Geometric crystals on flag varieties and unipotent subgroups of classical groups

For a classical simple algebraic group GG, we obtain the affirmative answer for the conjecture in Nakashima (2005) [3] that there exists an isomorphism between the geometric crystal on the flag variety and the one on the maximal unipotent subgroup U−U−.

► Let GG be a simple algebraic group and UmUm be its maximal unipotent subgroup.
► We give some criteria for the existence of a unipotent crystal on UmUm.
► The unipotent crystal induces the geometric crystal YY on UmUm.
► Let VV be the geometric crystal on the flag variety of GG.
► We describe the isomorphism between VV and YY.