Cacti and cells (Q2418885)

This paper presents the construction of an action of the cactus group of a Weyl group $W$ on $W$ that is nicely compatible with Kazhdan-Lusztig cells, which are relevant for the representation theory of the universal enveloping algebra. Moreover, this action can be realized by the wall-crossing bijections, which are combinatorial shadows of wall-crossing functors on the category $\mathcal{O}$. The author is motivated by the new (conjectural) approach to cells due to \textit{C. Bonnafe} and \textit{R. Rouquier} [``Cellules de Calogero-Moser'', Preprint, \url{arXiv:1302.2720}] and follow the technics of \textit{R. Bezrukavnikov} and \textit{I. Losev} [``Etinghof conjecture for quantized quiver varieties'', Preprint, \url{arXiv:1309.1716}] related to the braid group action on a category.