# 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

$$$$e_se_w= \begin{cases} e_{sw} & \text{, if } \left|sw\right|>\left|w\right|\\ q_{s}e_{sw}+\left(q_{s}-1\right)e_w & \text{, if } \left|sw\right|<\left|w\right| \end{cases} \text{,}$$$$ $$$$\nonumber \left(e_w\right)^{\ast}=e_{w^{-1}}$$$$ for $$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).

Section 68 of Curtis, Reiner "Methods of representation theory, volume II" explains this carefully. The active ingredient is the "Tits deformation theorem".

• Thank you for your answer! It seems like your reference formulates the statement I was asking for only in the special case of Hecke algebras coming from BN-pairs. Do you maybe have a reference which also covers "my" cases? Or is it possible that the statement I claimed follows by the same argument as in Theorem 68.21 in the book? Commented Oct 23, 2019 at 18:38
• Oh I see. The only time the BN pair is used is to see that the specialised Hecke algebra in question is separable. So you just want to know that your specialisations are separable. This is true since you are only specialising to positive real numbers and not roots of unity. You can find an explicit list of non-semisimple specialisations in Geck-Rouquier1997 as well as criteria for checking semisimplicity, though this is overkill in your situation. Commented Oct 23, 2019 at 23:56
• I'll add a comment about the second question. The argument above only says that the two algebras are both products of matrix rings of the same size, and thus obviously isomorphic. It doesn't give you a particular isomorphism, there are many. Commented Oct 24, 2019 at 0:13
• Lusztig defined an explicit isomorphism (see "On a theorem of Benson and Curtis" and later Geck, "On Iwahori-Hecke algebras... and Lusztig's isomorphism") but it is difficult for me to see at the moment whether it preserves the *-structure. I don't think it does, but it seems to be very close. Commented Oct 24, 2019 at 0:45