On the equivariant K-theory of the nilpotent cone in the general linear group

Let $G$ be a simple complex algebraic group. Lusztig and Vogan have conjectured the existence of a natural bijection between the set of dominant integral weights of $G$, and the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit. This conjecture has been proved by Bezrukavnikov. In this paper, we develop combinatorial algorithms for computing the bijection and its inverse in the case of $G = GL(n,{\mathbb C})$.