I am looking for an extension of a result of Riche-Soergel about a functor which maps Soergel bimodules to Soergel modules.
Fix a given Coxeter system $(W,S)$, together with a (reflection faithful) representation $\mathfrak{h}$ of $W$. As usual, let $R=\mathrm{Sym}(\mathfrak{h}^\ast)$ and consider the asssociated category of Soergel bimodules, denoted by $\mathbb{S}\mathrm{Bim}$, as well as the category of Soergel modules, denoted by $\overline{\mathbb{S}\mathrm{Bim}}$ . Soergel modules are defined as graded right $R$-modules which are (finite direct sums of shifts of) direct summands of Bott-Samelson modules $$ \overline{\mathrm{BS}}(s_1,\dots,s_k):=\mathbb{k}\otimes_R R\otimes_{R^{s_1}} \dots \otimes_{R^{s_k}} R$$ The functor $ \mathbb{k}\otimes_R (-):\mathbb{S}\mathrm{Bim}\to \overline{\mathbb{S}\mathrm{Bim}}$ has been studied by Soergel and it was shown that for finite Weyl groups, the map $$\mathbb{k}\otimes_R \mathrm{Hom}_{\mathbb{S}\mathrm{Bim}}^\bullet(B,B')\to\mathrm{Hom}_{\overline{\mathbb{S}\mathrm{Bim}}}^\bullet(\mathbb{k}\otimes_R B,\mathbb{k}\otimes_R B')$$ is an isomorphism. This result was extended by Riche-Soergel in the case of finite Coxeter groups. One can find the precise references for these proofs in Elias-Makisumi-Thiel-Williamson's book on Soergel bimodules at proposition 15.27.
Does the above isomorphism hold for more general Coxeter systems? For example, is this result known to hold even in the case of infinite dihedral type? Is there a simple counter-example that I have yet to find?
I've been digging through litterature and I can't seem to find anything. I should note that Riche-Soergel's proof makes use of the longest element of the Coxeter group, which means it is not easily extended to the general case.
Edit (August 2024).
After a bit of reading, I found some counter-examples to the above isomorphism in L. Patimo's thesis which one can find here. The counter-examples are given in section 3.6.
However, a reputable source told me that the result should still hold in the infinite dihedral case above. This leaves this seemingly much more subtle question :
When does the above isomorphism hold? Is there a simple criterion distinguishing Coxeter systems for which it holds?
Any help is appreciated.