A faithful canonical realization that is not reflection faithful: the affine rank-two system

The representation is faithful: $\rho$ is injective by clause (3) of The root-length criterion and faithfulness of the canonical reflection representation, applied to the system $S=\{s,t\}$, $m(s,t)=\mathrm{\infty }$. Hence $\rho$ is a faithful canonical realization of the system, and by clause (4)(a) of Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary the canonical construction is a realization in the sense of Elias–Williamson, with $\mathfrak{h}=V$, ${\alpha }_s^{\vee }=e_s$ and ${\alpha }_s$ the functional $2B(-,e_s)$.

The radical: since $m(s,t)=\mathrm{\infty }$ one has $B(e_s,e_s)=B(e_t,e_t)=1$ and $B(e_s,e_t)=-1$ (The real Coxeter form, its radical, reflections, and form-preserving maps). Hence $B(e_s+e_t,e_s)=1-1=0$ and $B(e_s+e_t,e_t)=-1+1=0$, so $\mathbb{R}(e_s+e_t)\subseteq \mathrm{rad}(B)$; for $v=a{e}_s+b{e}_t$, one has $B(v,e_s)=a-b$ and $B(v,e_t)=b-a$. Thus $v$ lies in the radical exactly when $a=b$. The radical is therefore the one-dimensional line $\mathrm{rad}(B)=\mathbb{R}(e_s+e_t)$, and is a hyperplane.

The radical is fixed pointwise by $W$: for $a=e_s$ or $e_t$ and $v\in \mathrm{rad}(B)$ one has $B(v,a)=0$, so the reflection formula gives $r_a(v)=v$ (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)); a composite of finitely many maps fixing a vector fixes it, and every element of $W$ is a finite product of the generators $s,t$ and their inverses by [F3]. Hence every element of $W$, in particular $\rho ((st{)}^k)$ for $k\mathrm{\ne }0$, fixes $\mathrm{rad}(B)$ pointwise.

There are infinitely many reflections: by part (4) of The rank-two block computation, exact dihedral orders, the signed reflection action, and ambient reducedness, $st$ has infinite order in $W$. Put $r:=st$; since $s{r}^{-1}=sts=rs$, induction gives $s{r}^{-m}=r^{m}s$ and hence $r^{m}s{r}^{-m}=r^{2m}s$ for every $m\in \mathbb{Z}$. Each $r^{2m}s$ lies in the reflection set $T=\{ws{w}^{-1}:w\in W,s\in S\}$ (The canonical reflection homomorphism, roots, reflections, and the positive cone (2)), and the elements $r^{2m}s$, $m\in \mathbb{Z}$, are pairwise distinct: if $r^{2m}s=r^{2m^{\mathrm{\prime }}}s$ then right multiplication by $s$ gives $r^{2m}=r^{2m^{\mathrm{\prime }}}$, hence $r^{2(m-m^{\mathrm{\prime }})}=1$ and $m=m^{\mathrm{\prime }}$ because $r$ has infinite order. So $W$ has infinitely many reflections. Moreover $(st{)}^k$ for $k\mathrm{\ne }0$ has infinite order, whereas every conjugate of $s$ or $t$ is an involution; hence these powers are not reflections.

The matrices of the two generators in the basis $(e_s,e_t)$ follow from the reflection formula with $2c(s,t)=2$ for $m=\mathrm{\infty }$: $r_s(e_s)=-e_s$ and $r_s(e_t)=e_t+2e_s$, and $r_t(e_t)=-e_t$, $r_t(e_s)=e_s+2e_t$ (The real Coxeter form, its radical, reflections, and form-preserving maps, Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (3)). By step 1.2 the fixed space of $r_s$ is $\ker B(-,e_s)=\{a{e}_s+b{e}_t:a-b=0\}=\mathbb{R}(e_s+e_t)=\mathrm{rad}(B)$, and similarly $\mathrm{Fix}(r_t)=\ker B(-,e_t)=\mathrm{rad}(B)$: the two distinct reflections $r_s\mathrm{\ne }{r}_t$ have the same fixed hyperplane. In coordinates [F2] $[r_s]=\left(\begin{array}{cc}-1& 2\\ 0& 1\end{array}\right),[r_t]=\left(\begin{array}{cc}1& 0\\ 2& -1\end{array}\right),$ differ in their $(1,1)$-entries, so indeed $r_s\mathrm{\ne }{r}_t$ while $\mathrm{Fix}(r_s)=\mathrm{Fix}(r_t)$.

