Do there exist explicit formulas for the action of Lusztig's canonical basis of
$U_q(\mathfrak n_+)$ in the Gelfand-Tsetlin basis of a Verma module for $sl(n)$?

should it exist ? What about classical limit q=1: is Verma basis for sl(3) related to GT (if Yes - it should be known I think). As far as I understand in sl(3) "Verma basis" is limit of canonical bases (not 100%sure)
– Alexander ChervovJan 24 '12 at 17:39

I don't know - but I would appreciate any information
– Alexander BravermanJan 24 '12 at 18:13

I would suggest to ask Anton Gerasimov - they worked a lot on GT, moreover I had very vague memories that at some seminar he discussed exactly this question - but I am completely not sure... Side remarks: GT - is eigenbasis for GT commutative subalgebra (=integrable system). There is result by Vinberg that GT is limit of "top"=Gaudin=Hitchin=argument shift=Mishenko-Fomenko... There is GLn-GLm duality between GT and Jucys-Murphy (mathoverflow.net/questions/83150/).I have not heard any relation with canonical basis of any of these object. Except some paper by I. Frenkel on Macdonald<->CB
– Alexander ChervovJan 26 '12 at 17:03