Geometric and Unipotent Crystals Ii: from Unipotent Bicrystals to Crystal Bases

For each reductive algebraic group G we introduce and study unipotent bicrystals which serve as a regular version of birational geometric and unipotent crystals introduced earlier by the authors. The framework of unipotent bicrystals allows, on the one hand, to study systematically such varieties as Bruhat cells in G and their convolution products and, on the other hand, to give a new construction of many normal Kashiwara crystals including those for G-modules, where G is the Langlands dual groups. In fact, our analogues of crystal bases (which we refer to as crystals associated to G-modules) are associated to G-modules directly, i.e., without quantum deformations. One of the main results of the present paper is an explicit construction of the crystal B0 for the coordinate ring of the dual flag variety X 0 = G /U based on the positive unipotent bicrystal on the open Bruhat cell X0 = Bw0B. Our general tropicalization procedure assigns to each strongly positive unipotent bicrystal a normal Kashiwara crystal B equipped with the multiplicity erasing homomorphism B → B0 and the combinatorial central charge B → Z which is invariant under all crystal operators. Applying the construction to B0 × B0 gives a crystal multiplication B0 × B0 → B0 and an invariant grading B0 × B0 → Z.