# Do global bases exist for quantum enveloping algebras at $q$ nonroot of unity?

Take $$\Bbbk$$ to be a field, $$q \in \Bbbk$$ a nonroot of unity, and $$U = U_q(\mathfrak g)$$ the quantized enveloping algebra of a complex finite dimensional simple Lie algebra, and write $$U^-$$ for its negative part.

SHORT VERSION: can you get a Global basis of $$U^-$$ in the sense of Kashiwara and Lusztig in these conditions (i.e. not just $$q$$ transcendental over $$q$$), say by specialization, or do you lose some of the nice properties that make this basis a distinguished one?

LONG VERSION: Kashiwara and Lusztig discovered a very nice basis for $$U^-$$, called the global basis, but in order to show it exists they need $$\Bbbk$$ to be of characteristic $$0$$ and $$q$$ transcendental over $$\mathbb Q$$. On the other hand, they prove that this basis is in fact a basis of a suitable $$\mathcal A = \mathbb Z[v,v^{-1}]$$-form of $$U^-$$, denoted by $$U^-_\mathcal A$$. So letting $$v$$ act as multiplication by $$q$$ on $$\Bbbk$$ we get a map $$U^- \to \Bbbk \otimes_{\mathcal A} U^-_{\mathcal A}$$ that is (a) surjective and (b) compatible with the weight decomposition of $$U^-$$, hence an isomorphism (this is the part that breaks down if $$q$$ is a root of unity). So if I got this correctly, the global basis in fact survives in the case where $$\Bbbk$$ is any old field containing a nonroot of unity $$q$$.

Now the global basis has some other nice properties related to highest weight representations of $$U$$. In that case, I am less sure but I think similar (rather roundabout) arguments apply to show that they also survive from the $$\mathbb Q(v)$$-context to the $$\Bbbk$$ context, but maybe I am missing some obvious reason for which things break down quickly in the second case. I have searched the literature for references for global bases for $$q$$ nonroot of unity but could not find any. Is this simply because no one has found a need for them, or is it because they are not useful in this context?