversion formula in Lemma 7, the contribution of all positive roots in such a plane is \[\sum_{\gamma\in\Phi_D^+}\varepsilon(\gamma) =\ell_D(y)-\ell_D(x)=2k_D(e)+1.\] Removing \(\alpha\) leaves the contribution \(2k_D(e)\). These contributions are nonnegative, and only finitely many can be nonzero, because each nonzero one meets the finite support of \(\varepsilon\) away from \(\alpha\). Summing the disjoint parts now gives \[d(x,y)-1=\sum_D 2k_D(e),\] as required. If there is no root independent of \(\alpha\), the sum is empty and the same identity follows directly from the ambient inversion count. ◻

Proof. Use the Hecke algebra convention \((T_s+1)(T_s-q)=0\). The bar involution sends \(q^{1/2}\) to \(q^{-1/2}\) and \(T_s\) to \(T_s^{-1}\) for each simple generator, preserving products. For \(w\in D\), put \[C_w^{\circ}=q^{-\ell_D(w)/2}\sum_{z\le_D w}T_z.\] We prove that these elements are bar invariant. This is immediate for \(w=1\) and for a simple reflection. Let \(w\) have an alternating reduced expression starting with \(s\), and set \(n=\ell_D(w)\ge2\). We claim that \[ C_s^{\circ}C_{sw}^{\circ} = \begin{cases} C_w^{\circ},&n=2,\\ C_w^{\circ}+C_p^{\circ},&n>2, \end{cases} \tag{9}\] where \(p\) is the prefix of length \(n-2\) of the chosen expression for \(w\).

To check this, partition \(D\) into left \(s\)-strings \(\{r,sr\}\) with \(\ell_D(sr)=\ell_D(r)+1\). A lower interval cannot contain \(sr\) without containing \(r\). If only \(r\) is present in \([1,sw]_D\), then \[(1+T_s)T_r=T_r+T_{sr}.\] If both elements are present, the quadratic relation gives \[(1+T_s)(T_r+T_{sr})=(1+q)(T_r+T_{sr}).\] The strings that meet \([1,sw]_D\) have union \([1,w]_D\). There are no doubly included strings when \(n=2\). When \(n>2\), their union is \([1,p]_D\): both alternating words occur in every length below \(n-2\), and the unique element of length \(n-2\) in this union is the prefix \(p\). This is also valid when \(w\) is the longest element of a finite dihedral group. Consequently \[(1+T_s)\sum_{z\le_D sw}T_z =\sum_{z\le_D w}T_z+ \begin{cases} 0,&n=2,\\ q\displaystyle\sum_{z\le_D p}T_z,&n>2. \end{cases}\] Multiplication by \(q^{-n/2}\) proves (9). Induction now proves bar invariance of \(C_w^{\circ}\).

Relative to the normalized standard basis \(\{q^{-\ell_D(z)/2}T_z:z\in D\}\), the leading coefficient of \(C_w^{\circ}\) is \(1\), and its lower coefficients are \[q^{-(\ell_D(w)-\ell_D(z))/2} \in q^{-1/2}\mathbb Z[q^{-1/2}] \qquad(z<_D w).\] Bar invariance and the triangular characterization therefore give \[C_w^{\circ} =q^{-\ell_D(w)/2}\sum_{z\le_D w}P^D_{z,w}(q)T_z.\] Comparison of coefficients proves the lemma. ◻

Proof. We first justify the character theorem for precisely this realization. Restrict the ambient finite-dimensional representation to \(D\). It is faithful. Every reflection of \(D\) has rank-one change from the identity. Conversely, suppose an element \(d\in D\) has rank-one change. Ambient reflection faithfulness implies that \(d\) is an ambient reflection. The image of \(d-1\) lies in \(\Pi\), since this is true for every product of reflections whose roots lie in \(\Pi\). Its root therefore lies in \(\Pi\), and the maximal plane-subgroup description makes it a reflection of \(D\). The restricted representation is consequently reflection faithful.

