The asymptotic algebra, filtered layers and the specialization homomorphism

Reversing words gives the anti-involution $T_w\mapsto T_{w^{-1}}$ of the Hecke presentation. It commutes with bar. Inversion preserves length and sends a reflection-chain step $u\to ut$ to $u^{-1}\to t{u}^{-1}=u^{-1}(ut{u}^{-1})$, again a length-increasing right-reflection step; hence it preserves Bruhat order. Thus it preserves Bruhat triangularity, and KL uniqueness gives $C_w\mapsto C_{w^{-1}}$. Hence $h_{x,y,z}=h_{y^{-1},x^{-1},z^{-1}}$, $a(z)=a(z^{-1})$, and ${\gamma }_{x,y,z}={\gamma }_{y^{-1},x^{-1},z^{-1}}$. Comparing coefficients in the bar-invariant products $C_x{C}_y$ also gives $\overline{h_{x,y,z}}=h_{x,y,z}$. P3 with $y=1$ forces $a(1)=0$ and $1\in \mathcal{D}$, since $C_1{C}_1=C_1$ makes the required nonzero coefficient possible only for $d=1$ and exponent zero. The assignment $T_s\mapsto -T_s^{-1}$ preserves the quadratic relation and braid relations (the signs agree at equal braid length and inverse reversed braid words agree), is involutive and is $A$-linear. Thus $D_x{D}_y$ has the same constants $h$.

P8 implies that products in $J$ vanish across distinct $a$-degrees. For a fixed degree $i$, multiply P15′ by $v^i$ and take its constant coefficient. Whenever a $\gamma$ term is nonzero, P8 forces every participating index to have degree $i$. The resulting identity is ${\sum }_u{\gamma }_{x,u,y^{-1}}{\gamma }_{w,x^{\mathrm{\prime }},u^{-1}}={\sum }_u{\gamma }_{x,w,u^{-1}}{\gamma }_{u,x^{\mathrm{\prime }},y^{-1}}$, exactly equality of the coefficients of $t_y$ in $t_x(t_w{t}_{x^{\mathrm{\prime }}})$ and $(t_x{t}_w)t_{x^{\mathrm{\prime }}}$. Other degree combinations vanish. This proves associativity without using any unlisted P property.

The coefficient of $t_y$ in $t_x{\sum }_d{n}_d{t}_d$ is ${\sum }_d{n}_d{\gamma }_{x,d,y^{-1}}={\sum }_d{n}_d{\gamma }_{y^{-1},x,d}$ by P7. P2 makes it zero unless $y=x$; then P3 and P5 give the unique coefficient $n_d^2=1$. The left identity is proved in the same way, using cyclic symmetry with ${\gamma }_{x,y^{-1},d}$; the appropriate unique $d$ comes from P3 applied to $x^{-1}$. Step 1.1 proves that ${\iota }_J$ reverses products. Since $d=d^{-1}$ by P6, $\mu (t_x{t}_y)={\sum }_d{\gamma }_{x,y,d}{n}_d$ is ${\delta }_{x,y^{-1}}$ by P2, P3 and P5. This symmetric permutation Gram matrix is invertible over $\mathbb{Z}$, proving every trace assertion.

P4 implies that each $I_i$ is a two-sided ideal. Its layer has exactly the stated basis since both ideals are spans of subsets of the $D$-basis. P8 and step 3.1 give the orthogonal ideal decomposition $J={⨁}_i{J}^i$, with central identities $1_i={\sum }_{d:a(d)=i}{n}_d{t}_d$. On the free layer $B_i$, identify $[D_w]$ with $t_w$ for $a(w)=i$ and give it the left regular $J_A^i$ action. Natural left and right $H$ multiplication are well defined on the layer.

Let $A_x$ be natural left multiplication by $D_x$ on $B_i$. The coefficient of $[D_y]$ in $A_x{R}_{t_{x^{\mathrm{\prime }}}}[D_w]$ is ${\sum }_u{h}_{x,u,y}{\gamma }_{w,x^{\mathrm{\prime }},u^{-1}}$; the coefficient in $R_{t_{x^{\mathrm{\prime }}}}{A}_x[D_w]$ is ${\sum }_u{h}_{x,w,u}{\gamma }_{u,x^{\mathrm{\prime }},y^{-1}}$. Here only indices of degree $i=a(w)=a(y)$ survive. P15′ gives equality. Thus $A_x$ commutes with every right regular $J_A^i$ multiplication and is left multiplication by its value on $1_i$. That value is ${\sum }_{z:a(z)=i}{\sum }_{d:a(d)=i}{h}_{x,d,z}{n}_d{t}_z$, the degree-$i$ component of the displayed $\varphi (D_x)$. This proves compatibility of the left $H$ action with $\varphi$.

Composition of natural left $H$ actions gives $L_{\varphi (h{h}^{\mathrm{\prime }})}=L_{\varphi (h)}{L}_{\varphi (h^{\mathrm{\prime }})}$ on ${⨁}_i{J}_A^i$. Evaluating on its identity proves $\varphi (h{h}^{\mathrm{\prime }})=\varphi (h)\varphi (h^{\mathrm{\prime }})$; evaluating the identity action proves $\varphi (1_H)=1_J$. For the right $H$ action to commute with left $J$, apply P15′ with $(x,x^{\mathrm{\prime }},w,y)$ replaced by $(x^{-1},z^{-1},w^{-1},y^{-1})$ and replace the summation variable by its inverse. Using step 1.1 gives ${\sum }_u{\gamma }_{z,w,u^{-1}}{h}_{u,x,y}={\sum }_u{h}_{w,x,u}{\gamma }_{z,u,y^{-1}}$, exactly the coefficient equality for $(t_z\cdot [D_w])D_x=t_z\cdot ([D_w]D_x)$. Thus $B_i$ is the asserted bimodule. This argument requires no equality between $\varphi$ and its opposite/adjoint version.

The bases, identities and multiplication equalities above are finite integral coefficient identities; they remain valid after every commutative base change. If $h\in \ker {\varphi }_R$, compatibility says that $h{I}_i\subseteq I_{i+1}$ for all $i$. A product of $N+1$ such elements therefore kills $I_0=H_R$ and in particular its unit, proving the nilpotence bound. Over $K$, the algebra $H_K$ is a product of full matrix algebras by the given split generic hypothesis. A nilpotent two-sided ideal of such a product is zero: its intersection with any simple matrix factor is either zero or that factor, and a nonzero factor contains its identity and is not nilpotent. Thus ${\varphi }_K$ is injective and, since both sides have dimension $\mathrm{\mid }W\mathrm{\mid }$, is an isomorphism.

Transport matrix units of $H_K$ through ${\varphi }_K$. Their coordinates in the integral basis $t_w$ are rational functions of $v$, and their coordinate determinant is nonzero. Choose a nonzero rational value avoiding the finitely many zeros of their denominators and that determinant; such a value exists because a nonzero polynomial has finitely many roots and ${\mathbb{Q}}^{\times }$ is infinite. Evaluating the coordinates preserves the finite matrix-unit and unit identities and retains their linear independence. They therefore give an isomorphism $J_{\mathbb{Q}}\cong {\prod }_E{\mathrm{Mat}}_{d_E}(\mathbb{Q})$. After scalar extension back to $K$, this decomposition has the same primitive central blocks: in the center of a split matrix product every idempotent has only zero/one coordinates. Relabel its factors accordingly. This gives rational models $V_E$ for all generic labels. Finally the central orthogonal identities $1_i$ act on a simple module through exactly one component; denote its index by ${\nu }_E$.