Geometric and unipotent crystals
Abstract
In this paper we introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal torus of G. The structure of a geometric G-crystal on an algebraic variety X consists of a rational morphism γ:X-->T and a compatible family ei:Gm× X-->X, i∈ I of rational actions of the multiplicative group Gm satisfying certain braid-like relations. Such a structure induces a rational action of W on X. Quite surprisingly, many interesting rational actions of the group W come from geometric crystals. Also all the known examples of the action of W which appear in the construction of gamma-functions for the representations of LG in the recent work by A. Braverman and D. Kazhdan come from geometric crystals. There are many examples of positive geometric crystals on (Gm)l, i.e., those geometric crystals for which the actions ei and the morphism γ are given by positive rational expressions. To each positive geometric crystal X we associate a Kashiwara's crystal corresponding to the Langlands dual group LG. An emergence of LG in the "crystal world" was observed earlier by G. Lusztig. Another application of geometric crystals is a construction of trivialization which is an W-equivariant isomorhism X-->γ-1(e) × T for any geometric SLn-crystal. Unipotent crystals are geometric analogues of normal Kashiwara crystals. They form a strict monoidal category. To any unipotent crystal built on a variety X we associate a certain gometric crystal.
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