If \(D\) is finite of type \(I_2(m)\), its normalized canonical simple roots have Gram matrix with diagonal entries one and off-diagonal entries \(-\cos(\pi/m)\) by Lemma 12. Its root plane is therefore positive definite and carries the usual real dihedral reflection representation. The nondegenerate ambient space splits into this plane and its orthogonal complement, and \(D\) fixes the latter pointwise. The real Soergel character theorem therefore applies to the plane representation [20]; adjoining the fixed polynomial directions preserves the character formula and the Braden–MacPherson construction. Indeed the relevant kernels and images commute with this flat polynomial extension, and graded minimal free covers remain minimal.

If \(D\) is infinite, it is a universal Coxeter system of rank two. Fiebig’s universal-group character theorem applies under its standing reflection-faithful hypotheses and orbit choices [23]. These requirements hold for the restricted real representation. A regular orbit exists by avoiding the countable union of proper fixed subspaces. The required simple-subregular orbits exist by the generic-hyperplane argument in Lemma 2, applied to the restricted orthogonal representation. In particular, no assertion that its simple-root pairing is the geometric pairing is needed. The moment-graph identification and duality give the same stalk normalization as above [23].

The coset identification preserves the relative orientation and the left root labels, so these statements apply on \(C\). Lemma 15 and the stalk character formula now show that every stalk below \(c\) is \(A\) with a generator in degree zero. Every nonzero edge module is likewise \(A/(\alpha_e)\) with its generator in degree zero.

Choose a homogeneous global section extending the generator at \(c\). We show that its value at every vertex of \([m,c]_D\) is a generator, where \(m\) is the relative minimum of \(C\). Descend along any path from \(c\) to the chosen vertex. If its value at the upper endpoint of an edge is a nonzero degree-zero generator, its restriction to that edge is nonzero, by the upper-end quotient rule. Compatibility forces the section value at the lower endpoint to be nonzero. Since that stalk is a rank-one free module generated in degree zero, a nonzero homogeneous value of degree zero is again a generator. This proves the assertion along the path, and every vertex below \(c\) lies on such a path.

Use these section values as the generators of the vertex stalks, and use their common restrictions as the generators of the edge modules. Both maps at every supported edge are then the quotient map. This identifies the sheaf with \(\mathcal O_c\). Generation and flabbiness are properties of the Braden–MacPherson sheaf, so they hold for this model. ◻

Proof. Freeness and the quotient rule persist under localization. The graph on \([1,b]\) is finite, and its section module is the kernel of a map between finite direct sums of stalk and edge modules. Formation of sections therefore commutes with flat localization. In particular, ambient flabbiness and generation at stalks persist. Restricting a global section to \(K\) proves generation there.

To prove relative flabbiness, let \(J\subseteq K\) be a relative upper set and prescribe a section on \(J\). Regard \(J\) as a subset of \([1,b]\). It is upper closed under all surviving edges: those edges stay in \(C\), and their orientations agree with the relative dihedral graph. We extend the section by adding missing vertices in decreasing ambient order.

When a missing vertex \(a\) is added, all ambient vertices strictly above \(a\) are already present. Put \[\Omega_a=\{z\in[1,b]:a<z\}.\] This is an ambient upper set. Ambient flabbiness extends the restriction of the prescribed section to \(\Omega_a\); use the resulting value at \(a\). It is compatible with every already prescribed upper neighbor. An already prescribed lower neighbor \(z\) cannot be joined to \(a\) by a surviving edge: upper closure under surviving edges would have required \(a\) to be present already. Edges with zero modules impose no condition. Consequently the chosen value at \(a\) is compatible with every previous value. Moreover, the enlarged set is still upper closed under surviving edges, since all vertices above \(a\) are present. The finite induction extends the original section to \([1,b]\). Restriction to \(K\) proves the required surjectivity. ◻

Proof. We construct an isomorphism in descending relative order. Suppose the sheaf has already been identified above a vertex \(a\) with a sum of the models in the statement. More precisely, take any descending linear extension of the finite poset \(K\); the isomorphism has been constructed on all previously processed vertices and on edges between them. All relative upper neighbors of \(a\) have already been processed.

