Crystals and shift of argument algebras
For a simple Lie algebra, a crystal is a combinatorial object which can essentially be thought of as a basis for a representation. It has Kashiwara operators which roughly encode the action of the Chevalley generators of the Lie algebra. The original constructions of crystals come from quantum groups.
In some cases crystals can be defined directly on a basis of a representation itself: for example, in type A one can do this using Gelfand-Tsetlin patterns. Shift of argument algebras are commutative algebras, defined for any simple Lie algebra, and include Gelfand-Tsetlin algebras as special limiting cases. We will describe on-going work with Halacheva, Kamnitzer and Rybnikov that shows how to put a crystal structure on eigenbases for shift of argument algebras, and can be thought of as generalizing the Gelfand-Tsetlin crystals to other types.