# Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic

Let $$(W,S)$$ be a Coxeter system. Let $$q=(q_s)_{s\in S} \in \mathbb{R}^{\text{#}S}$$ be a tuple of positive real numbers with $$q_s=q_t$$ whenever $$s$$ and $$t$$ are conjugate to each other. Follwing Davis', Proposition 19.1.1 we can build a (unique) $$\ast$$-algebra $$\mathbb{C}_qW$$ over $$\mathbb{C}$$ freely generated by basis elements $$(e_w)_{w\in W}$$ such that

$$\begin{equation} e_se_w= \begin{cases} e_{sw} & \text{, if } \left|sw\right|>\left|w\right|\\ q_{s}e_{sw}+\left(q_{s}-1\right)e_w & \text{, if } \left|sw\right|<\left|w\right| \end{cases} \text{,} \end{equation}$$ $$\begin{equation} \nonumber \left(e_w\right)^{\ast}=e_{w^{-1}} \end{equation}$$ for $$s\in S,w\in W$$. This is the multi-parameter (Iwahori-) Hecke algebra associated to $$q$$.

Several sources suggest that, if the Coxeter group $$W$$ is finite, the Hecke algebra $$\mathbb{C}_qW$$ is always (abstractly) isomorphic to the group algebra. But I can not find a good reference for that statement. Can you give a reliable reference?

I'm also wondering if the mentioned isomorphism preserves the $$\ast$$-structure of $$\mathbb{C}_q W$$ (i.e. if it is an isomorphism of $$\ast$$-algebras or only an isomorphism of algebras).