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Let $W$ be the Weyl group of a simple algebraic group $G$. The Artin braid group $Br_{\mathfrak{g}}$ is generated by the $T_i$ , $i \in I$ such that for all $i, j \in I$, \begin{align} \underbrace{T_i T_j \cdots}_{m_{ij}} = \underbrace{T_j T_i \cdots}_{m_{ij}}, \end{align} where $m_{ij}$ is the $(i,j)$-entry in the Coxeter matrix of $W$. To each $w \in W$ one associates the element $T_w \in Br_{\mathfrak{g}}$ such that \begin{align} T_w = T_{i_1} \cdots T_{i_m}, \end{align} for each $\mathbf{i}=(i_1, \ldots, i_m) \in R(w)$, where $R(w) = \{(i_1, \ldots, i_m) : w = s_{i_1} \cdots s_{i_m}\}$.

Let $w_0$ be the longest element in $W$. Then $T_{w_0 s_i}(F_i) = F_{i^*}$ and $T_i^{-1}(F_i)=-E_i$ by a result of Lusztig. Here $E_i, F_i \in U_q(\mathfrak{g})$, where $\mathfrak{g}$ is the Lie algebra of $G$. $i^* = \tau(i)$, $\tau$ is some isomorphism of the Dynkin diagram of $G$. I am not able to find this result in his book. Are there some references about this result? Thank you very much.

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    $\begingroup$ Have you looked at Lusztig's homepage math.mit.edu/~gyuri, where he lists all his papers and comments on some of them? Before and after his book was published, he wrote a number of papers on quantum groups involving braid group actions. In any case, it might be helpful if you indicated what your source for the question is or at least the context. $\endgroup$ Commented Dec 31, 2013 at 20:11
  • $\begingroup$ @JimHumphreys, thank you very much. $\endgroup$ Commented Jan 1, 2014 at 8:29

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The result $T_{w_0 s_i} F_i = F_{i^*} $ is a special case of Chari-Pressley, A Guide to Quantum Groups, Proposition 8.1.6. Presumably, if you look in the book you will find a reference to one of Lusztig's papers. In my paper with Peter Tingley, "The crystal commutor and Drinfeld's unitarized R-matrix", we used precisely this special case in the proof of our Lemma 5.4.

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I had good intentions of looking more closely at Lusztig's papers and book but didn't follow up for some time. If it's still relevant, the following "answer" to this old question may be useful. [EDITED to make references more precise, with links.]

In the couple of years leading up to 1990 (and thereafter), Lusztig wrote numerous papers on quantum groups and then canonical basis, modifying some of his notation as needed. His book came afterward, with Part VI being devoted to braid group action. As he points out, his normalizations changed along the way, so it's important to specify which source you rely on for your formulas ("a result of Lusztig"). His 1988 Advances in Mathematics paper here treated only the ADE types in $\S5$, superseded in his 1990 Geometriae Dedicata paper here by a more general and somewhat different set-up for braid group action in $\S3$. Part VI of his book adopts a more comprehensive viewpoint, incorporating in the operator notation a sign $\pm 1$. (See the notes and references at the end of Part VI.)

Your notation seems closest to what is used in the 1990 paper, but you've omitted the generators $K_i$ and $K_i^{-1}$ which play an essential role in the formulas for the action of $T_i$ on $E_i$ and $F_i$. So I can't yet reconcile what you've written with that paper of Lusztig. In any case, you need to be more precise about your references, since he wrote so many relevant papers including those on the canonical basis.

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