Is theorem B and it’s application not enough for a separate paper?

You understand that the Hecke algebra can be defined over $\mathbb{K}$, right?

It seems that if you can show that $\tau$ is a refinement of $\rho$, then $\rho$ is a refinement of $\tau$. Is there something special about the Euclidean topology that breaks this symmetry?

Ha. I almost gave this answer with a link to your website.

@DenisNardin In the group case the determinant is $\pm 1$. This is not going to carry over in general. It would be helpful if to see a list of properties that one might want to see in such a sign representation (besides giving the sign rep. in the group case).

Let us continue this discussion in chat.

The bar involution is an algebra automorphism defined by $\overline{E}=E$, $\overline{F}=F$ and $\overline{q}=q^{-1}$. The relation $[E,F]=(K^2-K^{-2})/(q-q^{-1})$ implies that $\overline{K}=K^{-1}$ since $\overline{[E,F]}=[E,F]$. I guess if ``the star is antilinear'' means take complex conjugates of coefficients, then, for $q$ a root of unity, this would be compatible.

Maybe I am misunderstanding what you mean by $U_qSL(2)_{\mathbb{C}}$. Is this the quantum group over $\mathbb{C}(q)$, or $U_qSL(2)\otimes_{\mathbb{Q}(q)}\mathbb{C}$ where $\mathbb{C}$ is regarded as a $\mathbb{Q}(q)$ module with $q$ acting as a scalar?

Are you sure both send $K\mapsto K$? My first thought was that they differ by the bar involution, but then one should be $K\mapsto K^{-1}$.