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](https://people.math.osu.edu/davis.12/davisbook.pdf) 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).



Reference request: Finite (multi-parameter) Iwahori-Hecke algebras are pairwise isomorphic
Section 68 of Curtis, Reiner "Methods of representation theory, volume II" explains this carefully. The active ingredient is the "Tits deformation theorem".


