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Losev, we construct an action of ${\\mathrm{Cact}}\\_W  \\times {\\mathrm{Cact}}\\_W$ on $W$ which has nice properties with respect to the partition of $W$ into left, right or two-sided cells (under some hypothesis, which hold for instance if $\\varphi$ is constant or if $W$ is finite of rank $\\textless{} 5$). 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