Finite Coxeter Hecke algebras are semisimple over the generic characteristic-zero field

Specialize every parameter $v_C$ to $1$ and call the resulting algebra $H_1$. By [F2], each specialized generator $T_s$ is an involution and the alternating braid relations hold. For involutions $a,b$, put $g:=ab$, so $ba=g^{-1}$. If $m=2r$, the braid relation is $g^r=g^{-r}$, equivalent to $g^{2r}=1$. If $m=2r+1$, its two sides are $g^{r}a$ and $(ba{)}^{r}b$; right multiplication by $a$ makes equality equivalent to $g^r=(ba{)}^{r+1}=g^{-(r+1)}$, which is equivalent to $g^{2r+1}=1$. Thus, for every finite braid label, braid and Coxeter relator are equivalent in the presence of the involution relations. The specialized generators satisfy exactly [F1].

If $\varphi :R_k\to K$ is a ring homomorphism to a field with $\varphi (D_W)\mathrm{\ne }0$, [F7] says that the regular-trace Gram matrix of $H_{\varphi }$ has nonzero determinant and is therefore nondegenerate. Applying [F6] shows that $H_{\varphi }$ is semisimple. This criterion does not infer splitness and makes no claim when $\varphi (D_W)=0$.

By the group universal property in [F1], $s\mapsto T_s$ extends to $W\to H_1^{\times }$, and linear extension gives a $k$-algebra map $kW\to H_1$. Conversely, the group generators in $kW$ satisfy the specialized quadratic and braid relations, so the quotient presentation in [F3] gives a $k$-algebra map $H_1\to kW$. The composites are identity on their algebra generators, hence $H_1\cong kW$ and $T_w$ corresponds to $w$. By [F4], $(T_w{)}_{w\in W}$ is a basis of $H_1$, so ${\dim }_k{H}_1=\mathrm{\mid }W\mathrm{\mid }$.

For $g\in W$, $L_g$ permutes the group basis by $w\mapsto gw$. If $g=1$, all $\mathrm{\mid }W\mathrm{\mid }$ basis vectors are fixed; if $g\mathrm{\ne }1$, $gw=w$ would imply $g=1$ by right cancellation, so none is fixed. Thus ${\mathrm{Tr}}_k(L_g)=\mathrm{\mid }W\mathrm{\mid }$ for $g=1$ and $0$ otherwise. The regular-trace Gram matrix at $v_C=1$ therefore has entries $\mathrm{\mid }W\mathrm{\mid }{\delta }_{y,x^{-1}}$; it is $\mathrm{\mid }W\mathrm{\mid }$ times a permutation matrix and has determinant $\pm \mathrm{\mid }W{\mathrm{\mid }}^{\mathrm{\mid }W\mathrm{\mid }}\mathrm{\ne }0$ in characteristic zero.

By [F7], this determinant is $D_W$ evaluated at $v_C=1$, so $D_W$ is not the zero Laurent polynomial. Since $k$ is a field, [F5] makes $R_k$ an integral domain and embeds it into its fraction field $F$; therefore $D_W$ remains nonzero in $F$. The regular-trace Gram matrix over $F$ is invertible, and [F6] proves that $F{\otimes }_{R_k}{H}_{R_k}(W)$ is semisimple.

When $\mathrm{\mid }W\mathrm{\mid }=1$, the matrix in step 3.1 is $[1]$, so the argument includes the trivial Coxeter group. All bases are finite and specified; no form of Choice is used.