Geometric Robinson–Schensted–Knuth correspondence and canonical bases

We define a geometric RSK correspondence for a Kac–Moody group and any reduced decompositon of element of its Weyl group. This correspondence is a biration map of tori of dimension equal to the length of a reduced decomposition. For classical Lie groups and the element of the Weyl group with the longest length, the tropicalization of this map turns out to be the crystal isomorphism between the Lusztig crystal on the canonical basis and the Kashiwara crystal on the dual canonical basis. The geometric RSK-corespondence provide us with a transformation of the corresponding superpotentials for geometric crystals.