# What is the current status for Lusztig's positivity conjecture for symmetric Cartan datum?

This is related to the earlier question here

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.

• Marty- this question would be a lot better if you provided a little more reference for those of us who don't have Lusztig's book at hand. Perhaps you could mention what object you're looking at the canonical basis of? There are quite a few. – Ben Webster Jun 27 '11 at 4:55
• I'm sorry - part of my trouble is that I'm not familiar with other literature on canonical bases. I'm talking about the canonical basis of Lusztig's modification of the universal enveloping algebra -- the basis $\dot B$ of $\dot U$. This is the modification in which there is a basis element $1_\lambda$ for each $\lambda$ in the cocharacter lattice $Y$. This is a system of idempotents for the nonunital algebra $\dot U$ (making it an algebra with approximate identity). – Marty Jun 27 '11 at 12:01

## 1 Answer

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 of Yiqiang Li.

For ADE type, a clever person can derive this from Theorem 1.19 in my paper KI-HRT II; 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; 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.

• Thanks Ben! Looking at two papers (Li's and yours) is much better than my previous hunting around. Fortunately for me, I realized yesterday that I can work around the positivity conjecture, so the stakes aren't so high. But it also seems like an incredibly interesting problem, and I'm glad progress is being made. – Marty Jun 30 '11 at 11:32