Let \(E_a\) be the set of relative upward edges from \(a\) in \(K\) and let \(K_{>a}=\{y\in K:a<_D y\}\). Define the boundary image \[N_a=\mathop{\mathrm{im}}\left( \Gamma(K_{>a},B_R) \longrightarrow\bigoplus_{e\in E_a}B_R^e \right),\] where the map restricts a section from the upper endpoint of each edge. The image of the stalk map \[\varphi:B_R^a\longrightarrow\bigoplus_{e\in E_a}B_R^e\] is exactly \(N_a\). For one inclusion, generation at \(a\) extends any stalk value to a section, whose upper restriction gives its boundary value. For the other, relative flabbiness extends any section on \(K_{>a}\) to \(K\), producing a preimage of its boundary value at \(a\). Thus \(\varphi\) is a surjection from the free stalk \(B_R^a\) onto \(N_a\).

We will show that the stalks of the models already constructed above \(a\) give a free cover of \(N_a\) which is an isomorphism modulo \(\mathfrak m\). Splitting the given stalk through this cover will leave a free summand with zero boundary map. That summand will supply the models with top \(a\).

The edge modules immediately above \(a\), together with their upper restriction maps, are already identified: the upper-end quotient rule identifies each with its upper stalk modulo the corresponding label. Hence the isomorphism already constructed above \(a\) also identifies \(N_a\) with the boundary image for the sum of the existing models. Only models with top \(c>_D a\) contribute to this image. Models whose tops are not above \(a\) have neither a stalk at \(a\) nor a nonzero edge module immediately above it.

Consider one contributing copy with top \(c\). Let \(E_{a,c}\) be its upward edges at \(a\) whose upper endpoints are at most \(c\) in relative order. This set is nonempty. Its stalk map is \[R\longrightarrow\bigoplus_{e\in E_{a,c}}R/(\alpha_e).\] Distinct labels at a vertex are nonassociate irreducibles in \(R\). Consequently its kernel is \[ \bigcap_{e\in E_{a,c}}\alpha_eR =p_{a,c}R, \qquad p_{a,c}=\prod_{e\in E_{a,c}}\alpha_e. \tag{10}\] The boundary image of this model is therefore \(R/(p_{a,c})\). Here we use its generation and flabbiness, as above, to identify the stalk image with the boundary image coming from sections above \(a\). Since all the labels lie in \(\mathfrak m\) and \(E_{a,c}\) is nonempty, \(p_{a,c}\in\mathfrak m\). Thus \[R\longrightarrow R/(p_{a,c})\] is a minimal free cover: reduction modulo \(\mathfrak m\) is an isomorphism.

The section and boundary maps of a finite direct sum split componentwise. Taking the sum of the existing model stalks at \(a\) therefore gives a free module \(P_a\) and a surjection \[\pi:P_a\longrightarrow N_a\] whose reduction modulo \(\mathfrak m\) is an isomorphism. If no model contributes, both \(P_a\) and \(N_a\) are zero and the same argument applies.

We now split the given stalk by an explicit lift. Because \(B_R^a\) is free and \(\pi\) is surjective, there exists \[g:B_R^a\longrightarrow P_a, \qquad \pi g=\varphi.\] Modulo \(\mathfrak m\), the map \(\pi\) is an isomorphism and \(\varphi\) is surjective, so \(g\) is surjective modulo \(\mathfrak m\). Nakayama’s lemma applied to its finitely generated cokernel implies that \(g\) is surjective. Since \(P_a\) is free, \(g\) splits. We obtain \[B_R^a\cong P_a\oplus Q_a, \qquad \varphi=(\pi,0),\] where \(Q_a=\ker g\) is finite projective and therefore free over the local ring \(R\).

The factor \(P_a\) extends the isomorphism for the existing models to the stalk at \(a\) and all its upward restrictions. Add one copy of \(\mathcal O_{a,R}\) for each basis element of \(Q_a\). These new models have zero outgoing edge maps at their top and vanish at every previously processed vertex. Thus they realize the factor \(Q_a\) without changing any previously constructed data. The incoming edge modules will be identified when their lower endpoints are processed, using the upper-end quotient rule. This completes the descending induction and proves the decomposition. ◻

Proof. Apply Theorem 1 to each restricted interval. For the second assertion, write reciprocity as \[q^{d(x,y)}P_{x,y}(q^{-1}) =\sum_{x\le z\le y}R_{x,z}(q)P_{z,y}(q).\] The term with \(z=y\) is \(R_{x,y}\), and all other terms use smaller rank differences for \(R\). Induction therefore recovers all \(R\)-polynomials from the order and the preserved \(P\)-polynomials. ◻