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.



Reference request: Lusztig's symmetries
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.