Every full fixed space of codimension one of an element of $W$ equals $\mathrm{rad}(B)$. Every element of $W$ is $r^k$ or $r^{k}s$ for some $k\in \mathbb{Z}$ with $r:=st$: repeatedly deleting adjacent equal letters from a word in $s,t$ (using $s^2=t^2=1$) leaves an alternating word, and the alternating words are $(st{)}^k$, $(st{)}^{k}s$, $(ts{)}^k=(st{)}^{-k}$ and $(ts{)}^{k}t=(st{)}^{-k-1}s$. For every reflection $wa{w}^{-1}$ of $W$, with $a\in \{s,t\}$, its image is $\rho (wa{w}^{-1})=r_{\rho (w)e_a}$ with $\rho (w)e_a\in \mathrm{\Phi }$ (Descent of the reflection representation, unit root norms, and conjugation of reflections (4)); its fixed space is $\ker B(-,\alpha )$, a hyperplane (Reflections: involutivity, form invariance, fixed hyperplane, and exact rank-two order (2)) containing $\mathrm{rad}(B)$ by step 1.3, hence equal to $\mathrm{rad}(B)$. For the elements $r^k$ write $A:=\rho (r)$, whose matrix is the product of the two matrices of step 2.1, $A=\left(\begin{array}{cc}3& -2\\ 2& -1\end{array}\right)$; then $N:=A-I=\left(\begin{array}{cc}2& -2\\ 2& -2\end{array}\right)$ satisfies $N^2=0$, so $A^k=(I+N{)}^k=I+kN$ for every $k\in \mathbb{Z}$ (the case $k=-1$ being $(I+N{)}^{-1}=I-N$ when $N^2=0$), and $(I+kN)v=v$ with $k\mathrm{\ne }0$ exactly when $Nv=0$, i.e. exactly when $v\in \ker (A-I)=\{a{e}_s+b{e}_t:a=b\}=\mathrm{rad}(B)$; the identity $A^0=I$ fixes all of $V$, of codimension zero. For the elements $r^{k}s$ one has $(r^{k}s{)}^2=r^k(s{r}^{k}s)=r^k{r}^{-k}=1$ because $srs=r^{-1}$, so $\rho (r^{k}s)$ is an involution; it fixes $\mathrm{rad}(B)$ pointwise by step 1.3, while $\rho (r^{k}s)=A^k{r}_s=(I+kN)r_s=\left(\begin{array}{cc}-1-2k& 2+2k\\ -2k& 1+2k\end{array}\right)\mathrm{\ne }I$, so its fixed space is a proper subspace containing the hyperplane $\mathrm{rad}(B)$, hence equals $\mathrm{rad}(B)$. Therefore the only full fixed space of codimension one of an element of $W$ is $\mathrm{rad}(B)$.

Conclusion: the assignment "reflection $\mapsto$ its fixed hyperplane" is not injective, since the distinct reflections $r_s\mathrm{\ne }{r}_t$ of step 2.1 have the same image $\mathrm{rad}(B)$; it also cannot be a bijection onto the codimension-one fixed subspaces, since by step 1.4 the set of reflections is infinite while by step 3.1 the set of full fixed spaces of codimension one of elements of $W$ is the singleton $\{\mathrm{rad}(B)\}$. Hence the faithful canonical realization $\rho$ is not reflection faithful in the sense of Elias–Williamson (Hecke and Lie seam contract compatibility: the Artin-to-Hecke map, normalization conversions, root-length matching, and the reflection-faithfulness boundary (4)), and any Soergel-calculus or Kazhdan–Lusztig argument invoking reflection faithfulness must supply a realization satisfying that hypothesis separately. The failure is caused by the degeneracy of $B$: the radical is a codimension-one fixed subspace, fixed by non-reflections and by all reflections alike.

Scope and choice: this is a counterexample in the rank-two system $m(s,t)=\mathrm{\infty }$; it constructs no Soergel bimodule, no reflection-faithful realization and no Kazhdan–Lusztig object, and it claims nothing beyond the failure of reflection faithfulness for this canonical realization. Every argument is a finite matrix and line computation plus the supplied order, faithfulness and reflection facts, and no choice is used.