Geometric lifting of the canonical basis and semitoric degenerations of Richardson varieties

In the sl_n case, A. Berenstein and A. Zelevinsky studied the Sch"{u}tzenberger involution in terms of Lusztig’s canonical basis, [3]. We generalize their construction and formulas for any semisimple Lie algebra. We use for this the geometric lifting of the canonical basis, on which an analogue of the Sch"{u}tzenberger involution can be given. As an application, we construct semitoric degenerations of Richardson varieties, following a method of P. Caldero, [6]