Quiver Varieties, Category O for Rational Cherednik Algebras, and Hecke Algebras

We relate the representations of the rational Cherednik algebras associated with the complex reflection group S(n) x (mu e)(n) to sheaves on Nakajima quiver varieties associated with extended Dynkin graphs via a Z-algebra construction. This is done so that as the parameters defining the Cherednik algebra vary, the stability conditions defining the quiver variety change. This construction motivates us to use the geometry of the quiver varieties to interpret the ordering function ( the c-function) used to define the highest weight structure on category O of the Cherednik algebra. This interpretation provides a natural partial ordering on O that we expect will respect the highest weight structure. This partial ordering has appeared in a conjecture of Yvonne on the composition factors in O and so our results provide a small step toward a geometric picture for that. We also interpret geometrically another ordering function ( the a-function) used in the study of Hecke algebras. ( The connection between Cherednik algebras and Hecke algebras is provided by the KZ-functor.) This is related to a conjecture of Bonnafe and Geck on equivalence classes of weight functions for Hecke algebras with unequal parameters since the classes should ( and do for type B) correspond to the G. I. T. chambers defining the quiver varieties. As a result, anything that can be defined via the quiver varieties, including the a-function, will be constant on these classes.