This is related to the earlier [question here](https://mathoverflow.net/questions/39934/when-does-lusztigs-canonical-basis-have-non-positive-structure-coefficients)


In Conjecture 25.4.2 in his "Introduction to Quantum Groups," Lusztig conjectures that "If the Cartan datum is symmetric, then the structure constants $m\_{a,b}^c$, $\hat m\_c^{a,b}$ are in $N[v,v^{-1}]$. 


In what cases is this proven? I only care about classical groups, specifically split reductive groups over $Z$ as constructed via Lusztig's canonical bases (specialize at $v=1$). I would love it if the structure constants were non-negative, so that there were no troublesome signs floating around in the antipode. When is this known to be the case? My googling leads me to suspect something is known in the simply-laced case. Anything more general yet? References? I'm not interested in the non-symmetric case. 



What is the current status for Lusztig's positivity conjecture for symmetric Cartan datum?
You should take this with a grain of salt, but I would guess that this is stated in the literature in type A and no other types. It follows in type A from the Beilinson-Lusztig-MacPherson construction, I believe. This is discussed a bit in [this paper](http://arxiv.org/abs/1007.5384) of Yiqiang Li.


For ADE type, a clever person can derive this from Theorem 1.19 in my paper [KI-HRT II](https://arxiv.org/abs/1005.4559); it isn't stated explicitly (I'm writing up a separate canonical basis paper at the moment), but it only requires a small twist on the current arguments.
(Essentially, one must show that Theorem 1.19 implies that the orthodox basis of $\dot U$ is canonical, and orthodox bases always have positive structure coefficients).


For arbitrary symmetric types, this is trickier (the discussion above uses very strongly that highest and lowest weight reps coincide in finite type). At the moment, I think the main roadblock is proving 4.13 of [Li's paper](http://arxiv.org/abs/1007.5384); I think I know how to do this, but it's not written up yet, and hasn't been completely vetted for dumb mistakes. That would establish this for all symmetric types.


