# Commutative subalgebra of Iwahori-Hecke algebra

I am currently reading the (french) book [1] by Matsumoto. In (2.1.11) the following statement (reformulated in my own words/notation) appears:

Let $$(W,S)$$ be a Coxeter system, let $$R$$ be a commutative ring, let $$q=(q_s)\in R^S$$ be an element with $$q_s=q_t$$ if $$s$$ and $$t$$ are conjugate in $$W$$ and denote by $$R[W,q]$$ the corresponding Iwahori-Hecke algebra, i.e. the unique $$R$$-algebra freely generated by a basis $$\{T_w \mid w\in W\}$$ with $$\begin{eqnarray} {T}_{s}{T}_{\mathbf{w}}=\begin{cases} {T}_{s\mathbf{w}} & \text{, if }\left|s\mathbf{w}\right|>\left|\mathbf{w}\right|\\ q_{s}{T}_{s\mathbf{w}}+(q_{s}-1){T}_{\mathbf{w}} & \text{, if }\left|s\mathbf{w}\right|<\left|\mathbf{w}\right| \end{cases} \end{eqnarray}$$ for all $$s\in S$$, $$w \in W$$. Let $$S^\prime \subseteq S$$ be a subset such that the special subgroup $$W_{S^\prime}:=\left\langle S^{\prime}\right\rangle \subseteq W$$ is finite. Further write $$R[W,q,W^\prime]$$ for the subalgebra of $$R[W,q]$$ of bi-invariant functions over $$W^\prime$$, i.e. $$R[W,q,W^\prime]$$ consists of all finite sums of the form $$x=\sum_{w\in W} x(w)T_w$$ with $$x(w)\in R$$ and $$x(uwu^\prime)=x(w)$$ for all $$u,u^\prime \in W^\prime$$, $$w \in W$$. Assume that there exists an automorphism $$\sigma$$ of $$W$$ such that $$\sigma(S)=S$$, such that $$q_{\sigma(s)}=q_s$$ for all $$s\in S$$ and such that $$\sigma(x^{-1}) \in W^\prime x W^\prime$$ for all $$x \in W$$. It is then claimed that $$R[W,q,W^\prime]$$ is commutative.

I don't see a proper argument for this statement and unfortunately the book doesn't provide a proof. It is only mentioned that the lemma is an analogue to Selberg's lemma. Can anyone give me a hint at how to prove this?

[1] H. Matsumoto, Analyse harmonique dans les systèmes de Tits bornologiques de type affine, Lecture Notes in Mathematics, Vol. 590, Springer-Verlag, Berlin-New York, 1977. i+219 pp.

If you want a general $$R$$, you can't use it directly, but you can take strong inspiration from Lemma 2.3 from Vershik–Okounkov - A New Approach to the Representation Theory of the Symmetric Groups. 2. It's used there to prove a special case of this where $$q=1$$, $$W=S_n$$ and $$W'=S_{n-1}$$.
The Hecke algebra has an anti-automorphism induced by $$T_s\mapsto T_{\sigma(s)}$$. That is, $$T_{w}\mapsto T_{\sigma(w)^{-1}}$$. By assumption, this preserves $$R[W,q,W']$$ as a subalgebra (since it sends each double coset $$W'wW'$$ to $$W'\sigma(w^{-1})W'$$). On this subalgebra it acts by the identity. If the identity of a ring is an anti-automorphism, then the ring is commutative.