Suppose $G$ is a complex reductive algebraic group, and ${\mathcal N}\subset {\frak g}^*$ is the nilpotent cone. A conjecture of Lusztig, proved by Bezrukavnikov, says that there is a natural bijection

irreducible $G$-eqvt vector bundles on $G$ orbits on ${\mathcal N}$ $\longleftrightarrow$ dominant weights for $G$.

(The coherent sheaves in the title arise because the left side is more or less obviously a basis for the Grothendieck group of $G$-equivariant coherent sheaves on ${\mathcal N}$.)

I'll explain a definition and computation of this bijection in terms of finite-dimensional representation theory; applications to infinite-dimensional representations that would follow from computing it; and generalizations to real reductive groups.