oh, and yes to 2)

Humphreys' book on finite reflection groups is better.

I would call it the bar involution.

let $p_1,\ldots,p_N$ be a complete list of primes. Since no $p_i$ divided $p_1\cdots p_N+1$, this can't be a complete list of primes.

I'll have to think about it more, but my first instinct is that the answer is no. The $k$-multipartition description comes from tensoring together $k$ level 1 representations, each parametarized by partitions. In my paper, I skipped the big space and defined the action in one shot. Then again, maybe the translation is not too bad? I'll think some more.

I guess I should say that it is straighforward to extend the $\mathfrak{gl}_\infty$-module I mentioned above to $a_\infty$ in a way analogous to the Kac-Raina level 1 extension. So, you get an action of $\hat{\mathfrak{sl}}_p$ on the Fock space. The problem is that it is not generally irreducible as a $\hat{\mathfrak{sl}}_p$-module.

I thought you were suggesting that the $\{b_g\}$ basis might only coincide with the canonical basis up to sign, though as I understand you now, this is not the case. I do not know of any alternative to Saito's result.

This is a very interesting point and maybe there is an issue? Leclerc constructs his basis $\{b_g\}$ using Lusztig's construction (so in some sense it is the canonical base by definition). First, he defines a PBW basis $\{E_g\}$ using an action of the braid group (there is more than one choice of action, but I don't know how much it matters). Then he obtains a unique bar invariant $b_g=E_g + \cdots$. The $E_g$ that Leclerc constructs are very natural in the sense that they have a well defined highest order term, and the coefficient of the leading terms are positive.

No problem. I find this whole discussion very interesting. The Lyndon basis theory works really well for these algebras. It seems we have two different canonically defined bases coming from the same dual PBW basis. How are they related? Is there some way to twist the isomorphism in Kashiwara's definition of the global basis to make them coincide? Or is the relation more complicated?

Okay, I changed it.